Methods of approximation theory

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Author: A. I. Stepanets

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Contents

PREFACE

xi PART I

1. REGULARITY OF LINEAR METHODS OF SUMMATION OF FOURIER SERIES 1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2. Nikol’skii and Nagy Theorems . . . . . . . . . . . . . . . . . . . . 3. Lebesgue Constants of Classical Linear Methods . . . . . . . . . . 4. Lower Bounds for Lebesgue Constants . . . . . . . . . . . . . . . . 5. Linear Methods Determined by Rectangular Matrices . . . . . . . . 6. Estimates for Integrals of Moduli of Functions Defined by Cosine and Sine Series . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7. Asymptotic Equality for Integrals of Moduli of Functions Defined by Trigonometric Series. Telyakovskii Theorem . . . . . . . . . . 8. Corollaries of Theorem 7.1. Regularity of Linear Methods of Summation of Fourier Series . . . . . . . . . . . . . . . . . . 2. SATURATION OF LINEAR METHODS 1. Statement of the Problem . . . . . . . . . . . . . 2. Sufficient Conditions for Saturation . . . . . . . 3. Saturation Classes . . . . . . . . . . . . . . . . . 4. Criterion for Uniform Boundedness of Multipliers 5. Saturation of Classical Linear Methods . . . . . .

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1 1 6 15 21 23

. 28 . 43 . 66

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79 79 81 84 90 98

3. CLASSES OF PERIODIC FUNCTIONS 101 1. Sets of Summable Functions. Moduli of Continuity . . . . . . . . . . 101 2. Classes Hω [a, b] and Hω . . . . . . . . . . . . . . . . . . . . . . . 108 3. Moduli of Continuity in Spaces Lp . Classes Hωp . . . . . . . . . . 110 v

vi

Contents

4. 5. 6. 7.

Classes of Differentiable Functions . . . Conjugate Functions and Their Classes . Weyl–Nagy Classes . . . . . . . . . . . Classes Lψ βN . . . . . . . . . . . . . .

8.

Classes Cβψ N . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126

9.

N . . . . . . . . . . . . . . . . Classes Lψ β¯ ¯ Order Relation for (ψ, β)-Derivatives . . . ¯ ψ-Integrals of Periodic Functions . . . . . Sets M0 , M∞ , and MC . . . . . . . . . . Set F . . . . . . . . . . . . . . . . . . . . Two Counterexamples . . . . . . . . . . . Function ηa (t) and Sets Defined by It . . . Sets B and M0 . . . . . . . . . . . . . .

10. 11. 12. 13. 14. 15. 16.

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112 116 119 120

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133 137 147 153 156 160 162

4. INTEGRAL REPRESENTATIONS OF DEVIATIONS OF POLYNOMIALS GENERATED BY LINEAR PROCESSES OF SUMMATION OF FOURIER SERIES 165 1. First Integral Representation . . . . . . . . . . . . . . . . . . . . . . 165 2. Second Integral Representation . . . . . . . . . . . . . . . . . . . . . 167 ¯ 3. Representation of Deviations of Fourier Sums on Sets C ψ M ¯ and Lψ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173 5. APPROXIMATION BY FOURIER SUMS IN SPACES C AND 1. Simplest Extremal Problems in Space C . . . . . . . . . . . . . 2. Simplest Extremal Problems in Space L1 . . . . . . . . . . . . . 3. Approximations of Functions of Small Smoothness by Fourier Sums . . . . . . . . . . . . . . . . . . . . . . . . . 4. Auxiliary Statements . . . . . . . . . . . . . . . . . . . . . . . . 5. Proofs of Theorems 3.1–3.3 . . . . . . . . . . . . . . . . . . . . 6. Approximation by Fourier Sums on Classes Hω . . . . . . . . . ˜ω . . . . . . . . . 7. Approximation by Fourier Sums on Classes H 8. Analogs of Theorems 3.1–3.3 in Integral Metric . . . . . . . . . 9. Analogs of Theorems 6.1 and 7.1 in Integral Metric . . . . . . . . 10. Approximations of Functions of High Smoothness by Fourier Sums in Uniform Metric . . . . . . . . . . . . . . . . . . . . 11. Auxiliary Statements . . . . . . . . . . . . . . . . . . . . . . . . 12. Proofs of Theorems 10.1–10.3 . . . . . . . . . . . . . . . . . . .

L1 187 . . 189 . . 198 . . . . . . .

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203 207 225 235 239 243 252

. . 253 . . 259 . . 271

Contents 13. 14. 15. 16. 17. 18. 19. 20. 21. 22. 23. 24.

vii

Analogs of Theorems 10.1–10.3 in Integral Metric . . . . . . . . . . 278 Remarks on the Solution of Kolmogorov–Nikol’skii Problem . . . . . 279 ¯ Approximation of ψ-Integrals That Generate Entire Functions by Fourier Sums . . . . . . . . . . . . . . . . . . . . . . . . . . . . 284 Approximation of Poisson Integrals by Fourier Sums . . . . . . . . . 294 Corollaries of Telyakovskii Theorem . . . . . . . . . . . . . . . . . . 303 Solution of Kolmogorov–Nikol’skii Problem for Poisson Integrals of Continuous Functions . . . . . . . . . . . . . . . . . . . . . . . . 310 Lebesgue Inequalities for Poisson Integrals . . . . . . . . . . . . . . 338 Approximation by Fourier Sums on Classes of Analytic Functions . . 345 Convergence Rate of Group of Deviations . . . . . . . . . . . . . . . 363 Corollaries of Theorems 21.1 and 21.2. Orders of Best Approximations374 Analogs of Theorems 21.1 and 21.2 and Best Approximations in Integral Metric . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 378 Strong Summability of Fourier Series . . . . . . . . . . . . . . . . . 383

BIBLIOGRAPHICAL NOTES (Part I)

393

REFERENCES (Part I)

399

6. 0. 1. 2. 3. 4. 5. 6. 7. 8. 9.

PART II CONVERGENCE RATE OF FOURIER SERIES AND THE BEST APPROXIMATIONS IN THE SPACES Lp Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . Approximations in the Space L2 . . . . . . . . . . . . . . . . Direct and Inverse Theorems in the Space L2 . . . . . . . . . Extension to the Case of Complete Orthonormal Systems . . . Jackson Inequalities in the Space L2 . . . . . . . . . . . . . . Marcinkiewicz, Riesz, and Hardy–Littlewood Theorems . . . ¯ Imbedding Theorems for the Sets Lψ Lp . . . . . . . . . . . . ¯ Approximations of Functions from the Sets Lψ Lp by Fourier Sums . . . . . . . . . . . . . . . . . . . . . . . . . . . . Best Approximations of Infinitely Differentiable Functions . . Jackson Inequalities in the Spaces C and Lp . . . . . . . . .

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429 429 432 437 439 444 448 452

. . . . 455 . . . . 466 . . . . 481

7. BEST APPROXIMATIONS IN THE SPACES C AND L 1. Chebyshev and de la Vallée Poussin Theorems . . . . . . . . . . . . . 2. Polynomial of the Best Approximation in the Space L . . . . . . . . 3. General Facts on the Approximations of Classes of Convolutions . . .

489 490 492 495

viii 4. 5. 6. 7. 8. 9.

Contents Orders of the Best Approximations . . . . . . . . . . . . . . Exact Values of the Upper Bounds of Best Approximations . Dzyadyk–Stechkin–Xiung Yungshen Theorem. Korneichuk Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . Serdyuk Theorem . . . . . . . . . . . . . . . . . . . . . . . Bernstein Inequalities for Polynomials . . . . . . . . . . . . Inverse Theorems . . . . . . . . . . . . . . . . . . . . . . .

. . . . . 505 . . . . . 510 . . . .

8. INTERPOLATION 1. Interpolation Trigonometric Polynomials . . . . . . . . . . . . 2. Lebesgue Constants and Nikol’skii Theorems . . . . . . . . . 3. Approximation by Interpolation Polynomials in the Classes of Infinitely Differentiable Functions . . . . . . . . . . . . 4. Approximation by Interpolation Polynomials on the Classes of Analytic Functions . . . . . . . . . . . . . . . . . . . . . 5. Summable Analog of the Favard Method . . . . . . . . . . . .

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13.

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522 525 539 545

553 . . . . 553 . . . . 557 . . . . 560 . . . . 572 . . . . 586

9. APPROXIMATIONS IN THE SPACES OF LOCALLY SUMMABLE FUNCTIONS ˆp . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1. Spaces L 2. Order Relation for (ψ, β)-Derivatives . . . . . . . . . . . . . . 3. Approximating Functions . . . . . . . . . . . . . . . . . . . . . 4. General Estimates . . . . . . . . . . . . . . . . . . . . . . . . . ˆψ . . . . . . . . . 5. On the Functions ψ(·) Specifying the Sets L β 6. Estimates of the Quantities ˆ rσc (t, β)1 for c = σ − h and h > 0 . . . . . . . . . . . . . . . . . . . . . . . . . . 7. Estimates of the Quantities ˆ rσc (t, β)1 for c = θσ, 0 ≤ θ ≤ 1, and ψ ∈ Ac . . . . . . . . . . . . . . . . . . . . . . . . . . 8. Estimates of the Quantities ˆ rσc (t, β)1 for c = 2σ − η(σ) and ψ ∈ A∞ . . . . . . . . . . . . . . . . . . . . . . . . . 9. Estimates of the Quantities ˆ rcσ (t, 0)1 for c = θσ, 0 ≤ θ < 1, and ψ ∈ A0 . . . . . . . . . . . . . . . . . . . . . . . . . . 10. Estimates of the Quantities δˆσ,c (t, β)1 . . . . . . . . . . . . 11. Basic Results . . . . . . . . . . . . . . . . . . . . . . . . . . . ψ 12. Upper Bounds of the Deviations ρσ (f ; ·) in the Classes Cˆβ,∞ and Cˆ ψ Hω . . . . . . . . . . . . . . . . . . . . . . . . . . β

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597 598 601 607 615 624

. . . 626 . . . 632 . . . 634 . . . 635 . . . 636 . . . 639 . . . 648

Some Remarks on the Approximation of Functions of High Smoothness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 666

Contents 14.

ix

Strong Means of Deviations of the Operators Fσ (f ; x) . . . . . . . . 668

10. APPROXIMATION OF CAUCHY-TYPE INTEGRALS 1. Definitions and Auxiliary Statements . . . . . . . . . . . 2. Sets of ψ-Integrals . . . . . . . . . . . . . . . . . . . . 3. Approximation of Functions from the Classes C ψ (T)+ . 4. Landau Constants . . . . . . . . . . . . . . . . . . . . . 5. Asymptotic Equalities . . . . . . . . . . . . . . . . . . . 6. Lebesgue–Landau Inequalities . . . . . . . . . . . . . . 7. Approximation of Cauchy-Type Integrals . . . . . . . .

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11. APPROXIMATIONS IN THE SPACES S p 1. Spaces Sϕp . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2. ψ-Integrals and Characteristic Sequences . . . . . . . . . . . . 3. Best Approximations and Widths of p-Ellipsoids . . . . . . . . 4. Approximations of Individual Elements from the Sets ψSϕp . . . 5. Best n-Term Approximations . . . . . . . . . . . . . . . . . . 6. Best n-Term Approximations (q > p) . . . . . . . . . . . . . 7. Proof of Lemma 6.1 . . . . . . . . . . . . . . . . . . . . . . . . 8. Best Approximations by q-Ellipsoids in the Spaces Sϕp . . . . . 9. Application of Obtained Results to Problems of Approximation of Periodic Functions of Many Variables . . . . . . . . . . . 10. Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11. Theorems of Jackson and Bernstein in the Spaces S p . . . . . . 12. APPROXIMATIONS BY ZYGMUND AND ´ POUSSIN SUMS DE LA VALLEE 1. Fejér Sums: Survey of Known Results . . . . . . . . . . . 2. Riesz Sums: A Survey of Available Results . . . . . . . . 3. Zygmund Sums: A Survey of Available Results . . . . . . ψ 4. Zygmund Sums on the Classes Cβ,∞ . . . . . . . . . . . 5. De la Vallée Poussin Sums on the Classes Wβr and Wβr Hω 6.

¯

De la Vallée Poussin Sums on the Classes Cβψ N and C ψ N

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679 680 696 702 713 716 723 727

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741 741 745 747 750 755 771 777 808

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847 848 860 863 866 871

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BIBLIOGRAPHICAL NOTES (Part II)

881

REFERENCES (Part II)

885

Index

917

PREFACE What is new in this monograph? First of all, it is worth noting methods that enable one to solve, within the framework of a common approach, traditional problems of approximation theory for large collections of functions, including the well-known Weyl–Nagy and Sobolev classes as particular cases as well as classes of functions defined by convolutions with arbitrary summable kernels. The results obtained with the use of these methods are also new. Systematic investigations in this direction were originated in the 1980s under the influence of works by B. Nagy, S. M. Nikol’skii, S. B. Stechkin, V. K. Dzyadyk, N. I. Akhiezer, N. P. Korneichuk, A. V. Efimov, S. A. Telyakovskii, etc. In the same years, the concept of (ψ, β)-derivative defined for a given function f by a given sequence of numbers ψ = ψ(k), k = 1, 2, . . . , and numbers β was formed. The ordinary rth derivative, r = 1, 2, . . . , of a periodic function is a particular case of the (ψ, β)-derivative for ψ(k) = k −r and β = r. Derivatives in the Weyl and Weyl–Nagy sense are also particular cases of (ψ, β)-derivatives. Generally speaking, the sequences ψ(k) in the definition of derivatives may be arbitrary. However, it turned out that, in many cases, it suffices to consider only sequences convex downwards (in the present monograph, the set of such sequences is denoted by M ). This simplifies investigations without considerable loss of generality. It turns out that any summable (or continuous) 2π-periodic function f necessarily has the (ψ, β)-derivative, which remains summable (continuous), and, furthermore, ψ ∈ M. Moreover, if the set of periodic functions for which (ψ, β)-derivatives exist for given ψ and β is denoted by Lψ β , then the following equality is true: ∪ Lψ β = L, ψ∈M

where L denotes the set of 2π-periodic functions integrable over the period. This equality means, in particular, that, using the notion of (ψ, β)-derivatives, one can classify all functions from L. It was established that the common part of all sets Lψ β for ψ ∈ M consists solely of trigonometric polynomials. This xi

xii

Preface

implies that, under this classification, every function f ∈ L falls only into its “own” set Lψ β , and only polynomials remain indistinguishable (they are contained in each of these sets). Clearly, any classification of functions has the right to exist only if its efficiency is confirmed. Apparently, the main problem of approximation theory is to establish the properties of approximation characteristics of a function on the basis of the postulated properties of this function. In the case of approximation of 2π-periodic functions, it is customary to use the convergence rates of Fourier series, best approximations by trigonometric polynomials, approximations by polynomials generated by linear methods of summation of Fourier series, approximations by interpolation polynomials, etc., as such characteristics. Functions with the same a priori properties are combined into classes, so that the facts established for a given class are valid for every representative of it. In this case, there arises the possibility of formulating new problems for entire classes of functions. Among them, there are problems of the properties of upper bounds for a given class of deviations of Fourier sums, best approximations, approximations by linear methods, etc., problems of various widths of given classes, etc. Therefore, the success of any classification of functions depends, first of all, on whether the properties used as its basis are adequate to the objective chosen. In this respect, the classification considered can withstand even strong criticism because all approximation characteristics investigated are completely expressed (most often, in explicit form) in terms of parameters defining classes of functions. For example, the orders of upper bounds of the best uniform approximations by trigonometric polynomials of degree n − 1 on classes of functions for which (ψ, β)-derivatives are bounded are equal to ψ(n). An analogous quantity in the Lebesgue spaces Lp has the same value for any p ∈ [1, ∞]. If a function f is the (ψ, β)-derivative of a function F, then it is quite natural to call F the (ψ, β)-integral of the function f. In this case, the set Lψ β is the collection of (ψ, β)-integrals of all functions f ∈ L. Obviously, it makes no difference which concept (derivative or integral) is regarded as primary. For a long time, the concept of derivative has been primary in the definition of classes of functions. However, in recent years, a preference has been given to the concept of integral (indeed, the approximated objects are, as a rule, integrals). This concept dominates in the present monograph. As a rule, the main results are formulated for entire families of classes whose parameters are sequences running through a given set (e.g., the set M or a certain part of it). Since they are quite general, their proof is not simple. All cumbersome

Preface

xiii

proofs are divided into several steps, the first of which, as a rule, is the derivation of integral representations for the quantities investigated, and the second step is devoted to the simplification of the obtained expressions according to the objective of the problem considered. Problems of approximation of 2π-periodic functions of one variable occupy the main place in the monograph. The most complete and final results are obtained in the case of approximation of these functions by their Fourier sums in the uniform metric and in the metric of the space L = L1 . These results are presented in Chapter 5. Chapters 1–4 can be regarded as preparatory, though, undoubtedly, they are of independent interest. For, example, Chapter 1 is devoted to general problems of the theory of summation of Fourier series, in particular, problems of their regularity. In this chapter, we present the well-known Nagy and Nikol’skii statements that give sufficient regularity conditions and (apparently, for the first time in monographs) the known Telyakovskii theorem that enables one to determine the principal parts of the integrals of summable functions in terms of their Fourier coefficients in fairly general cases. General problems of the theory of summation of Fourier series (problems of saturation of linear methods) are considered in Chapter 2. An important place in the monograph is occupied by Chapter 3. One may begin reading this monograph with exactly this chapter, where problems of classification of periodic functions (from the notion of modulus of continuity to prop¯ erties of ψ-integrals) are systematically presented. Formulas for the integral representation of deviations of polynomials gener¯ ated by linear processes of summation of Fourier series on sets of ψ-integrals are deduced in Chapter 4. Parallel with traditional representations of these deviations in terms of convolutions with integrals over the period, we also present here representations with the use of convolutions with integrals taken over the entire real axis. Exactly these representations are most often used in what follows. In the derivation of these representations, an important role is played by functions that are integrable and continuous on the entire axis and orthogonal to every function from L. Approximations by Fourier sums and the best approximations in the spaces Lp for p > 1 are considered in Chapter 6. While, in the case of the spaces C and L1 , asymptotic equalities for the upper bounds of deviations of Fourier sums are obtained in Chapter 5 for large collections of functions, in the spaces Lp , with rare exceptions, one has to be content with exact order relations. It is clear that one of these positive exceptions corresponds to the case p = 2. Time will show whether this is a consequence of inadequate analysis or of the disharmony

xiv

Preface

between the original data and the objectives formulated. Arguments in favor of the latter are given, in particular, by the constructions presented in Chapter 11, where it is proposed to consider functions (including functions of many variables) as elements of linear spaces with a norm defined in a special way. Chapter 7 is devoted to the determination of orders and exact values of the best ¯ polynomial approximations in the spaces C and L1 on the sets of ψ-integrals. Orders are readily determined for large collections of functions. However, as for exact values, only a few new results are added to the well-known Favard– Akhiezer–Krein, Nagy, Dzyadyk–Stechkin–Xiung Yungshen, and Korneichuk results, and, probably, the main achievements in this direction are yet to come. In Chapter 8, interpolation problems are considered. The Nikol’skii results on the approximation of differentiable functions by interpolation polynomials and ¯ new results for sets of ψ-integrals are presented here. It is shown that, in fact, approximations by interpolation polynomials are not inferior in quality to approximations by Fourier sums. ¯ The methods for the investigation of approximations of ψ-integrals developed in Chapters 5–7 are fairly universal. In particular, these methods and new facts (which are actually at the level of definitions) enable one to investigate approximation characteristics of classes of functions locally integrable on the entire axis and defined by Cauchy-type integrals in Jordan domains with rectifiable boundary. The corresponding results are presented in Chapters 9 and 10. In Chapter 9, we investigate problems of approximation of locally integrable functions defined on the entire axis (not necessarily periodic) with the use of entire functions of the exponential type. Here, in fact, we construct a theory analogous to that for the periodic case presented in Chapters 5–7: the notion of the (ψ, β)ˆ is integral of a function f locally integrable on the numerical axis (f ∈ L) ψ ψ ˆ introduced, sets Lβ analogous to the sets Lβ are defined, and approximations of ˆ ψ by entire functions of the exponential type are studied. Main functions f ∈ L β

attention is given to approximations by so-called Fourier operators (analogs of Fourier sums in the periodic case). Here, analogs of the spaces Lp are the sets ˆ p of functions f with the finite norm f pˆ defined as follows: L

f pˆ =

⎧ ⎞1/p ⎛ a+2π ⎪ ⎪ ⎪ ⎪ ⎪ |f (t)|p dt⎠ , 1 ≤ p < ∞, ⎨ sup ⎝ a∈R1

a

⎪ ⎪ ⎪ ⎪ ⎪ ⎩ess sup |f (t)|, t∈R1

p = ∞.

Preface

xv

ˆ ψ involve the corresponding Therefore, the statements obtained for the sets L β results for the sets Lψ β. The results presented in Chapter 11 are completely new (they were published in journals in 2001–2002), and, apparently, their analysis will be performed in the future. This material is a result of a search for new approaches to problems of the theory of approximation of functions of many variables and, in particular, functions periodic in each variable. There are many problems here, but the following may be regarded as the most important ones: the choice of approximating aggregates, the choice of classes of functions, and the choice of approximation characteristics. The extension of one-dimensional results to the multidimensional case immediately encounters the problem of the choice of approximating aggregates. In the one-dimensional case, the form of the simplest aggregate is determined by the natural order of a natural series, whereas in the multidimensional case, i.e., in the case where there is a set X [a Banach space of functions f (t) = f (t1 , . . . , tm ), t ∈ Rm , of m variables], the choice of the simplest aggregates becomes problematic; it is well known that the first difficulties here begin with the problem of the choice of an analog of a partial sum for the multiple series

ck , k = (k1 , . . . , km ),

(1)

k∈Z m

where Z m is the integer lattice in Rm . It seems quite natural to introduce “rectangular” sums and the corresponding approximating aggregates in the periodic case, namely trigonometric polynomials of the form n1 nm ... ck1 ,...,km ei(k1 t1 +...+km tm ) . (2) k1 =−n1

km =−nm

However, partial sums of a multiple series can be introduced in many ways, e.g., as follows: Let {Gα } be a family of bounded domains in Rm that depend on the numerical parameter α and are such that any vector n ∈ Z m belongs to all domains Gα for sufficiently large values of α. Then the expression k∈Gα

ck

xvi

Preface

is called a partial sum of series (1) corresponding to the domain Gα . By analogy, one introduces the corresponding partial sums of the trigonometric series: ck eikx = ck1 ,...,km ei(k1 x1 +...+km xm ) . (3) k∈Gα

k∈Gα

It turned out quite soon that, in the case of approximation of functions from the Sobolev classes Wpr (Rm ), it is “more productive” to use sums of the form (3) constructed on the basis of domains formed by certain hyperbolas instead of the rectangular sums (2). These domains, first introduced by K. I. Babenko in 1960, were called hyperbolic crosses. The appearance of the notion of hyperbolic cross has made a considerable impact on the development of the theory of approximation of functions of many variables. However, this notion turned out to be of little use for classes of functions different from Sobolev classes . In this connection, there naturally arise assumptions that, for every individual class of functions N (or any family of these classes), it is necessary to select “its own” family of domains Gα defined by the parameters of this class and to construct approximating aggregates of the form (3) on the basis of this family. In this case, as in the case of hyperbolic crosses, the values of the Kolmogorov widths of these functional classes may serve as a measure of the applicability of approximating aggregates. These arguments motivated the investigations described in Chapter 11. A few words should be said about the metric used in that chapter, which is alternative to the metric used in the definition of the Lebesgue spaces Lp . Let L2 be the set of 2π-periodic square-integrable functions. For such functions, a norm is traditionally defined in two different ways, namely,

f L2

⎛ 2π ⎞1/2

1/2 ∞ = f 2 = ⎝ f 2 (t)dt⎠ = c2k = cl2 , k=−∞

0

where c = {ck }k∈Z 1 are the Fourier coefficients of the function f. For this reason, the notion of norm for different values of p = 2 can be generalized in at least two ways: in the first case, the quantities ⎛ 2π ⎞1/p f p = ⎝ |f (t)|p dt⎠ 0

Preface

xvii

are taken as a norm, and in the second case a norm is defined as follows:

∞

1/p |ck |

p

= clp .

k=−∞

In Chapter 11, the second approach is used. In particular, this enables one to consider functions from the general positions of analysis and leads to fairly interesting results. It is also necessary to make some remarks concerning Chapter 12. Initially, it was planned to present all assertions in this chapter with complete proofs because there is no systematic presentation of this material available in the monographic literature. However, later, it became clear that an additional volume is required just for this purpose. For this reason, it was decided to present basic results obtained in this field without proofs. Since there are many papers and more than 30 theses devoted to investigations in this field, the author had the problem of the choice of material. The monograph mainly contains results obtained with the participation of the author. Unfortunately, the size of the monograph does not allow the author to include the results of the doctoral thesis of A. K. Kushpel’ devoted to the investigation of the widths of classes of (ψ, β)-differentiable functions, the doctoral thesis of N. L. Pachulia, where problems of strong summability of Fourier series on the sets Lψ β were studied, and the doctoral theses of P. V. Zaderei and A. S. Romanyuk, where approximations of (ψ, β)-differentiable functions of many variables were investigated. Early results on these subjects were presented by the author in his monograph Classification and Approximation of Periodic Functions [Naukova Dumka, Kiev (1987); English version: Kluwer, Dordrecht (1995)]. The main results of the present monograph have been obtained after 1987. In the presentation of the main material, as a rule, we do not refer the reader to the literature; all references are given in the bibliographical notes at the end of the book. The chapters of the monograph are divided into sections, which, in turn, are divided into subsections enumerated by two numbers: the first stands for the number of the section and the second for the number of the subsection itself. The same enumeration is used for all kinds of statements and formulas. If we refer to statements or formulas from a different chapter, then we use triple enumeration, where the first number stands for the chapter and the other two have the meaning described above.

xviii

Preface

My first words of deep and sincere gratitude go to T. A. Andreeva, who patiently carried out the large and difficult work of computer typesetting and composition of the monograph. I am grateful to my colleagues A. S. Serdyuk and V. V. Savchuk for a great help in the preparation of the manuscript and to N. L. Bilokon’ for her kind help in the preparation of the English edition. The warmest words of gratitude are addressed to my wife and colleague N. I. Stepanets, who is the first technical and scientific editor of this book. I am grateful to the entire staff of the Institute of Mathematics of the National Academy of Sciences of Ukraine, where I have been working since 1965, for a creative and friendly atmosphere. A. Stepanets January, 2002–December, 2004

PART I 1. REGULARITY OF LINEAR METHODS OF SUMMATION OF FOURIER SERIES

1.

Introduction

1.1. Let f (x) be a 2π-periodic function Lebesgue summable on the period (f ∈ L) and let S[f ] =

∞

∞

k=1

k=0

a0 (ak cos kx + bk sin kx) ≡ Ak (f ; x) + 2

be its Fourier series, i.e., for any k = 0, 1, 2, . . . , π π 1 1 ak = ak (f ) = f (t) cos ktdt, bk = bk (f ) = f (t) sin ktdt; π π −π

(1.1)

(1.2)

−π

the nth-order partial sum Sn (f ; x) of series (1.1) is the trigonometric polynomial called the nth-order Fourier sum for the function f (x). (n) Further, let Λ = λk , n = 0, 1, . . . , k = 0, 1, . . . , n, be an arbitrary infinite triangular numerical matrix. We associate every function f ∈ L, on the basis of its expansion in a Fourier series, with a sequence of polynomials Un (f ; x; Λ) of the form a0 (n) (n) Un (f ; x; Λ) = λ0 + λk (ak cos kx + bk sin kx). 2 n

(1.3)

k=1

Thus, every triangular matrix Λ determines a method for the construction of polynomials Un (f ; x; Λ), or, in other words, a certain sequence of polynomial operators Un (f ; Λ) is defined on the set L. In this case, one says that the matrix Λ 1

2

Regularity of Linear Methods of Summation of Fourier Series

Chapter 1

determines a certain linear method, or a process, of summation of Fourier series. It is clear that, for any n ∈ N, the operators Un (f ; Λ) are linear. For this reason, the Λ-methods are also called linear methods, or processes, of summation of Fourier series. Substituting the representation of Fourier coefficients (1.2) in (1.3), we obtain 1 Un (f ; x) = Un (f ; x; Λ) = π where

π f (t + x)Un (t; Λ)dt, −π

(n) λo λk cos kt + 2 (n)

Un (t) = Un (t; Λ) =

(1.4)

n

(1.5)

k=1

is a trigonometric polynomial of degree n, which is called the kernel of the operator (method) Un (f ; Λ). (n) If the matrix Λ is such that λk ≡ 1, then, obviously, Un (f ; x; Λ) ≡ Sn (f ; x). According to the accepted terminology, this method is called the method of partial Fourier sums. In this case, 1 cos kt + 2 n

Un (t; Λ) = Dn (t) =

(1.5 )

k=1

is the Dirichlet kernel of order n. Among linear methods different from the method of partial Fourier sums, one should, first of all, mention the classical arithmetic-mean method (Fejér method). (n) This method is determined by the matrix Λ in which λk = 1 − k/n, k = 0, 1, . . . , n − 1. In this case, the polynomials Un (f ; x; Λ) have the form 1 Sk (f ; x) n n−1

Un (f ; x; Λ) = σn (f ; x) =

(1.6)

k=0

and are called Fejér sums. The well-known Fejér theorem states the following: If f (x) is a continuous 2π-periodic function (f ∈ C), then its Fourier series is uniformly summable by the Fejér method, i.e., for any f ∈ C, the following relation holds uniformly on the period: lim σn (f ; x) = f (x). (1.7) n→∞

This theorem put an end to scepticism existed in the theory of Fourier series in the last century and connected with the fact that functions whose Fourier series

Section 1

Introduction

3

diverge at certain points exist even in the set C. The Fejér theorem is a starting point in the contemporary theory of summation of Fourier series. Later, various authors proposed other linear summation methods, among which methods of de la Vallée Poussin, Riesz, Rogosinski, Zygmund, Bernstein, etc. (n) The de la Vallée Poussin method is determined by the matrix Λ = λk in which ⎧ 0 ≤ k ≤ m, ⎨1, (n) λk = n − k (1.8) ⎩ , m < k ≤ n. n−m In this case, the polynomials Un (f ; x; Λ) = Vmn (f ; x) =

n−1 1 Sk (f ; x) n−m

(1.9)

k=m

are called de la Vallée Poussin sums. It is clear that, for m = 0, we have V0n (f ; x) = σn (f ; x). If 2 σ k (n) , k = 0, 1, . . . , n − 1, δ > 0, λk = 1 − n then Un (f ; x; Λ) ≡

Snδ (f ; x)

δ n k2 a0 1 − 2 (ak cos kx+bk sin kx). (1.10) = + 2 n k=1

This method of summation is called the Riesz method, and the polynomials Snδ (f ; x) are called Riesz sums. kπ (n) In the case where λk = cos , k = 0, 1, . . . , n − 1, we obtain the Ro2n gosinski method. Here, 1 Un (f ; x; Λ) ≡ Rn (f ; x) = (Sn (f ; x + π/2n) + Sn (f ; x − π/2n)). 2 The polynomials Rn (f ; x) are called Rogosinski sums.

(1.11)

4

Regularity of Linear Methods of Summation of Fourier Series Setting

(n) λk

Chapter 1

s k = 1− , s > 0, we obtain the Zygmund method. The n

polynomials Un (f ; x; Λ) ≡ Zn(s) (f ; x) s n−1 k a0 1− (ak cos kx + bk sin kx) = + 2 n

(1.12)

k=1

(1)

are called Zygmund sums. For s = 1, we have Zn (f ; x) = σn (f ; x). 1.2. Each of these methods became a milestone in the theory of approximation of functions and the theory of Fourier series. Their different properties were investigated by greatest mathematicians for many decades. Main attention was given to properties that characterize the approximation capability of the methods under study. As a result of the investigation of certain linear methods, new approaches were developed that enable one to correctly formulate and successfully solve the corresponding problems also for general linear methods, i.e., in the case where a summation method is determined by an arbitrary numerical matrix Λ, (n) and it is only known that its elements λk satisfy certain conditions. In the course of investigation of general linear methods, the following question naturally (n) arises: What conditions should be imposed on the numbers λk in order that the sequence Un (f ; x; Λ) converge uniformly on the entire space C, i.e., in order that the following relation hold for all f ∈ C : lim f (x) − Un (f ; x; Λ)C = 0.

n→∞

(1.13)

If this relation is true, then the method Un (f ; Λ) is called regular in the space C, or simply regular. An exhaustive answer to this question is given by the following theorem: Theorem 1.1. In order that a sequence of polynomials Un (f ; x; Λ) converge uniformly to f (x) on the entire space C of continuous 2π-periodic functions, it is necessary and sufficient that, for any fixed k = 0, 1, . . . , n, (n)

lim λk = 1,

(1.14)

n→∞

and, moreover, that the sequence of numbers 2 Ln (Λ) = π

π |Un (t; Λ)|dt, 0

n = 0, 1, . . . ,

(1.15)

Section 1

Introduction

5

which are called the Lebesgue constants of the given method, be bounded: Ln (Λ) = O(1),

n → ∞.

(1.16)

Proof. If equality (1.13) holds for any f ∈ C, then it also holds in the space Tm of trigonometric polynomials Pm (x) of degree m, i.e., for any Pm ∈ Tm , one has lim Un (Pm ; x; Λ) − Pm (x)C = 0. (1.17) n→∞

However, this is obviously true only if (1.14) holds. Furthermore, according to the known Banach theorem, in order that a sequence of linear operators Un (f ) that map a complete normed vector space F into its part possess the property lim f − Un (f )F = 0,

n→∞

it is necessary that Un F = sup{Un (f )F : f ∈ F, f F ≤ 1} = O(1),

n → ∞.

(1.18)

The space C is complete, and, by virtue of (1.4), it is obvious that π 1 f (t)Un (t − x; Λ)dt Un (Λ)C = sup Un (f ; x; Λ)C = sup f C ≤1 f C ≤1 π −π

=

1 π

C

π |Un (t; Λ)|dt = Ln (Λ).

(1.19)

−π

Thus, the conditions of the theorem are necessary. Let us show that they are sufficient. According to the classical Weierstrass theorem, any function f ∈ C can be approximated by trigonometric polynomials with arbitrary accuracy. In (ε) other words, for any ε > 0 and f ∈ C, there exists a polynomial Pm (f ; x) such that (ε) f (x) − Pm (f ; x)C < ε. (1.20) Consequently, for any Λ and n, the following relation is true: f (x) − Un (f ; x; Λ)C (ε) (ε) ≤ f (x) − Pm (f ; x)C + Pm (f ; x) − Un (f ; x; Λ)C (ε) (ε) (ε) ≤ ε + Pm (f ; x) − Un (Pm ; x; Λ)C + Un (Pm − f ; x; Λ)C . (1.21)

6

Regularity of Linear Methods of Summation of Fourier Series

Chapter 1

However, by virtue of (1.4) and (1.20), we have π 2ε (ε) |Un (t; Λ)|dt = εLn (Λ). Un (Pm − f ; x; Λ)C ≤ π 0

Hence, (ε) (ε) f (x) − Un (f ; x; Λ)C ≤ ε(1 + Ln (Λ)) + Pm (f ; x) − Un (Pm ; x; Λ).

Therefore, conditions (1.14) and (1.16) indeed guarantee the validity of relation (1.13). The theorem is proved. As a rule, condition (1.14) can be fairly easily verified. In particular, its validity is obvious for all examples in Subsection 1.2. The verification of condition (1.16), in many cases, encounters serious difficulties. There are numerous deep investigations devoted to this problem, in which sufficient (and often necessary) (n) conditions on the numbers λk that guarantee the validity of relation (1.16) are obtained. The results related to this problem are presented in Sections 2–8. It is shown in Section 3 that the Lebesgue constants for Fourier sums increase unboundedly. Therefore, by virtue of Theorem 1.1, the sums Sn (f ; x) cannot converge uniformly on the entire class C. n (f ; x) = S If we set m = n − 1 in (1.9), then we obtain Vn−1 n−1 (f ; x). Hence, for arbitrarily chosen numbers m, de la Vallée Poussin sums are also not convergent on the entire space C. For the other methods considered in Subsection 1.1, the sequences of Lebesgue constants are bounded, and, thus, these methods induce sequences of polynomials that are uniformly convergent for any f ∈ C. This is also true for the sums Vmn (f ; x) with properly chosen m, e.g., if one sets n − m = [kn], where k ∈ (0, 1).

2.

Nikol’skii and Nagy Theorems

2.1. The first result that establishes conditions for the boundedness of the Lebesgue constants, and, consequently, conditions for the regularity of a given (n) general method Un (f ; Λ) in terms of the elements λk of the matrix Λ belongs to S. M. Nikol’skii. He proved the following theorem: (n)

Theorem 2.1. Suppose that, for any n ∈ N, the system of numbers λk , (n) (n) k = 0, n, λ0 = 1, λn+1 = 0, is convex upwards or downwards, i.e., (n)

(n)

(n)

(n)

Δ2k = Δ2 λk = λk − 2λk+1 + λk+2 ≥ 0 or

Δ2k ≤ 0, k = 0, n − 1. (2.1)

Section 2

Nikol’skii and Nagy Theorems

7

Then the conditions (n)

|λk | ≤ K, and

k = 1, 2, . . . , n, n ∈ N,

n λ(n) k ≤ K1 n − k + 1

(2.2)

(2.3)

k=1

are necessary and sufficient for the following inequality to be true: 2 Ln (Λ) = π

π n 1 (n) λk cos kt dt ≤ K2 , + 2

(2.4)

k=1

0

where K2 is a constant independent of n ∈ N. The necessity of condition (2.2) is obvious:

n 2 π 1 (n) (n) λm cos mt cos ktdt ≤ Ln (Λ). + λk = π 2 m=1

0

Further, let fn (t) = sign sin(n + 1)t

n+1 k=1

n+1 ∞ sin kt 4 sin kt sin(2m − 1)(n + 1)t = . k π k 2m − 1 m=1

k=1

The expansion of this function in a Fourier series has the form S[fn ] =

n ∞ 2 cos kt αk cos kt, + π n−k+1 k=0

k=n+1

where αk are certain numbers. Therefore, π π

n n 2 1 cos kt (n) Un (t; Λ)fn (t)dt = λk cos kt + dt π 2 n−k+1 k=1 k=0 0 0 n λ(n) k = . n − k + 1 k=0

8

Regularity of Linear Methods of Summation of Fourier Series

Chapter 1

At the same time, π π Un (t; Λ)fn (t)dt ≤ fn (t)C |Un (t; Λ)|dt = fn (t)C π Ln (Λ). 2 0

0

However, it is well known that, for any n ∈ N, n sin kt k=1

Hence, we always have

k

π
0 there exists δ > 0 such that Φx (t) < εt for any t ∈ [0, δ]. For such δ, in view of (2.11) we get δ δ δ ∗ ∗ δ ϕx (t)Un (t)dt ≤ |ϕx (t)|Un (t)dt = (Φx (t)Un (t))0 − Φx (t)dUn∗ (t) 0

0

≤ εδUn∗ (δ) − ε

0

δ 0

tdUn∗ (t) = ε

δ 0

Un∗ (t)dt ≤ K1 ε.

(2.17)

Section 2

Nikol’skii and Nagy Theorems

11

At the same time, according to (2.10), for sufficiently large n we have π π ϕx (t)Un (t)dt ≤ mn (δ) |ϕx (t)|dt ≤ K2 ε.

(2.17 )

0

δ

Combining these estimates, we arrive at equality (2.15). The lemma is proved. It is well known that every Lebesgue point of the function f (·) satisfies condition (2.15). Moreover, by virtue of (1.4) and (1.5), 1 Un (f ; x; Λ) − f (x) = π

π ϕx (t)Un (t; Λ)dt. 0

Therefore, according to the lemma, to prove Theorem 2.2 it suffices to show that if inequality (2.9) is true, then the kernel Un (t) = Un (t; Λ) satisfies conditions (a) and (b). Denoting, as before, the Dirichlet kernel by Dk (t), we set Mk (t) =

k

Di (t),

Nk (t) = Mk (t) − Mn (t),

k = 0, 1, . . . , n.

(2.18)

i=0

Since, for any k ∈ N, Dk (t) = (sin(2k + 1)t/2)/(2 sin t/2), we conclude that −1

Mk (t) = (2 sin(t/2)) =

t 4 sin 2 2

=

4 sin2

t 2

−1

−1

t 3t 2k + 1 sin + sin + . . . + sin t 2 2 2

(2.19)

(1 − cos t + (cos t − cos 2t) + . . . + (cos kt − cos(k + 1)t)) (1 − cos(k + 1)t),

i.e., Mk (t) = (sin2 (k + 1)t/2)/2 sin2

t 2

(2.20)

12

Regularity of Linear Methods of Summation of Fourier Series

Chapter 1

and, hence,

t t Nk (t) = − sin(n + k + 2) sin(n − k) 2 2

t /2 sin2 . 2

(2.21)

Applying twice the Abel transformation, according to which, for arbitrary real u0 , u1 , . . . and v0 , v1 , . . . and any n ∈ N, one has n

uk vk =

k=0

n−1

(uk − uk+1 )Vk + un Vn ,

Vk =

k

vi ,

(2.22)

i=0

k=0

and setting ν = [(n + 1)/2], we obtain

Un (t; Λ) =

n

Dk (t)Δk =

k=0

=

ν−1

n−1

Mk (t)Δ2k + Mn (t)Δn

k=0

Mk (t)Δ2k

k=0

+

n−1

Nk (t)Δ2k + Mn (t)Δν .

(2.23)

k=ν

It follows from (2.20) and (2.21) that, for any δ > 0, the functions |Mk (t)| and δ |Nk (t)| are bounded on the segment [δ, π] by sin−2 . Thus, 2 mn (δ) = max |Un (t; Λ)| ≤ sin−2 δ≤t≤π

δ 2

n−1

|Δ2k | + Δν

.

(2.24)

k=0

Let us verify that, under condition (2.9), the right-hand side of (2.24) tends to zero as n → ∞. Since n n−k k+1 ≥ (n − k) > i n

i=n−k

(k + 1)/2

if 0 ≤ k < ν,

(n − k)/2 if ν ≤ k ≤ n,

relation (2.9) yields ν−1 k=0

(k −

1)|Δ2k |

< K1 ,

n−1 k=ν

(n − k)|Δ2k | ≤ K2 .

(2.25)

Section 2

Nikol’skii and Nagy Theorems

13

Here and in what follows, K and Ki , i ∈ N, are absolute positive constants. Further, using the Abel transformation, we get 1 = λ0 − λn+1 =

n

ν−1

Δk =

k=0

=

ν−1

(k +

1)Δ2k

Δk +

k=0

n

Δk

k=ν

+ νΔν + (n − ν + 1)Δν −

k=0

n−1

(n − k)Δ2k .

k=ν

Hence, (n + 1)Δν = 1 −

ν−1

(k +

1)Δ2k

+

k=0

n−1

(n − k)Δ2k .

k=ν

Therefore, in view of (2.25), (n + 1)|Δν | ≤ K3 .

(2.26)

Let us show that, in the case considered, lim

n−1

n→∞

|Δ2k | = lim

n→∞

k=0

n−1

(n)

|Δ2 λk | = 0.

(2.27)

k=0

Let r be an arbitrary natural number. We define a number n0 = n0 (r) so that, for n ≥ n0 and all m ≤ r, the following inequality is true: n

i−1 > r.

(2.28)

i=m

It is clear that such a number n0 (r) always exists. For n ≥ n0 (r), in view of (2.25) and (2.28), we have n−1

|Δ2k |

≤

k=0

r−1 k=0

|Δ2k |

ν−1 n−r 1 1 2 + (k + 1)|Δk | + (n − k)|Δ2k | r r k=r k=ν

n n−1 n−k 1 |Δ2k | + r i k=n−r+1

≤

r−1 k=0

|Δ2 λk | + r−1 K4 . (n)

i=n−k

(2.29)

14

Regularity of Linear Methods of Summation of Fourier Series

Chapter 1

This yields equality (2.27) because, for any ε > 0, one can find r such that r−1 K4 < ε/2 and then, according to (1.14), choose n so large that r−1

(n)

|Δ2 λk | < ε/2.

k=0

Combining (2.24), (2.26), and (2.27), we get lim mn (δ) = 0 ∀δ ∈ (0, π].

n→∞

To prove Theorem 2.2, it remains to show that the kernel Un (t; Λ) satisfies condition (b). To this end, we set 2 π (k + 1)2 /8, 0 ≤ t ≤ 2/(k + 1), ∗ Mk (t) = 2/(k + 1) ≤ t ≤ π. π 2 /2t2 , It is clear that, for any t ∈ [0, π], we have |Mk (t)| ≤ Mk∗ (t) and π

Mk∗ (t)dt ≤ K5 (k + 1).

(2.30)

0

Also let ⎧ 2 π (n + k + 2)(n − k)/4, 0 ≤ t ≤ 2/(n + k + 2), ⎪ ⎪ ⎨ Nk∗ (t) = π 2 (n − k)/2t, 2/(n + k + 2) ≤ 2/(n − k), ⎪ ⎪ ⎩ 2 2 π /t , 2/(n − k) ≤ t ≤ π. Then, for any t ∈ [0, π], one has |Nk (t)| ≤ Nk∗ (t) and π 0

Nk∗ (t)dt

π2 n+k+2 −π = (n − k) 2 + ln 2 n−k π2 n+1 < (n − k) 2 + ln 2 + ln 2 n−k

n i−1 . < K6 (n − k) 1 + i=n−k

(2.31)

Section 3

Lebesgue Constants of Classical Linear Methods

15

We define the required function Un∗ (t; Λ) by the formula Un∗ (t; Λ) =

ν−1

Mk∗ (t)|Δ2k | +

k=0

n−1

Nk∗ (t)|Δ2k | + Mn∗ (t)|Δν |.

k=ν

It is easy to see that it decreases monotonically and satisfies the condition |Un (t; Λ)| ≤ Un∗ (t; Λ), and, furthermore, by virtue of (2.30), (2.31), (2.25), and (2.26), we have π Un∗ (t; Λ)dt ≤ K. 0

Theorems 2.2 and 2.1 are proved.

3.

Lebesgue Constants of Classical Linear Methods

The statements established in Section 2 enable one to solve the problem of the boundedness of Lebesgue constants for a wide range of linear methods. At the same time, more complete information about these constants is available for several classical linear methods. 3.1. According to Subsection 1.1, the kernel of the Fourier method is the trigonometric polynomial 1 sin(2n + 1)t/2 cos kt = + . 2 2 sin(t/2) n

Dn (t) =

(3.1)

k=1

Therefore, in view of (1.15), the Lebesgue constant Ln of the Fourier method has the form 2 Ln = π

π

| sin((2n + 1)t/2)| 2 dt = sin(t/2) π

0

=

2 π

π/2

π/2

| sin(2n + 1)t| dt sin t

0

| sin(2n + 1)t| dt + r1 (n), t

0

where 2 r1 (n) = π

π/2 t − sin t | sin(2n + 1)t| dt. t sin t 0

(3.2)

16

Regularity of Linear Methods of Summation of Fourier Series

Chapter 1

For t ∈ [0, π/2], we have sin t > t − t3 /6 and sin t > 2t/π, whence 2 r1 (n) ≤ π

π/2

πt π2 dt = . 12 48

(3.3)

0

Further, according to (3.2), we get n−1 2 Ln = π

π/(2n+1)

sin(2n + 1)t

k=0

2 = π

0

π sin t

n−1 k=0

0

2 = π

dt + r1 (n) t + kπ/(2n + 1)

π sin t

n−1 k=1

0

1 dt + r1 (n) t + kπ 1 2 dt + t + kπ π

π

sin t dt + r1 (n). t

(3.4)

0

For t ∈ [0, π], we have n n−1 n−1 n−1 1 11 1 11 ≤ ≤ ≤ , π k (k + 1)π t + kπ π k k=2

i.e.,

n−1 k=1

k=1

k=1

1 1 = t + kπ π

k=1

n−1

1 − θn , k

θn ∈ [0, 1].

(3.5)

k=1

At the same time, for any n ∈ N, we get ln n
0,

(3.26)

k=1

and the kernel of the Zygmund method is the polynomial Zns (t)

s n−1 k 1 1− cos kt, = + 2 n

s > 0.

k=1

Representations of these kernels in a form similar to (3.1), (3.13), (3.17), and (3.23) are unknown, except for the case where s = 1 in the Zygmund method. This complicates the direct determination of upper bounds for their Lebesgue constants. At the same time, the boundedness of these constants follows from Theorem 2.1. Indeed, setting 2 δ k 1− , n

(n) λk

=

Section 4

Lower Bounds for Lebesgue Constants (n)

21

(n)

we conclude that the system of numbers λk , k = 0, n, λn+1 = 0, satisfies conditions (2.1), (2.2), and (3.3) because, in this case, the expression n (1 − (k/n)2 )δ k=1

n−k+1

=

n−1 k=1

(1 − (k/n)2 )δ 1 , 1 − (k − 1)/n n

δ > 0,

is the integral sum for the integral 1 0

(1 − u2 )δ du = 1−u

1

(1 − (1 − t)2 )δ dt t

0

1

1 t

=

δ−1

(2 − t) dt ≤ 2 δ

0

δ

tδ−1 dt < δ −1 .

0

Thus, the constants Rn of the Riesz method are uniformly bounded. By analogy, we establish the boundedness of the Lebesgue constants Zns of (n) (n) the Zygmund method. If λk = 1 − (k/n)s , k = 0, n, λn+1 = 0, then conditions (2.1) and (2.2) are obvious. The expression n 1 − (k/n)s k=1

n−k+1

=

n−1 k=1

1 − (k/n)s 1 , 1 − (k − 1)/n n

s > 0,

is the integral sum for the integral 1 0

1 − us du 1−u

of a bounded function. Thus, condition (2.3) is satisfied, and, hence, the constants Zns are uniformly bounded.

4.

Lower Bounds for Lebesgue Constants 4.1. Inequality (2.6) means that, for an arbitrary trigonometric polynomial a0 ak cos kt, Kn (t) = + 2 n

k=1

22

Regularity of Linear Methods of Summation of Fourier Series

we have

π n a k ≤ C1 |Kn (t)|dt, n − k + 1 k=0

Chapter 1

C1 = siπ.

(4.1)

0

Along with this estimate, the following statement is often useful: Theorem 4.1. If ak ≡ 0, then, for any n ∈ N, n k=0

|ak | , π n−k+1

n ∈ N.

(4.5)

k=1

Linear Methods Determined by Rectangular Matrices

5.1. In the previous sections, the matrix Λ was regarded as triangular, i.e., ≡ 0 for k > n. However, it is reasonable to omit this condition and consider (n) matrices Λ with arbitrary elements λk , n = 0, 1, . . . , k = 0, 1, . . . . With every function f ∈ L with Fourier series S[f ], we associate the sequence of series (n) λk

(n)

Un (f ; x; Λ) =

a0 λ1 2

+

∞

(n)

λk (ak cos kx + bk sin kx), n = 0, 1, . . . , (5.1)

k=1

which, in the case considered, may not be trigonometric polynomials; generally speaking, they may not even be Fourier series of certain functions.

24

Regularity of Linear Methods of Summation of Fourier Series

Chapter 1

Thus, the following question naturally arises: For f ∈ C, what conditions (n) should be satisfied by the numbers λk in order that series (5.1) converge uniformly for any n ∈ N and lim f (x) − Un (f ; x; Λ)C = 0

n→∞

(5.2)

for every f ∈ C ? In this case, the summation method Un (Λ) determined by a given matrix Λ is also called regular in the space C, or simply regular. An answer to this question is given by the following theorem: Theorem 5.1. In order that the summation method Un (Λ) be regular in the space C, it is necessary and sufficient that the following conditions be satisfied: (i) for all k = 0, 1, 2, . . . , (n)

lim λk = 1;

(5.3)

n→∞

(ii) every function (n) λ0 λk cos kt, + 2 m

(n)

Kn(m) (t) =

n = 0, 1, . . . , m ∈ N

k=1

satisfies the condition π |Kn(m) (t)|dt ≤ Mn ,

n = 0, 1, . . . ,

(5.4)

0

where Mn is a constant independent of m; (iii) the total variation of the function x ¯ n (x) = lim K

(n)

Kn(m) (t)dt =

m→∞

xλ0 2

0

+

∞ 1 (n) λ sin kx k k k=1

is uniformly bounded on the segment [0, π] : π

π

¯n = VK 0

¯ n (t)| ≤ C. |dK 0

(5.5)

Section 5

Linear Methods Determined by Rectangular Matrices

25

Proof. The proof of this theorem is based on the following well-known statement from functional analysis: Theorem 5.2. In order that a sequence {Fm } of linear functionals be convergent to a functional F0 at every point f of a Banach space E, it is necessary and sufficient that the following conditions be satisfied: (i) the norms of the functionals Fm are uniformly bounded: Fm ≤ C;

(5.6)

(ii) if M is the set of functions ϕi , i = 0, 1, . . . , whose linear combinations are dense everywhere in E, then, for any ϕi ∈ M, lim Fm (ϕi ) = F0 (ϕi ),

i = 0, 1, 2, . . . .

m→∞

(5.7)

Conditions (i) and (ii) guarantee the validity of the following equality: lim Fm (f ) = F0 (f ) ∀f ∈ E.

m→∞

(5.8)

Furthermore, if f (v) is continuous on (a + α, b + β), 0 < α < β, and fx (t) = f (x + t), then (5.9) lim Fm (fx ) = F0 (fx ) m→∞

uniformly in t ∈ (α, β). If series (5.1) converge uniformly for any f ∈ C, then, setting (n)

(n) Fm (f )

=

=

(n) Fm (f ; x)

1 π

1 = π

a0 λ0 = 2

+

m

(n)

λk Ak (f ; x)

k=1

π f (x + t)Kn(m) (t)dt −π

π ¯ (m) (t), f (x + t)dK n −π

n = 0, 1, 2, . . . ,

(5.10)

26

Regularity of Linear Methods of Summation of Fourier Series

Chapter 1

we conclude that, according to Theorem 5.2, for this purpose it is necessary and sufficient that, first, the norms of the functionals Fm (f ) be bounded, i.e., π 1 (m) f (x + t)Kn (t)dt Fm (f ) = sup f C ≤1 π −π

π

2 π

=

|Kn(m) (t)|dt ≤ Cn ,

(5.11)

0

and, second, any function from the trigonometric system 1, cos x, sin x, cos 2x, sin 2x, . . .

(5.12)

satisfy condition (5.7), in particular, (n) (cos kx) lim Fm

m→∞

(n)

(n)

= lim λk cos kx = F0 (cos kx), m→∞

k = 0, 1, 2, . . . . (5.13)

(n)

In addition, the functional F0 (f ) satisfies the condition (n) F0 (f )

1 = lim m→∞ π

π ¯ nm (t). f (x + t)dK −π

However, in the space C, every linear functional has the form 1 π

π f (x + t)dK(t),

(5.14)

−π

where K(t) is a certain periodic function of bounded variation on the period. Therefore, taking into account the Helly theorem on the limit transition under the sign of Stieltjes integral, we get (n) F0 (f )

1 = π

π ¯ n (t) = Un (f ; x; Λ), f (x + t)dK −π

¯ n (t) is a function of bounded variation on the period. where K

(5.15)

Section 5

Linear Methods Determined by Rectangular Matrices

27

Thus, the uniform convergence of series (5.1) yields relations (5.11), (5.13), and (5.15), which immediately implies the necessity of conditions (5.3) and (5.4). Further, setting 1 Fn (f ; x) = Um (f ; x; Λ) = π

π ¯ n (t) f (x + t)dK −π

in Theorem 5.2, we conclude that the condition 2 Fm (f ; x) = π

π ¯ n (t)| < C, |dK

(5.16)

0

together with (5.3), is necessary and sufficient for equality (5.2) to be valid. Therefore, condition (5.5) is also necessary. The sufficiency of conditions (5.3)–(5.5) is also obvious, Indeed, if conditions (5.3) and (5.4) are satisfied, then relations (5.11), (5.13), and (5.15) are also true. Hence, series in (5.1) converge uniformly, and if, in addition, relation (5.5) holds, then, by virtue of (5.16), equality (5.2) is also true. 5.2. The simplest sufficient conditions for the regularity of a rectangular summation method are given by the following Karamata–Tomic theorem: Theorem 5.3. In order that the method Un (Λ) be regular in the space C, it (n) is sufficient that the elements λk of the matrix Λ satisfy the following conditions: (i) for all k = 0, 1, 2, . . . , one has (n)

lim λk = 1,

n→∞

(5.17)

and, for a fixed n, (n)

λk = O(1/ ln k),

k → ∞;

(5.18)

(ii) for every n, one can indicate a number m (fixed or tending to infinity with n) such that ∞ m+k m+k (n) |m − k| ln |Δ2 λk | ≤ C, m |m − k|

k=0 k =m

(5.19)

28

Regularity of Linear Methods of Summation of Fourier Series

Chapter 1

where (n)

(n)

(n)

(n)

Δ2 λk = λk − 2λk+1 + λk+2 and C is a constant independent of n. Proof. This theorem is a consequence of the Telyakovskii theorem presented in Subsection 8.5. We only note here that, for any fixed m, condition (5.19) is equivalent to the inequality ∞

(n)

(k + 1)|Δ2 λk | ≤ C1 ,

(5.20)

k=0 (n)

where the constant C1 is independent of n. In this case, the sequence λk , k = 0, 1, . . . , is called quasiconvex with respect to the index k uniformly in n ∈ N.

6.

Estimates for Integrals of Moduli of Functions Defined by Cosine and Sine Series

6.1. Many problems in approximation theory lead to the necessity of obtaining estimates for integrals of the form π ∞ a0 (ak cos kx + bk sin kx) dx + 2

−π

(6.1)

k=1

in terms of the coefficients ak and bk , k ∈ N. In this section, we obtain estimates for integral (6.1) in the case where the series in it consists only of cosines or only of sines, i.e., for the series ∞

a0 ak cos kx + 2

(6.2)

k=1

or

∞

ak sin kx.

(6.3)

k=1

First, we present several well-known facts from the general theory of trigonometric series related to series (6.2) and (6.3), which will be used in what follows.

Section 6

Estimates for Integrals of Moduli of Functions

29

The sequence {ak } is of bounded variation if ∞

|Δak | = V (a) < ∞,

(6.4)

k=0

where Δak = ak − ak+1 . Theorem 6.1. If a sequence {ak } is of bounded variation and lim ak = 0,

k→∞

(6.5)

then both series (6.2) and (6.3) converge uniformly on every segment ε ≤ x ≤ 2π−ε for any ε > 0. Furthermore, series (6.3) converges everywhere, and series (6.2) converges for all x ∈ [−π, π], except, possibly, the point x = 0. Theorem 6.2. If a trigonometric series converges in the interval (0, 2π) to some integrable function f (x), then it is the Fourier series of this function. Combining these theorems, we obtain the following statement: Corollary 6.1. If conditions (6.4) and (6.5) are satisfied and series (6.2) and (6.3) converge to integrable functions, then these series are the Fourier series of their sums. Theorem 6.3. If a sequence {ak } satisfies condition (6.5) and, for any k = 0, 1, . . . , one has Δ2 ak = ak − 2ak+1 + ak+2 ≥ 0, (6.6) then series (6.2) converges everywhere, except, possibly, the point x = 0, to some integrable function c(x) and is its Fourier series; moreover, c(x) ≥ 0. Condition (6.6) is essential: one can indicate a series of the form (6.2) with coefficients monotonically tending to zero whose sum does not belong to L. Theorem 6.4. Suppose that the sequence {ak } tends monotonically to zero as k → ∞. Then the sum g(x) of series (6.3) is integrable if and only if ∞

ak /k < ∞.

(6.7)

k=1

If condition (6.7) is satisfied, then series (6.3) is the Fourier series of the function g(x), and series (6.2) is the Fourier series of some function c ∈ L.

30

Regularity of Linear Methods of Summation of Fourier Series

Chapter 1

Theorem 6.4. If both series ∞

a0 (ak cos kx + bk sin kx) + 2 k=1

and

∞

(ak sin kx − bk cos kx)

k=1

are Fourier series, then

∞ a2n + b2n /n < ∞. n=1

6.2. The main result for series of the form (6.2) to be proved here is the following theorem: Theorem 6.5. Suppose that series (6.2) satisfies conditions (6.4) and (6.5) and, furthermore, n/2 ∞ Δan−k − Δan+k | | = B(a) < ∞, k

df

n/2 =[n/2].

(6.8)

n=2 k=1

Then series (6.2) is a Fourier series and the following estimate holds: π I=

∞

a0 | + an cos nx|dx ≤ C(V (a) + B(a)), 2

(6.9)

n=1

0

where C is a certain absolute constant. Proof. By virtue of Corollary 6.1, the theorem will be proved if we establish estimate (6.9). Performing the Abel transformation, setting Δak = Δk , and taking relation (2.19) into account, we get π ∞ I= | Δk sin(k + 1/2)x| k=0

0

2 ≤ π

π | 0

∞ k=0

dx 2 sin x/2

Δk sin(2k + 1)x|

dx . x

(6.10)

Section 6

Estimates for Integrals of Moduli of Functions

31

Further, we use the following statement of Boas [1] well known in the theory of Fourier series: Lemma 6.1. Let f ∈ L and ϕ = f (x)sign x for x ∈ [−π, π]. If the Fourier series of the functions f (·) and ϕ(·) are absolutely convergent, then the integral π dx |f (x)| , (6.11) x −π

converges, and, moreover, π −π

dx |f (x)| ≤C x

∞ |a0 | + |a0 | (|an | + |bn | + |an | + |bn |) , + 2

(6.12)

n=1

where {an , bn } and {an , bn } are the Fourier coefficients of the functions f (·) and ϕ(·), respectively, and C is an absolute constant. Setting f (x) =

∞

Δk sin(2k + 1)x, we conclude that, by virtue of (6.4),

k=0

the Fourier series of the function f (x) is absolutely convergent. Further, since π sin(2k + 1)x cos nxdx 0

=

0,

n = 2p + 1,

(2k + 1 + n)−1 + (2k + 1 − n)−1 , n = 2p, p = 0, 1, . . . ,

the Fourier series of the function ϕ(x) = f (x)sign x is the series

∞ ∞ 2 Δk (2k + 1 + 2n)−1 + (2k + 1 − 2n)−1 cos 2nx, π n=0

(6.13)

k=0

which is obviously absolutely convergent. Therefore, by virtue of (6.10) and (6.12), we get I≤C

∞ k=0

∞ ∞ −1 −1 |Δk | + ((2k + 1 + 2n) + (2k + 1 − 2n) ) . (6.14) n=0 k=0

32

Regularity of Linear Methods of Summation of Fourier Series

Chapter 1

∞ ∞ 2 Δk |Δk | ≤2 2k + 1

Since

k=0

k=0

∞ ∞ 1 Δn |Δk |, +1 ≤C 4n + 1

and

n=0

k=0

relation (6.14) can be rewritten as follows: ∞ ∞ −1 −1 I ≤ C V (a) + Δk ((2k + 1 + 2n) + (2k + 1 − 2n) ) . (6.15) n=1 k=0 k =n

If (2k + 1 + 2n)−1 + (2k + 1 − 2n)−1 in the last sum is replaced by ((k + n)−1 + ∞ (k − n)−1 )/2, then this sum changes by a value that does not exceed C |Δn |. n=0

Therefore, according to (6.15), we have ⎞ ⎛ ∞ ⎟ ∞ ⎜ −1 −1 Δk ((k + n) + (k − n) )⎟ I ≤C⎜ ⎠. ⎝V (a) + n=1 k=0 k =n

(6.16)

(0)

We represent the inner sum Sn in the form Δk ((k + n)−1 + (k − n)−1 ) + Δk /(k + n) Sn(0) = 0≤k n, then Δbi−k = Δai−k . However, i − [i/2] > n for all i > 2n, and, hence, i/2 2n Δbi−k − Δai−k Rn ≤ | |. (8.39) k i=n k=1

Section 8

Corollaries of Theorem 7.1

75

Since i/2 Δbi−k − Δai−k

k

k=1

k=1

+ by setting di,k

i/2−1

= −(bi − ai ) +

1 1 (bi−k − ai−k )( − ) k k+1

1 − ai−[i/2] ), (b [i/2] i−[i/2]

⎧ −1, k = 0, ⎪ ⎪ ⎨ = k −1 − (k + 1)−1 , k = 1, . . . , [i/2] − 1, ⎪ ⎪ ⎩ [i/2]−1 , k = [i/2],

we obtain i/2 Δbi−k − Δai−k k=1

k

=

i/2

(bi−k − ai−k )dik .

k=0

However, bi−k = 0 for k ≥ i − n and bi−k = ai−k for i − n > k. Therefore, i/2 i/2 2n 2n Δbi−k − Δai−k | | ai−k di,k | |= k i=n k=1

i=n k=i−n

≤

=

max |ak |

n/2≤k≤n

|di,k |

i=n k=i−n

max |ak |(2 +

n/2≤k≤n

≤

i/2 2n

2n

(i − n)−1 )

i=n+1

max |ak |(3 + log n).

n/2≤k≤n

(8.40)

Comparing relations (8.38)–(8.40), we obtain estimate (8.36) and thus establish equality (8.32). Now note that, for any n ∈ N, V (b) =

∞ k=0

|Δbk | = |an+1 | +

∞

|Δak | ≤ |an+1 | + V (a).

(8.41)

k=n+1

Therefore, combining relations (8.31), (8.32), (8.41), and (8.23)–(8.25), we obtain estimate (8.28). Lemma 8.2 and, hence, Theorem 8.1 are proved.

76

Regularity of Linear Methods of Summation of Fourier Series

Chapter 1

8.5. By virtue of estimates (8.6) and (8.12), relations (8.16) and (8.17) hold if (n)

(n) |+ |λ0 | + |λm

∞ k|m − k|

m+k

k=1

(n)

|Δ2 λk−1 | ≤ C.

(8.42)

Therefore, the following theorem is true: Theorem 8.1. Suppose that the elements λk , n = 0, 1, . . . , k = 0, 1, . . . , of a matrix Λ satisfy the following conditions: (n)

(i) for any number n, there exists a number m such that inequality (8.42) holds; (ii) for any n, there exists a number Mn dependent only on n and such that estimate (8.18) holds for all k; Then, in order that the summation method Un (Λ) be regular in the space C, it is necessary and sufficient that conditions (8.19) and (8.20) be satisfied. Let us show that Theorem 8.1 yields the statement of Theorem 5.3. To this end, it suffices to show that conditions (5.17)–(5.19) yield estimates (8.42) and (8.20). Let 1+ξ 1+ξ Φ(ξ) = |1 − ξ| log , ξ > 0, ξ = 1. ξ |1 − ξ| If ξ ∈ (0, 1), then Φ(ξ) > 2|1 − ξ|, and since Φ(ξ) = Φ(1/ξ), a similar inequality holds for ξ ∈ (1, ∞). Therefore, if (5.19) is true, then C≥

∞ k=1 k =m+1

∞

k|m − k + 1| k−1 (n) (n) (k − 1)Φ( )|Δ2 λk−1 | > |Δ2 λk−1 | m m k=1

1 1 (n) k|Δ2 λk−1 | + ≥ 2 2 m/2

k=1

∞

(n)

|m − k||Δ2 λk−1 |

k=m/2+1

∞

>

1 k|m − k| 2 (n) |Δ λk−1 |. 8 m+k

(8.43)

k=1

Thus, for any m ∈ N, we have ∞ k|m − k| k=1

m+k

(n)

|Δ2 λk−1 | ≤ 8C.

(8.44)

Section 8

Corollaries of Theorem 7.1

77

Further, by virtue of (5.18), we get (n)

lim λk = 0.

k→∞

Therefore, for any m ∈ N, λ(n) m =

∞

(n)

∞

(n)

Δλk , Δλk =

k=m

(n)

Δ2 λi ,

i=k

whence λ(n) m =

∞ ∞

(n)

Δ2 λi

i ∞

=

(n)

Δ2 λi

=

i=m k=m

k=m i=k

∞

(n)

(i − m + 1)Δ2 λi .

i=m

In view of estimate (8.43), this yields (n) | |λm

≤

∞

(k − m +

(n) 1)|Δ2 λk |

=

k=m

∞

(n)

(k − m)|Δ2 λk−1 | ≤ C.

(8.45)

k=m+1

Combining relations (8.44)–(8.45) and (5.17), we obtain estimate (8.42). Now let us verify condition (8.20). We have m m m+k−1 1 (n) 1 (n) (n) Δλi |. |λm−k − λm+k | = | k k k=1

k=1

(8.46)

i=m−k

However, m+k−1

(n) Δλi

=−

i=m−k

m+k−2

(n)

(m − i)Δ2 λi

(n)

(n)

+ (k + 1)Δλm−k + (k − 1)Δλm+k−1 .

i=m−k

Thus, we must estimate the following three sums: m m+k−2 1 (n) (m − i)Δ2 λi |, | S1 = k k=1

S2 =

i=m−k

S3 =

m k−1 k=1

k

m k+1 k=1

(n)

Δλm+k−1 .

k

(n)

Δλm−k ,

(8.47)

78

Regularity of Linear Methods of Summation of Fourier Series

Chapter 1

Changing the order of summation and taking (5.19) into account, we get S1 ≤

m

(n)

(m − i)|Δ2 λi |

i=1

≤

m

m 2m m 1 1 (n) |m − i||Δ2 λi | + k k i=m+1

k=m−i

(m − i) ln

i=1

+

k=i−m

m (n) |Δ2 λi | m−i 2m

|m − i| ln

i=m+1

It is clear that S2 + S3 ≤ 3

m (n) |Δ2 λi | < C. |m − i|

∞

(8.48)

(n)

|Δλk |.

k=0

Hence, according to (8.12) and (8.42), we obtain S2 + S3 ≤

(n) C(|λ0 |

+

m−1 k=1

(n)

∞

k(m − k) 2 (n) (n) k|Δ2 λk+m−1 |) |Δ λk−1 | + |λ(n) m |+ m

(n) |+ ≤ C(|λ0 | + |λm

k=1

∞ k=1

k|m − k| 2 (n) |Δ λk−1 |) < C. m+k

(8.49)

Combining relations (8.46)–(8.49), we obtain estimate (8.20). Thus, Theorem 5.3 is proved.

2. SATURATION OF LINEAR METHODS

1.

Statement of the Problem

1.1. Together with the notion of regularity considered in Chapter 1, the notion of saturation of linear method refers to the general problems of the theory of linear methods of summation of Fourier series. In order to pass to this notion, we make several remarks. First of all, note that we use here the notation of Chapter 1 and, in particular, the notation accepted in Subsection 1.1.1. If, for some k ∈ N, we have (n)

f (x) − Un (f ; x; Λ)C = o(|1 − λk |), n → ∞,

(1.1)

then the Fourier coefficients ak and bk of the function f (·) are equal to zero. Indeed, since π 1 (n) [f (x) − Un (f ; x; Λ)]e−ikx dx = (1 − λk )(ak − ibk ), (1.2) π −π

(n) |1 − λk | a2k + b2k ≤ 2f (x) − Un (f ; x; Λ)C ,

we have

and, consequently, relation (1.1) holds only if ak = bk = 0. Therefore, if a given method Un (Λ) satisfies relation (1.1) for all k beginning with a certain k0 , then f (·) is a trigonometric polynomial of degree not higher than k0 − 1. In particular, if k0 = 1, then f (·) is a constant. Consequently, if f (·) is not identically a constant, then the order of decrease in f (x) − Un (f ; x; Λ)C to zero as n → ∞ is not larger than the maximal order of decrease in the differences (n) (n) 1 − λk , k ∈ N, to zero. Thus, Fejér sums satisfy the relation 1 − λk = k/n, (n) and, hence, min(1 − λk ) = n−1 . Therefore, k

f (x) − σn (f ; x)C = o(n−1 ), n → ∞, 79

80

Saturation of Linear Methods

Chapter 2

only if f (x) ≡ const. In other words, for any f ∈ C, f ≡ const, we have f (x) − σn (f ; x)C > Kn−1 , where K is independent of n. Similarly, we conclude that, in the case of Zygmund sums, we have f (x) − Zn(s) (f ; x)C > Kn−s , f ≡ const, and, in the case of Rogosinski sums, we have f (x) − Rn (f ; x)C > Kn−2 , f ≡ const. (n)

For partial Fourier sums, we have 1−λk = 0. Therefore, equality (1.1) does not really impose any restrictions on f (x) − Sn (f ; x)C . For de la Vallée Poussin sums, the restrictions depend on the choice of the number p. Thus, in the case of Fejér, Rogosinski, and Zygmund sums and some other linear methods, there exists a function ϕΛ (n) such that the approximations induced by a given method do not decrease faster than ϕΛ (n). In this connection, we have the following problem of saturation in the theory of linear methods of summation of Fourier series: Given a concrete Λ-method and certain properties of the elements of the matrix Λ, establish whether there exists a function ϕΛ (n) described above, determine it, and describe the class of functions for which the orders of approximation obtained by this method are equal to ϕΛ (n). 1.2. We now give the main definition. Let Lp , p ≥ 1, be the space of 2πperiodic functions ϕ(·) with finite norm ϕp , where, for p ∈ [1, ∞), we have π ϕLp = ϕp = (

|ϕ(t)|p dt)1/p ,

(1.3)

−π

and, for p = ∞, we have ϕ∞ = esssup |ϕ(t)|, i.e., L1 = L. As above, let C be the set of 2π-periodic continuous functions ϕ(t) with norm ϕ(t)C = max |ϕ(t)|. Definition 1.1. Let X be either C or Lp , p ∈ [1, ∞), and let Un (Λ) be a linear method of summation of Fourier series that generates polynomials Un (f ; x; Λ) of the form (1.1.3). Suppose that there exists a positive function ϕΛ (n), n ∈ N, that monotonically decreases to zero and is such that if f (x) − Un (f ; x; Λ)X = o(ϕΛ (n)), n → ∞,

(1.4)

Section 2

Sufficient Conditions for Saturation

81

then, for X = C, f (x) ≡ const, and, for X = Lp , f (x) = const almost everywhere, and there exists at least one function f (·) not identically equal to a constant and such that f (x) − Un (f ; x; Λ)X = O(ϕΛ (n)), n → ∞.

(1.5)

Then the method Un (Λ) is called saturated in the space X. The set of functions Φ(Λ)X satisfying equality (1.5) is called the saturation class, and the function ϕΛ (n) is called the order of saturation of the method Un (Λ). In view of this definition, the problem of saturation of a certain method Un (Λ) is to establish whether it is saturated in a given space X, and if so, then to determine the order and the class of saturation.

2.

Sufficient Conditions for Saturation 2.1. We cite a few more definitions that will be needed below. Let V be the set of functions F (x) of bounded variation with a finite norm π |dF (t)|

F V =

(2.1)

−π

such that F (x + 2π) = F (x) + (F (π) − F (−π)) and S[dF ] = a0 /2 +

∞

(ak (dF ) cos kx + bk (dF ) sin kx),

(2.2)

(2.3)

k=1

where 1 ak (dF ) = π

π −π

1 cos ktdF (t), bk (dF ) = π

π sin ktdF (t),

(2.4)

−π

is the Fourier–Stieltjes series for the function F ∈ V. A symbol ϕ(x)dF (x) denotes the Lebesgue–Stieltjes integral for the function ϕ(·). Further, let ψ(k), k ∈ N, be an arbitrary function of a natural variable. Let Lψ Lp be the set of functions f ∈ Lp such that the series ∞ k=1

1 (ak (f ) cos kx + bk (f ) sin kx) ψ(k)

(2.5)

82

Saturation of Linear Methods

Chapter 2

is the Fourier series of some function f ψ (·) from Lp , C ψ L∞ is the set of continuous functions f (·) such that series (2.5) is the Fourier series of a function f ψ (·) from L∞ , and Lψ V is the set of functions f ∈ L1 such that (2.5) is the Fourier–Stieltjes series of some function ϕ ∈ V. Using these definitions, we obtain the following result: Theorem 2.1. Suppose that, for the elements of a matrix Λ, there exists a positive function ϕΛ (n) monotonically decreasing to zero and such that (n)

1 − λk c = , ψ(k) > 0, k = 1, 2, . . . , n→∞ ϕΛ (n) ψ(k) lim

(n)

λ0

= 1, |c| > 0,

(2.6)

Then the method Un (Λ) is saturated in the spaces C and Lp , p ∈ [1, ∞), with order of saturation ϕΛ (n). Moreover, the saturation classes satisfy the relation Φ(Λ)C ⊆ C ψ L∞ ; Φ(Λ)Lp ⊆ Lψ Lp , p ∈ (1, ∞), Φ(Λ)L1 ⊆ Lψ V.

(2.7)

(n)

Proof. First of all, we note that, by virtue of (1.2), if λk = 1, then

a2k (f )

+

b2k (f )

= |[π(1 −

(n) λk )]−1

π

(f (t) − Un (f ; t; Λ))e−ikt dt|

−π

=|

ϕΛ (n) π(1 −

π

(n) λk ) −π

f (t) − Un (f ; t; Λ) −ikt dt|. e ϕΛ (n)

Therefore, if (2.6) and (1.4) are satisfied, then the right-hand side of the last equality tends to zero as n → ∞. Hence, ak (f ) = 0 and bk (f ) = 0 for all k ∈ N, i.e., f (x) ≡ const. (n) On the other hand, if f0 = cos x, then f0 (x)−Un (f ; x; Λ) = (1−λ1 ) cos x. Thus, in view of (2.6), we conclude that if X denotes either C or Lp , then f0 (x) − Un (f ; x; Λ)X = ϕΛ (n)

(n)

1 − λ1 cos xX = O(ϕΛ (n)). ϕΛ (n)

This means that the method Un (Λ) is saturated and the order of its saturation is equal to ϕΛ (n). It remains to show that equality (1.5) yields the corresponding inclusion (2.7). To this end, we show that if equality (1.5) holds, then the Fejér sums of series

Section 2

Sufficient Conditions for Saturation

83

(2.5) satisfy the uniform estimates

N −1

(1 − k/N )

k=1

1 Ak (f ; x)Y ≤ A, ψ(k)

(2.8)

where Y = L∞ if X = C in (1.5), and Y = Lp , p ∈ [1, ∞), if X = Lp . We now use the following fact, which is proved, e.g., in the book of Bari [1, pp. 165–170]: Lemma 2.1. A trigonometric series α0 /2 +

∞

(αk cos kx + βk sin kx)

k=1

is the Fourier series of a continuous function if and only if the sequence of its Fejér means σN (x) = α0 /2 +

N −1

(1 − k/N )(αk cos kx + βk sin kx)

k=1

converges uniformly. This series is the Fourier series of some function f ∈ Lp , 1 < p ≤ ∞, if and only if its Fejér means are uniformly bounded in Lp : σN (x)p ≤ A. A trigonometric series is the Fourier–Stieltjes series of some function F ∈ V if and only if σN (x)1 ≤ K, N = 1, 2, . . . . This immediately yields inclusions (2.7). Thus, let us prove estimate (2.8). By setting Fn (x; Λ) = (f (x) − Un (f ; x; Λ))/ϕΛ (n), n = 1, 2, . . . , according to (1.5), we conclude that, for any n ∈ N, Fn (x; Λ)X ≤ A. Therefore, the Fejér sums σN (Fn (·; Λ); x) of the functions Fn (x; Λ), which are equal to N −1 (n) 1 − λk σN (Fn (·; Λ); x) = (1 − k/N ) Ak (f ; x), ϕΛ (n) k=1

are uniformly bounded in X, i.e., σN (Fn (·; Λ); x)X ≤ A.

(2.9)

84

Saturation of Linear Methods

Chapter 2

However, relation (2.6) yields lim σN (Fn (·; Λ); x) = c

n→∞

N −1

(1 − k/N )

k=1

1 Ak (f ; x) ψ(k)

Therefore, (2.9) yields (2.8). The theorem is proved. 2.2. Remark 2.1. The conclusion of Theorem 2.1 remains true and its proof remains practically the same if condition (2.6) is replaced by the conditions (n)

(n)

1 − λk 1 − λk = α(k, n) + β(n) or = α(k, n)β(n), ϕΛ (n) ϕΛ (n)

(2.6 )

c , c = const, c = 0, and β(n) ψ(k) does not depend on k and is uniformly bounded in n, i.e., |β(n)| < A and c β(n) = − in the first case and |β(n)| > a0 > 0 in the second one. ψ(k) where α(k, n) is such that lim α(k, n) = n→∞

3.

Saturation Classes

3.1. Theorem 2.1 gives sufficient conditions for the saturation of a given method Un () and indicates the order of saturation and the set of functions containing the saturation class. The next result we wish to prove contains more precise information about saturation classes. Prior to its formulation, we give some additional information. Let T be the set of trigonometric series of the form df

y(x) = a0 /2 +

∞

(ak cos kx + bk sin kx)

(3.1)

k=1

and let μ = μ(k), k = 0, 1, . . . , be a fixed numerical sequence. For every y ∈ T, we define an element z ∈ T as follows: ∞

a0 μ0 μ(k)(ak cos kx + bk sin kx). z(x) = + 2 df

(3.1 )

k=1

Thus, every sequence μ = μ(k) generates an operator M acting from T into T, which is called a multiplier.

Section 3

Saturation Classes

85

If A and B are some subsets of T, then we denote by M (A, B) the set of multipliers M acting from A into B. In particular, if A = B = Lp , p ∈ [1, ∞], and series (3.1) and (3.1 ) are the Fourier series of functions y(x) and z(x), then we set M (Lp , Lp ) = Mp ; if A = B = V and series (3.1) and (3.1 ) are the Fourier–Stieltjes series of functions y(x) and z(x), then we set M (V, V ) = MV . If y and z are related by equalities (3.1) and (3.1 ) and the series in these equalities are either Fourier or Fourier–Stieltjes series of functions y(x) and z(x), then we write z(x) = M y(x). In addition, if M ∈ Mp , then let M p be the norm of the multiplier M as an operator from Lp into Lp : M p = sup M y(x)p ; yp ≤1

if M ∈ MV , then let M V be the quantity π M V = sup

yV ≤1 −π

|dM y(x)|.

If, as n → ∞, a sequence of multipliers M = {M (n)} satisfies the relation M (n)X = O(1), where X can be either Lp , 1 ≤ p ≤ ∞, or V and O(1) is a quantity uniformly bounded in n, then this sequence is called uniformly bounded in X. We have the following theorem: Theorem 3.1. Let ϕ(k) and ψ(k) be arbitrary functions of a natural vari(n) able and let Λ be a matrix consisting of elements λk such that the sequence 1, k = 0, (n) μ(n) = μk = (3.2) (n) (1 − λk )ψ(k)/ϕ(n), k = 1, 2, . . . , n = 1, 2, . . . , generates a sequence of multipliers M μn uniformly bounded in X. If X = Lp , then, for p = ∞ and any f ∈ C ψ L∞ , we have f (x) − Un (f ; x; Λ) ≤ A, (3.3) ϕ(n) p where A does not depend on n. If p ∈ [1, ∞), then (3.3) holds for any f ∈ Lψ p. If X = V, then (3.3) holds for p = 1 for any f ∈ Lψ V.

86

Saturation of Linear Methods

Chapter 2

If a function ψ(k) is such that the series ∞

ψ(k) cos kx

(3.4)

k=1

is the Fourier series of some function ψ ∈ L1 and inequality (3.3) holds for some p ∈ [1, ∞] for any f ∈ Lψ Lp , then sequence (3.2) generates a sequence of multipliers M μn uniformly bounded in Lp . Moreover, if (3.3) holds for p = 1 for any f ∈ Lψ V, then the multipliers μ n are uniformly bounded in the space V. M Proof. Let f ∈ C ψ L∞ . Then the series ∞ k=1

1 Ak (f ; x) ψ(k)

(3.5)

is the Fourier series of some function y ∈ L∞ . If the operators M (n) = M μn are uniformly bounded in L∞ , then, for every fixed n, the series ∞ k=1

(n)

μk

1 Ak (f ; x) ψ(k)

(3.6)

is the Fourier series of some function z ∈ L∞ such that z∞ = M μn y∞ ≤ M μn ∞ y∞ = O(1).

(3.7)

However, S[z(x)] =

∞ k=1

(n)

μk

f (x) − Un (f ; x; Λ) 1 Ak (f ; x) = S[ ]. ψ(k) ϕ(n)

Therefore, z(x) = (f (x) − Un (f ; x))/ϕ(n) almost everywhere, which, together with (3.7), implies that relation (3.3) holds for p = ∞. Relation (3.3) for p ∈ [1, ∞) is proved by analogy with the case X = Lp . The same method also proves it in the case where p = 1 and X = V. Indeed, if f ∈ Lψ V, then series (3.5) is the Fourier–Stieltjes series of some function y ∈ V, series (3.6) is the Fourier–Stieltjes series of some function z ∈ V, and, since the multipliers M μn are uniformly bounded in the space V, we have zV = M μn yV ≤ M μn V yV = O(1).

(3.7 )

Section 3

Saturation Classes

87

In this case, S[dz] =

∞ k=1

(n) 1 μk Ak (f ; t) ψ(k)

Hence,

t z(t) = −π

t = S[d −π

f (x) − Un (f ; x; Λ) dx]. ϕ(n)

f (x) − Un (f ; x; Λ) dx + c ϕ(n)

almost everywhere, and, consequently, in view of (2.1), we have π

t |d

zV = −π

−π

Thus, by virtue of

f (x) − Un (f ; x; Λ) 1 |= ϕ(n) ϕ(n)

(3.7 ),

π |f (x) − Un (f ; x; Λ)|dx. −π

we have

f (x) − Un (f ; x; Λ)L1 = O(1)ϕ(n), which proves the first part of the theorem. 3.2. Now assume that series (3.4) is the Fourier series of some function ψ ∈ L1 and relation (3.3) holds for all f ∈ Lψ Lp and some p ∈ [1, ∞). Let us show that, in this case, the multipliers M μn induced by sequence (3.2) belong to Mp , i.e., for any g ∈ Lp , we have M μn g ∈ Lp . a0 (g) Assume that g ∈ Lp , g(x) = + g1 (x) and 2 1 fg (x) = (g1 ∗ Ψ)(x) = π

2π g1 (x − t)Ψ(t)dt. 0

Then, considering the Fourier series of the function fg (·), we conclude that fg ∈ Lψ Lp and g1 (x) = f ψ (x) almost everywhere, and, moreover, ak (fg ) = ψ(k)ak (f ψ ) = ψ(k)ak (g), bk (fg ) = ψ(k)bk (f ψ ) = ψ(k)bk (g), k = 1, 2, . . . Thus, by setting Rn (f ; x) = (fg (x) − Un (fg ; x; Λ))/ϕ(n), we obtain (n)

1 − λk (n) ak (Rn (fg ; ·) + a0 (g)/2) = ψ(k)ak (g) = μk ak (g), ϕ(n)

(3.8)

88

Saturation of Linear Methods

Chapter 2

and, by analogy, (n)

bk (Rn (fg ; ·)) = μk bk (g).

(3.8 )

The function fg (·) satisfies relation (3.3). Hence, Rn (fg ; ·) + a0 (g)/2p ≤ c(g), n = 1, 2, . . . , and, since g ∈ Lp , equalities (3.8) and (3.8 ) imply that, for all n ∈ N, the (n) sequence μk generates a multiplier M μn mapping Lp into Lp and such that, for any g ∈ Lp , we get M μn g(x)p = Rn (fg ; x) + a0 (g)/2p ≤ c(g),

(3.9)

where c(g) does not depend on n. Further, we use the following well-known result by Banach and Steinhauz: If a sequence of linear operators Tn defined on a Banach space E is bounded for all x ∈ E, then the sequence of their norms Tn is uniformly bounded in n. Applying this fact to relation (3.9), we conclude that the sequence M μn g is uniformly bounded, i.e., the multipliers M μn are uniformly bounded in Lp , p ∈ [1, ∞). 3.3. Finally, we consider the case where relation (3.3) holds for p = 1 for any f ∈ Lψ V. We show that the multiplier M μn is then uniformly bounded in the space V. Let g(·) be an arbitrary function from the set V and let 2π fg (x) = (Ψ ∗ dg)(x) =

Ψ(x + t)dg(t).

(3.10)

0

Then fg ∈ L1 and, moreover, ak (fg ) = ψ(k)ak (dg), bk (fg ) = ψ(k)bk (dg). Hence, ak (dg) = ak (fg )/ψ(k), bk (dg) = bk (fg )/ψ(k). Therefore, the series ∞ ak (fg ) bk (fg ) ( cos kx + sin kx) ψ(k) ψ(k) k=1

(3.11)

Section 3

Saturation Classes

89

is the Fourier–Stieltjes series of the function g. Thus, fg ∈ Lψ V. The rest of the proof is similar to the case studied above. We have t ak (d −π

(n)

1 − λk fg (x) − Un (fg ; x; Λ) ak (fg ) (n) dx) = ψ(k) = μk ak (dg), ϕ(n) ϕ(n) ψ(k) t

bk (d −π

fg (x) − Un (fg ; x; Λ) (n) dx) = μk bk (dg), k = 1, 2, . . . , ϕ(n)

and since relation (3.3) holds, we obtain t −π

fg (x) − Un (fg ; x; Λ) fg (x) − Un (fg ; x; Λ) dxV = L1 ≤ c(g). ϕ(n) ϕ(n) (n)

Therefore, the multiplier M μn induced by the sequence μk V into V so that, for all n, we get M μn gV ≤ c(g).

maps the space

(3.12)

Let V ∗ be the space of equivalent classes of g containing all functions from the space V that differ by a constant, i.e., V ∗ = {g : g = {F + c, F ∈ V, c = const}}, with the norm

π |dF (t)|.

gV0∗ =

(3.13)

−π

V0∗

= {F : F ∈ g ∈ V ∗ , F (0) = 0}. It is easy to see that the Furthermore, let space V0∗ is Banach with respect to norm (3.13), and since V0∗ ⊂ V, by virtue of (3.12), for any F ∈ V0∗ , we have M μn (F )V0∗ = M μn (F )V ≤ c(F ). By using the Banach–Steinhauz theorem, we see that the multipliers M μn are uniformly bounded in the space V0∗ and, therefore, in the space V. The theorem is proved.

90

Saturation of Linear Methods

Chapter 2

Corollary 3.1. Let Un (Λ) be a saturated linear method, let ϕ(n) = ϕΛ (n) be its order of saturation, and let ψ ∗ (k) be a sequence such that the sequence μ(n) =

(n) μk

1, =

(1 −

k = 0, (n) λk )ψ ∗ (k)/ϕ(n),

(3.2 )

k = 1, 2, . . . , n = 1, 2, . . . ,

induces multipliers M μn uniformly bounded in the space X. If X = Lp and p = ∞, then ∗ (3.14) C ψ L∞ ⊆ Φ(Λ)C ; if p ∈ [1, ∞), then

∗

Lψ Lp ⊆ Φ(Λ)Lp ;

(3.15)

Lψ V ⊆ Φ(Λ)L1 .

(3.16)

if X = V, then In particular, if ψ ∗ (k) = ψ(k), where ψ(k) is a sequence satisfying condition (2.6), then inclusions (3.14) and (3.16) as well as (3.15) for p ∈ (1, ∞) become equalities.

4.

Criterion for Uniform Boundedness of Multipliers

4.1. Theorem 2.1 indicates that, in order to prove that a given linear method Un (Λ) is saturated and determine its order of saturation, it suffices to show that conditions (2.6) are satisfied, which is straightforward in most cases. The same conditions also provide us with some preliminary information about saturation classes. More detailed information about the classes Φ(Λ)X is provided by Theorem 3.1, or more precisely, by its corollary. In order to use it, we have to prove the uniform boundedness of multipliers M μn generated by sequences (3.2 ). For this purpose, we need the following result: Theorem 4.1. The set of multipliers M μn generated by a sequence (n) μk , k = 1, 2, . . . , n = 1, 2, . . . , is uniformly bounded in C, L∞ , or V if and only if, for any N = 1, 2, . . . , we have Mn =

N −1

(n)

(1 − k/N )μk cos kx1 < A,

k=0

where A is a quantity uniformly bounded in n.

(4.1)

Section 4

Criterion for Uniform Boundedness of Multipliers

91

Proof. Consider the trigonometric series ∞

a0 (ak cos kx + bk sin kx), + 2

(4.2)

k=1

∞

(n)

μk cos kx,

(4.3)

k=1 ∞

a0 (n) μk (ak cos kx + bk sin kx) + 2

(4.4)

k=1

(n)

(n)

and let σN (x), lN (x), and σN (x) be their Fejér sums of degree N − 1. We now prove the necessity of the conditions indicated in the statement of the theorem. First, assume that X = C or X = L∞ and series (4.2) is the Fourier series of some function f ∈ X. If the multipliers M μn are uniformly bounded in X, then series (4.4) are the Fourier series of some functions ϕn ∈ X, and, moreover, for any n = 1, 2, . . . , we have ϕn X = M μn f X ≤ M μn X f X ≤ Kf X . (n)

Therefore, taking into account the fact that, for any ϕn ∈ X, σN (ϕn ; x)X ≤ ϕn X , we find (n)

(n)

σN (x)X = σN (ϕn ; x)X ≤ ϕn X ≤ Kf X . However, (n) σN (x)

1 = π

π

(4.5)

(n)

lN (t − x)f (t)dt. −π

Thus, in view of (4.5), for any f ∈ X, we have 1 π

π

(n)

lN (t − x)f (t)dtX ≤ Kf X ,

n = 1, 2, . . . .

(4.6)

−π

We now use the following fact, which follows from the Banach–Steinhauz theorem: If the sequence b un (x) = x(t)yn (t)dt a

92

Saturation of Linear Methods

Chapter 2

is bounded for every bounded (or even every continuous) function x(t), t ∈ [a, b], then yn 1 ≤ c, n = 1, 2, . . . . Applying this result to inequalities (4.6), we establish that (4.1) holds. We now show the necessity of the conditions of the theorem for the space ∞ cos kx is the Fourier–Stieltjes series of some V. It is known that the series k=0

function G ∈ V. Therefore, if the set M μn is uniformly bounded in V, then the ∞ (n) μk cos kx are the Fourier–Stieltjes series of some function Gn ∈ V, series k=0

and, moreover, Gn V = M μn GV ≤ M μn V GV < K. (n)

Therefore, the Fejér means σN (x) of the Fourier–Stieltjes series of functions Gn satisfy the following relation (here, FN (·) is the Fejér kernel of degree N − 1): (n) σN (x)1

=

N −1

(n)

(1 − k/N )μk cos kx1

k=0

π = −π

1 | π

π −π

1 FN (t − x)dGn (t)|dx ≤ FN (·)1 π

π |dGn (t)| ≤ K.

−π

This proves the necessity of the conditions of the theorem. Assume that (4.1) holds and (4.2) is the Fourier series of some function f ∈ C. First of all, note that, in this case, (4.4) is also the Fourier series of some continuous function. Indeed, Lemma 2.1 implies that series (4.3) is the Fourier– Stieltjes series of a function Φn ∈ V. Therefore, we have (n) σN (x)

1 = π

π σN (x − t)dΦn (t)

(4.7)

−π

and, for any p ∈ Z, we get (n) σN +p (x)

−

(n) σN (x)C

1 ≤ π

π |σN +p (x − t) − σN (x − t)||dΦn (t)|.

(4.8)

−π

In view of (1.3.14), the sums σN (x) converge uniformly to f (x). Therefore, (n) the last inequality implies that, as N → ∞, the sums σn (x) also converge

Section 4

Criterion for Uniform Boundedness of Multipliers

93

uniformly. Then Lemma 2.1 implies that series (4.4) is indeed the Fourier series of some continuous function fn (x). Thus, every multiplier M μn acts from C into C and its norm satisfies the relation M μn C = sup{Mn (f )C : f C ≤ 1} (n)

≤ sup{(fn (x) − σN (fn ; x)C (n)

+ σN (fn ; x)C : fn C ≤ 1}.

(4.9)

Since (n) σN (fn ; x)C

1 = π

π

(n)

f (x − t)lN (t)dtC −π (n)

≤ f C lN 1 ≤ cf C ,

(4.10)

by taking the limit as N → ∞ and using (4.9), we obtain M μn C ≤ A. Now assume that (4.1) holds and (4.2) is the Fourier series of a function f ∈ L∞ . Then, by virtue of Lemma 2.1, series (4.4) is also the Fourier series of some function fn ∈ L∞ , i.e., the multiplier M μn acts from L∞ into L∞ . It is known that, for any f ∈ L∞ , f ∞ = lim f p . Therefore, we have p→∞

(n)

(n)

M μn f ∞ = fn (x)∞ = lim lim fn (x) − σN (x) + σN (x)p p→∞ N →∞

(n)

(n)

≤ lim lim fn (x) − σN p + lim lim σN (x)p . p→∞ N →∞

p→∞ N →∞

(4.11)

The Fejér sums of a function ϕ ∈ Lp converge to ϕ(·) in the space Lp . Hence, (n)

lim fn (x) − σN (x)p = 0.

N →∞

(4.12)

On the other hand, by virtue of Minkowski inequality, for any N and p > 1, we have (n) σN (x)p

π = σN (fn ; x)p = ( −π

≤

1 | π

π

(n)

f (t)lN (x − t)|p dtdx)1/p −π

1 (n) (n) f p lN (x)1 ≤ 2f ∞ lN (x)1 . π

(4.13)

94

Saturation of Linear Methods

Chapter 2

In view of inequality (4.1), relations (4.11)–(4.13) imply that, for any f ∈ L∞ , we have M μn f ∞ ≤ 2Af ∞ , which is obviously equivalent to the uniform boundedness of the set of multipliers M μn in the set L∞ . 4.2. Finally, assume that (4.1) holds and (4.5) is the Fourier–Stieltjes series of some function F ∈ V. Then, for any N and n, we have (n) σN (x)

1 =| π

π

(n)

(n)

lN (x − t)dF (x)| ≤ lN 1 F V ≤ AF V . −π

Hence, Lemma 2.1 implies that series (4.4) is the Fourier–Stieltjes series of some function Fn ∈ V. Consider the functionals π s(x)dFn (x), s ∈ C, n = 1, 2, . . . .

Φn (s) = −π

We have π |Φn (s)| ≤ lim | N →∞

π s(x)dFn (x) −

−π

(n)

s(x)σN (x)dx|

−π

π + lim | N →∞

(n)

s(x)σN (x)dx|.

−π

However, by passing to the limit under the sign of Stieltjes integral, we obtain π lim

N →∞ −π

(n) s(x)σN (x)dx

π = lim

N →∞ −π

π = lim

N →∞ −π

KN (x − t)dFn (t)dt

1 ( π

π s(x)KN (x − t)dx)dFn (t) −π

π s(t)dFn (t).

= −π

Section 4

Criterion for Uniform Boundedness of Multipliers

95

Moreover, for any N, π |

(n)

(n)

s(x)σN (x)dx| ≤ sC σN 1 .

−π

Therefore, for any n and any s ∈ C, we have |Φn (s)| ≤ KsC , where K is a quantity uniformly bounded in n. Using the Banach–Steinhauz theorem, we conclude that the sequence of norms Φn C of functionals Φn (s) is bounded. In our case, we have Φn C = Fn V . Consequently, the sequences Fn V are uniformly bounded, and, hence, M μn V =

sup M μn F V =

F V ≤1

sup Fn V ≤ K.

F V 0. This method is saturated with an order of saturation ϕΛ (n) = n−s and Φ(Λ)C = C ψ L∞ , Φ(Λ)L1 = Lψ V, Φ(Λ)Lp = Lψ Lp , 1 < p < ∞, ψ(k) = k −s .

(5.1)

Indeed, since, in the case considered, (n)

1 − λk = k −s , k = 1, 2, . . . , n→∞ ϕΛ (n) lim

according to Theorem 2.1 this method is saturated with the order of saturation ϕΛ (n) = n−s , and, moreover, Φ(Λ)C ⊆ C ψ L∞ , Φ(Λ)Lp ⊆ Lψ Lp , and Φ(Λ)L ⊆ Lψ V. In order to show that the last inclusions are equalities, by virtue of Lemma 4.1 it suffices to prove that the sequences 1, k = 0, 1, . . . , n, (n) μk = s −s n k , k > n, (n)

are uniformly bounded and quasiconvex. It is clear that |μk | ≤ 1 and (n) lim μk = 0 as k → ∞. Let us prove that the second condition in (4.16) is also satisfied. We have ∞

(n)

(k + 1)|Δ2 μk | = n|Δ2 μn−1 | + (n + 1)|Δμ(n) n |+

k=0

(n)

|Δ2 μk |.

(5.2)

k=n+1 (n)

For k > n, the sequences μk ∞

∞

(n)

are convex, and, hence, (n)

(n)

(k + 1)|Δ2 μk | = (n + 2)Δμn+1 + μn+2

k=n+1

= ns (n + 2)((n + 1)−s − (n + 2)−s ) + ns (n + 2)−s ≤ s(n + 2)l−s (n + θ)s−1 + 1 ≤ 2s + 1, 0 ≤ θ ≤ 1.

Section 5

Saturation of Classical Linear Methods

99

Performing elementary estimates of the first two terms on the right-hand side of (5.2), we conclude that conditions (4.16) are satisfied. This proves equalities (5.1). As noted above, for s = 1 the Zygmund method is equivalent to the Fejér method. Consequently, the Fejér method is saturated, its order of saturation is ϕΛ (n) = n−1 , Φ(Λ)C = C ψ L∞ , Φ(Λ)L1 = Lψ V, and Φ(Λ)Lp = Lψ Lp for 1 < p < ∞. 5.3. Rogosinski method. Recall that, in this case, the means are defined by the matrix kπ n−1 (n) Λ = {λk }n−1 = cos . k=0 2n k=0 This method is saturated with the order of saturation ϕΛ (n) = n−2 , Φ(Λ)C = C ψ L∞ , Φ(Λ)L1 = Lψ V, and Φ(Λ)Lp = Lψ Lp , 1 ≤ p < ∞, where ψ(k) = k −2 . Indeed, kπ (n) 1 − cos 2 2 1 − λk 2n = π k . lim = lim n→∞ ϕΛ (n) n→∞ n−2 8 Then, by virtue of Theorem 2.1, this method is saturated and its order of saturation is n−2 . Moreover, Φ(Λ) ⊆ C ψ L∞ , Φ(Λ)L1 ⊆ Lψ V, and Φ(Λ)Lp ⊆ Lψ Lp , p ∈ (1, ∞). To prove that these inclusions are equalities, it suffices to show that the sequences ⎧ 1, k = 0, ⎪ ⎪ ⎪ ⎪ ⎪ kπ ⎨ 1 − cos (n) μk = 2n , k = 1, 2, . . . , n − 1, ⎪ −2 ⎪ k ⎪ ⎪ ⎪ ⎩ 2 −2 n k , k ≥ n, are uniformly bounded (which is obvious) and uniformly quasiconvex (which can easily be verified by direct computation). 5.4. De la Vallée Poussin method. In this case, we have ⎧ ⎪ 1, k = 0, 1, . . . , n − p, ⎪ ⎪ ⎪ ⎨ k−n+p (n) , k = n − p + 1, . . . , n, λk = 1 − ⎪ p+1 ⎪ ⎪ ⎪ ⎩0, k > n,

100

Saturation of Linear Methods

Chapter 2

where p = p(n) is a natural parameter, 1 ≤ p(n) ≤ n, which may depend on n. It is easy to show that if the limit lim p(n)/n = θ exists and θ ∈ [0, 1), n→∞ then the de la Vallée Poussin method is not saturated. The same conclusion is also true if θ = 1, but n − p(n) → ∞ as n → ∞. If θ = 1 and n − p(n) = cn , cn < c = 0, then the method is not necessarily saturated: relation (1.4) may be satisfied not only if f ≡ const, but also if f is an arbitrary trigonometric polynomial of degree less than cn . On the other hand, in this case, de la Vallée Poussin sums cannot approximate functions that differ from such subsets with rate better than n−1 , i.e., in this case, the sums Vn,p (f ; ·) approximate with the same quality as the Fejér sums, which is clear because Vn,n (f ; ·) ≡ σn (f ; ·). 5.5. Favard method. In this case, we have ⎧ ⎪ ⎨ kπ cot kπ , k = 1, 2, . . . , n − 1, (n) 2n λk = 2n ⎪ ⎩0, k ≥ n. This method is saturated with the order of saturation n−2 . Indeed, (n) 1 − λk π2 2 lim = k . n→∞ n−2 12 Therefore, according to Theorem 2.1, ϕΛ (n) = n−2 , Φ(Λ)C ⊆ C ψ L∞ , Φ(Λ)Lp ⊆ Lψ Lp , 1 < p < ∞, Φ(Λ)L1 ⊆ Lψ V, and ψ(k) = k −2 . Further, taking into account that the function (1 − x cot x)/x2 is convex on the interval (0, π/2), we conclude, as before, that the sequences ⎧ 1, k = 0, ⎪ ⎪ ⎪ ⎪ ⎨ kπ kπ n2 (n) μk = (1 − cot ) , k = 1, 2, . . . , n − 1, ⎪ 2n 2n k 2 ⎪ ⎪ ⎪ ⎩ 2 2 n /k , k ≥ n, are uniformly bounded and quasiconvex. Consequently, Φ(Λ)C = C ψ L∞ , Φ(Λ)Lp = Lψ Lp , 1 < p < ∞, Φ(Λ)L1 = Lψ V, and ψ(k) = k −2 .

3. CLASSES OF PERIODIC FUNCTIONS

In their famous works, D. Jackson and S. N. Bernstein proved the first theorems, which are now called direct and inverse theorems of approximation theory. After this, it became clear that the approximative properties of functions, i.e., their capability of being approximated by algebraic (or trigonometric) polynomials, are completely determined by some other properties, which can be called smoothness properties. Therefore, the investigation of these properties and the classification of functions according to their properties is one of the most important elements of approximation theory. In this chapter, we consider definitions and simplest properties of classes of functions traditional for approximation theory and introduce new classes of periodic functions.

1.

Sets of Summable Functions. Moduli of Continuity

1.1. Let L(0, 2π) be the set of 2π-periodic functions summable on (0, 2π). The main functional spaces considered here are the following subsets of L(0, 2π): the space C of 2π-periodic functions f (t) continuous on the entire axis with the norm f C = max |f (t)|; t

the space M of 2π-periodic essentially bounded functions f (t) with the norm f M = ess sup |f (t)|; t

the space Lp , 1 ≤ p < ∞, of 2π-periodic functions f (t) summable to the pth power on (0, 2π) with the norm 101

102

Classes of Periodic Functions

f Lp

Chapter 3

2π = ( |f (t)|p dt)1/p . 0

In what follows, we write f p instead of f Lp and f ∞ instead of df

df

f M , bearing in mind that f ∞ = f L∞ = f M , i.e., the space M is formally identified with the space L∞ . It is clear that, for any p, p , 1 < p < p < ∞, the following imbedding holds: C ⊂ M = L∞ ⊂ Lp ⊂ Lp ⊂ L1 .

(1.1)

1.2. Let X denote one of the spaces C or Lp , 1 ≤ p ≤ ∞. The simplest classification of functions f (·) from X is as follows: Let ρSX denote a ball of radius ρ, ρ > 0, in X, ρSX = {f : f X ≤ ρ}.

(1.2)

Then every value of ρ is associated with a certain set of functions f (·) belonging to the ball ρSX . Thus, the space X is decomposed into the sets ρSX , which are called classes. It is clear that, for any ρ and ρ , 0 < ρ < ρ , we have ρSX ⊂ ρ SX .

(1.3)

For ρ = 1, we denote the ball ρSX by SX and write Sp instead of SLp . 1.3. A more detailed classification of functions can be carried out by using moduli of continuity. The modulus of continuity of a function f (x) continuous on a segment [a, b], f ∈ C[a, b], is a function ω(t) = ω(f, t) defined for t ∈ [0, b − a] by the relation ω(t) = ω(f, t) = sup

max |f (x + h) − f (x)|

0≤h≤t a≤x≤b−h

= sup{|f (x ) − f (x )| : |x − x | ≤ t; x , x ∈ [a, b]}.

(1.4)

This definition yields the following basic properties of the modulus of continuity for f ∈ C[a, b] : (i) ω(0) = 0;

Section 1

Sets of Summable Functions. Moduli of Continuity

103

(ii) the function ω(t) does not decrease on the segment t ∈ [0, b − a]; (iii) the function ω(t) is semiadditive, i.e., ω(t1 + t2 ) ≤ ω(t1 ) + ω(t2 ), (t1 + t2 ) ∈ [0, b − a];

(1.5)

(iv) the function ω(t) is continuous on [0, b − a]. Indeed, if we denote by · the norm in the space C[α, β] of functions ϕ(x) continuous on the segment [α, β], i.e., ϕ = max |ϕ(x)|, x∈[α,β]

(1.6)

then equality (1.4) can be rewritten as follows: ω(t) = ω(f, t) = sup f (x + h) − f (x), α = a, β = b − h.

(1.7)

0≤h≤t

Properties (i) and (ii) are now obvious. Property (iii) can be established, e.g., in the following way: ω(t1 + t2 ) =

sup 0≤h≤t1 +t2

f (x + h) − f (x)

= sup{f (x + h1 + h2 ) − f (x) : h1 ∈ [0, t1 ], h2 ∈ [0, t2 ]} ≤

sup f (x + h1 + h2 ) − f (x + h2 )

0≤h1 ≤t1

+

sup f (x + h2 ) − f (x)

0≤h2 ≤t2

= ω(t1 ) + ω(t2 ). To prove property (iv), we first note that if 0 ≤ t1 ≤ t2 ≤ b − a, then (1.5) yields ω(t2 ) = ω(t2 − t1 + t1 ) ≤ ω(t2 − t1 ) + ω(t1 ), i.e., ω(t2 ) − ω(t1 ) ≤ ω(t2 − t1 ).

(1.8)

Hence, for any t, t + Δt ∈ [0, b − a], we have ω(t + Δt) − ω(t) ≤ ω(|Δt|).

(1.8 )

104

Classes of Periodic Functions

Chapter 3

However, by the uniform continuity of functions continuous on a closed segment, we get lim ω(t) = 0. (1.9) t→+0

Thus, the function ω(t) is continuous from the right at the point t = 0. This and (1.8 ) prove the continuity of ω(t) for all t ∈ [0, b − a]. 1.4. It follows from the property of semiadditivity of the modulus of continuity ω(t) that, for any n ∈ N, ω(nt) ≤ nω(t),

(1.10)

and, for arbitrary λ > 0, (λ + 1)t ∈ [0, b − a], ω(λt) ≤ (λ + 1)ω(t).

(1.10 )

Indeed, for n = 1, inequality (1.10) is obvious. Assume that it is valid for n = k, k > 1. Then, according to (1.5), we have ω((k + 1)t) ≤ ω(kt) + ω(t) ≤ (k + 1)ω(t), i.e., in this case, (1.10) also holds for n = k + 1. Hence, (1.10) is valid for any natural n. If λ is an arbitrary positive number, then, denoting by [λ] its integer part and using the monotonicity property of the function ω(t) and relation (1.10), we get ω(λt) ≤ ω(([λ] + 1)t) ≤ ([λ] + 1)ω(t) ≤ (λ + 1)ω(t).

1.5. Note that if a function ϕ(t) possesses properties (i)–(iv) on the segment [0, c] (with b − a = c ), then its modulus of continuity ω(ϕ, t) coincides with it, i.e., ω(ϕ, t) ≡ ϕ(t). (1.11) Indeed, for any t ∈ [0, c], we have ϕ(t) = ϕ(t) − ϕ(0) ≤ ω(ϕ, t).

(1.12)

On the other hand, parallel with conditions (i)–(iv), the function ϕ(t) also satisfies inequality (1.8). Therefore, ω(ϕ, t) = sup ϕ(x + h) − ϕ(x) ≤ sup ϕ(h) = ϕ(t). 0≤h≤t

0≤h≤t

(1.13)

Section 1

Sets of Summable Functions. Moduli of Continuity

105

Relations (1.12) and (1.13) are equivalent to (1.11). Thus, properties (i)–(iv) completely characterize the function ω(t) as a modulus of continuity. Therefore, in what follows, any function defined on t ∈ [0, c] and having properties (i)–(iv) is called a modulus of continuity. As an example, we consider a function ϕ(t) = Ktα , where K is a fixed positive constant and 0 < α ≤ 1. This function is a modulus of continuity for any c > 0. Properties (i), (ii), and (iv) are obvious for it and property (iii) can easily be verified. In view of (1.10 ), the following inequality holds for any modulus of continuity ω(t) which is not identically equal to zero: ω(b − a) = ω(t

b−a b−a t b−a )≤ (1 + )ω(t) ≤ 2 ω(t), t t b−a t

i.e., we always have ω(t) ≥

ω(b − a) · t. 2(b − a)

This means that none of moduli of continuity ω(t) ≡ 0 can be an infinitesimal value of order higher than one in t when t → 0. In particular, the function Ktα with α > 1 cannot be a modulus of continuity for any K > 0. Note that a function ω(t) satisfies condition (iii) for t ∈ [0, c] whenever the ratio ω(t)/t does not increase. Indeed, in this case, for all t1 , t2 ∈ [0, c] such that (t1 + t2 ) ∈ [0, c], we have ω(t1 + t2 ) = t1 ≤ t1

ω(t1 + t2 ) ω(t1 + t2 ) + t2 t 1 + t2 t1 + t2 ω(t1 ) ω(t2 ) + t2 = ω(t1 ) + ω(t2 ). t1 t2

If a function ω(t) is convex on [0, c], i.e., for any t1 , t2 ∈ [0, c], ω(

t 1 + t2 1 ) ≥ (ω(t1 ) + ω(t2 )), 2 2

then the ratio ω(t)/t does not increase (see, e.g., Natanson [1, p. 476]). Therefore, every function ω(t) convex for t ∈ [0, c], ω(0) = 0, is a modulus of continuity.

106

Classes of Periodic Functions

Chapter 3

It is not difficult to construct an example of a modulus of continuity that is not a convex function. Assume that h ∈ (0, 1), c ≥ 2, and ⎧ t/h, t ∈ [0, h], ⎪ ⎪ ⎪ ⎪ ⎪ ⎨1, t ∈ [h, 1], (1.14) ω(t) = ⎪ (t − 1 + h)/h, t ∈ [1, 1 + h], ⎪ ⎪ ⎪ ⎪ ⎩ 2, t ∈ [1 + h, c]. One can easily check that this function is not convex. Nevertheless, it satisfies conditions (i)–(iv), i.e., it is a modulus of continuity. 1.6. Convex moduli of continuity play a special role in what follows. In this connection, the following lemma on a convex majorant is often useful: Lemma 1.1. For any modulus of continuity ω = ω(t) ≡ 0 defined on [0, c], there exists a convex modulus of continuity ω∗ (t) such that, for any t ∈ [0, c], ω(t) ≤ ω∗ (t) < 2ω(t),

(1.15)

and, moreover, the constant 2 on the right–hand side of this inequality cannot be reduced. Proof. As ω∗ (t), we take a function whose graph bounds from above the smallest convex domain containing the curvilinear trapezoid {0 ≤ t ≤ c, 0 ≤ y ≤ ω(t)}. It is clear that ω∗ (0) = 0 and ω∗ (t) is a function continuous and convex on [0, c], i.e., ω∗ (t) is a convex modulus of continuity. The inequality ω(t) ≤ ω∗ (t) follows immediately from the definition. Let us show that ω∗ (t) ≤ 2ω(t) for any t ∈ [0, c]. For this purpose, clearly, it suffices to examine only the points t for which ω(t ) < ω∗ (t ). However, if t is a point of this sort, then there exist points t1 and t2 such that ω∗ (t1 ) = ω(t1 ) and ω∗ (t2 ) = ω(t2 ), t ∈ (t1 , t2 ), and ω∗ (t) is a linear function for any t ∈ [t1 , t2 ], ω∗ (t) = ω(t1 ) + Consequently,

ω(t2 ) − ω(t1 ) t2 − t t − t1 (t − t1 ) = ω(t1 ) + ω(t2 ). t2 − t1 t2 − t1 t2 − t1

ω∗ (t ) t2 − t ω(t1 ) t − t1 ω(t2 ) = + . ω(t ) t2 − t1 ω(t ) t2 − t1 ω(t )

(1.16)

Section 1

Sets of Summable Functions. Moduli of Continuity

107

However, t1 < t and, hence, ω(t1 )/ω(t ) ≤ 1. By virtue of (1.10), we have ω(t2 ) = ω(

t2 t2 t ) ≤ ( + 1)ω(t ), t t

and, therefore, ω(t2 )/ω(t ) ≤ (t2 + t )/t . By using these estimates and (1.16), we get ω∗ (t ) t − t1 t2 + t t2 t1 t2 t2 − t + · =1+ − · < 2. ≤ ω(t ) t2 − t 1 t2 − t 1 t t 2 − t1 t 2 − t1 t Inequalities (1.15) are proved. If we take the modulus of continuity (1.14) as ω(t) and construct its least convex majorant ω∗ (t), then, for t = 1, we get ω∗ (1) =2−h ω(1)

∀h ∈ (0, 1).

Since the number h is arbitrary, this equality means that, in the general case, the constant 2 in (1.15) cannot be reduced. 1.7. We now formulate a statement that characterizes convex functions (see, e.g., Natanson [1, p. 477]). Proposition 1.1. If f (x) is a function convex on [a, b], then it can be represented in the form x f (x) = ϕ(a) + ϕ(t)dt, a

where ϕ(t) is a function nonincreasing on [a, b]. Conversely, any function f (x) that can be represented in this form is convex on [a, b]. Hence, any convex modulus of continuity ω(t) can be represented as t ω(t) =

ϕ(τ )dτ, 0

where the function ϕ(τ ) is nonincreasing and ω (t) = ϕ(t) almost everywhere. In defining the modulus of continuity of a function f (x) and establishing its properties, we have used the finiteness of a segment [a, b] only when proving property (iv). In that case, we have used, in fact, only the uniform continuity of a function f (x). Therefore, the definition of the modulus of continuity ω(f, t)

108

Classes of Periodic Functions

Chapter 3

and all its properties remain valid for an infinite interval, provided that a function f (x) is uniformly continuous on it (this is true, e.g., for a 2π-periodic function continuous on a period). 1.8. According to property (ii), the modulus of continuity of any function f (x) continuous on [a, b] does not decrease on [0, b − a]. In the general case, one cannot characterize its growth. However, if a function f (x) is periodic, then one can obtain additional information, namely, if the period of f (x) is 2π, then ω(f, t) = ω(f, π) for t ≥ π. Indeed, let x and x be arbitrary points of the real axis. Clearly, there always exist an integer k and a point x0 the distance between which and x is at most π such that x = x0 + 2kπ. Therefore, for t ≥ π, we have ω(t) =

sup |f (x) − f (x )| =

|x−x |≤t

sup |x−x0 |≤π

|f (x) − f (x0 )| = ω(π).

Thus, for any f ∈ C, the modulus of continuity has the form ω(t), 0 ≤ t ≤ π, ω(f, t) = ω(π), t ≥ π,

(1.17)

where ω(t) is the modulus of continuity defined in Subsection 1.5. This implies, in particular, that, in studying moduli of continuity of 2πperiodic functions, it suffices to consider these functions on any segment of length 3π.

2.

Classes Hω [a, b] and Hω

2.1. Let C[a, b] be the set of functions f (·) continuous on [a, b] and let ω = ω(t) be an arbitrary fixed modulus of continuity defined on [0, b − a]. We say that a function f ∈ C[a, b] belongs to the class Hω [a, b] (f ∈ Hω [a, b]) if its modulus of continuity ω(f, t) satisfies the condition ω(f, t) ≤ ω(t) ∀t ∈ [0, b − a].

(2.1)

It is clear that, with this definition, any function f0 ∈ C[a, b] belongs to at least one of the classes Hω [a, b]. To verify this it suffices to note that we can take the modulus of continuity of a function f0 (·) itself as a modulus of continuity ω(t) that determines a class. As a result, we obtain the inclusion f0 ∈ Hω0 [a, b], where ω0 (t) = ω(f0 , t).

Classes Hω [a, b] and Hω

Section 2

109

Thus, the entire set C[a, b] can be decomposed into classes Hω [a, b]. In this case, obviously, the imbedding Hω(1) [a, b] ⊆ Hω(2) [a, b]

(2.2)

takes place, provided that ω (1) (t) ≤ ω (2) (t) for any t ∈ [0, b − a]. If f ∈ Hω [a, b], then ∀t1 , t2 ∈ [a, b] |f (t1 ) − f (t2 )| ≤ ω(|t1 − t2 |).

(2.3)

The converse statement is also true, namely, if inequality (2.3) holds for any t1 , t2 ∈ [a, b], then f ∈ Hω [a, b]. Indeed, ω(f, t) =

sup |t1 −t2 |≤t

|f (t1 ) − f (t2 )| ≤

sup |t1 −t2 | 0 and any collection of disjoint segments [ak , bk ] ⊂ [a, b] with nk=1 (bk − ak ) < ε/K, we have n

|f (bk ) − f (ak )| ≤

k=1

n

K(bk − ak ) < ε.

k=1

This means that, in fact, the function f (x) is absolutely continuous and, hence, almost everywhere on [a, b], it has a finite derivative f (x) satisfying the following inequality (at points where it exists): f (x + Δx) − f (x) KΔx |f (x)| = lim = K. ≤ lim Δx→0 Δx→0 Δx Δx On the other hand, if f (x) is absolutely continuous on [a, b] and |f (x)| ≤ K almost everywhere, then, for any x , x ∈ [a, b], we have x |f (x ) − f (x )| = f (t)dt ≤ K · |x − x |, x

110

Classes of Periodic Functions

Chapter 3

i.e., f ∈ KH 1 [a, b]. It is clear that classes KH α [a, b] defined analogously for α > 1 contain only constants. 2.3. By using moduli of continuity, we can similarly decompose the space C of continuous 2π-periodic functions f (·) into classes. Namely, if ω(t) is a modulus of continuity defined for all t ≥ 0, then Hω (or KH α , respectively) denotes the class of functions for which ω(f, t) ≤ ω(t) (ω(f, t) ≤ Ktα ). In addition, according to Subsection 1.8, it suffices to define a modulus of continuity ω(t) only on the segment [0, π]. It is clear that an imbedding of the type (2.2) is valid for classes Hω and the statement of Subsection 2.2 holds for the class KH 1 (this class coincides with the class of 2π-periodic absolutely continuous functions f (x) for which |f (x )| ≤ K almost everywhere (this class is also denoted by KW 1 )).

3.

Moduli of Continuity in Spaces Lp . Classes Hωp

3.1. Every function f ∈ Lp (a, b), 1 ≤ p < ∞, is continuous in the metric of this space in the sense of the following assertion: Proposition 3.1. Let f ∈ Lp (a, b), 1 ≤ p < ∞. Then lim f (· + h) − f (·)Lp (a,b) = 0.

h→0

(3.1)

Proof. Let f (·) be an arbitrary function from Lp (a, b). As is well known, the space C(a, b) is everywhere dense in Lp (a, b). Therefore, for any ε > 0, there exists ϕ ∈ C such that f (·) − ϕ(·)p < ε. Equality (3.1) is obviously satisfied for the function ϕ(·). Hence, we can find δ = δ(ε) such that ϕ(· + h) − ϕ(·)Lp (a,b) < ε for |h| < δ. Consequently, for |h| < δ, we get f (· + h) − f (·)Lp (a,b) ≤ f (· + h) − ϕ(· + h)Lp (a,b) + ϕ(· + h) − ϕ(·)Lp (a,b) + ϕ(·) − f (·)Lp (a,b) < 3ε. Proposition 3.1 allows one to introduce the notion of a modulus of continuity by analogy with a modulus of continuity in the space C.

Section 3

Moduli of Continuity in Spaces Lp . Classes Hωp

111

The modulus of continuity of a function f ∈ Lp , 1 ≤ p < ∞, in the space Lp is the function ωp (t) = ωp (f, t) = sup f (· + h) − f (·)p , 0 ≤ t < ∞. |h|≤t

(3.2)

The function ωp (t) possesses all the basic properties characterizing moduli of continuity. Indeed, relation (3.2) implies that ωp (0) = 0 and ωp (t) does not decrease on the interval 0 ≤ t < ∞. In order to verify the inequality ωp (t1 + t2 ) ≤ ωp (t1 ) + ωp (t2 ), t1 , t2 > 0,

(3.3)

it suffices to use the arguments presented in Subsection 1.3 in the proof of property (iii), assuming that · = · p . The continuity of the function ωp (t) from the right at the point t = 0 follows from (3.1). If 0 < t < ∞, then, for any h such that t + h ≥ 0, we have |ωp (t + h) − ωp (t)| ≤ ωp (|h|) by virtue of (3.3). Therefore, lim |ωp (t + h) − ωp (t)| ≤ lim ωp (t) = 0.

|h|→0

t→+0

It is clear that the functions ωp (f, t), f ∈ Lp , possess all the other properties of ω(ϕ, t) with ϕ ∈ C. In particular, for any n ∈ N, ωp (nt) ≤ nωp (t) and, for any λ > 0, ωp (λt) ≤ (λ + 1)ωp (t). An analog of equality (1.17) also takes place, namely, for any f ∈ Lp , 1 ≤ p < ∞, the modulus of continuity ωp (f, t) has the form ω(t), 0 ≤ t ≤ π, (3.4) ωp (f, t) = ω(π), t ≥ π, where ω(t) is a modulus of continuity. 3.2. By analogy with the spaces C, the spaces Lp , 1 ≤ p < ∞, can be decomposed into the classes Hωp as follows: Let ω(t) be a modulus of continuity defined for all t ≥ 0. Then the class Hωp contains the functions f ∈ Lp for which ωp (f, t) = ω(t). In this case, by virtue of (3.4), it suffices to define the modulus of continuity ω(t) only on the segment 0 ≤ t ≤ π. It is clear that the class Hωp is completely characterized by the fact that, for all f ∈ Hωp , the following analog of inequality (2.3) holds: f (· + t1 ) − f (· + t2 )p ≤ ωp (|t1 − t2 |)

∀t1 , t2 ∈ R.

(3.5)

112

Classes of Periodic Functions

Chapter 3

Indeed, since functions f ∈ Hωp , are periodic, we have f (· + t1 ) − f (· + t2 )p = f (· + |t1 − t2 |) − f (·)p

(3.6)

for any t1 , t2 ∈ R. Therefore, if f ∈ Hωp , then (3.5) follows immediately from (3.6). Conversely, if (3.5) holds, then, according to (3.2) and (3.6), we have ωp (f, t) = =

sup |t1 −t2 |≤t

sup |t1 −t2 |≤t

f (· + |t1 − t2 |) − f (·)p f (· + t1 ) − f (· + t2 )p ≤

sup |t1 −t2 |≤t

ωp (|t1 − t2 |) = ωp (t).

It is also clear that, for α > 1, the inequality ωp (f, t) ≤ Ktα can be satisfied only for functions that are different from constants on a period only on a set of measure zero. (1) (2) The imbeddings Hω(1) ⊆ Hω(2) with ωp (t) ≤ ωp (t) for any t ≥ 0 are p p also obvious.

4.

Classes of Differentiable Functions

4.1. It has been shown in Subsection 1.5 that, for any modulus of continuity ω(t), the inequality ω(t) ≥ Kt, where K is a constant, must hold. Therefore, according to relation (2.2), the narrowest set that can be extracted from L(0, 2π) by using moduli of continuity is the set KH 1 , i.e., the set of absolutely continuous functions. A subsequent decomposition of the set A of absolutely continuous functions into classes is carried out by the existence of a certain number of their derivatives as follows: Let Ar denote the set of functions f ∈ L(0, 2π) that have absolutely continuous derivatives up to the (r − 1)th order (r ∈ N ) and let N denote some other class of functions from L(0, 2π), e.g., one of the classes that have been considered above. In this case, if f ∈ Ar for some r ∈ N and, in addition, f (r) ∈ N, then we say that f (·) belongs to the class A(r) N. For example, Ar SM is the class of 2π-periodic functions whose rth derivatives do not exceed one in modulus almost everywhere. This class is usually denoted by W r . Furthermore, Ar Hω is the class of 2π-periodic functions whose rth derivatives belong to the class Hω . This class is usually denoted by W r Hω . Below, following the tradition, we write W r instead of Ar in combinations with specific classes N. For r = 0, we assume that A0 N = N by definition.

Section 4

Classes of Differentiable Functions

113

4.2. Let L0 be the set of functions f ∈ L(0, 2π) with mean value zero on a period: π f (t)dt = 0. −π

Let us prove the following statement: Proposition 4.1. The set Ar , r ∈ N, coincides with the set A¯r of functions f ∈ L(0, 2π) that can be represented by the equality π

∞

a0 (f ) 1 f (x) = + 2 πk r k=1

ϕ(x − t) cos(kt − −π

where 1 a0 (f ) = π and ϕ(·) is a function from

rπ )dt, x ∈ R, 2

(4.1)

π f (t)dt −π

L0 .

Proof. The imbedding Ar ⊆ A¯r is implied by the following reasoning: Assume that f ∈ Ar and ∞

∞

k=1

k=0

a0 (f ) df S[f ] = (ak (f ) cos kx + bk (f ) sin kx) = Ak (f ; x) + 2

(4.2)

is the Fourier series of this function. In the case under consideration, this series is obviously convergent for all x ∈ R, i.e., S[f ] = f (x). Therefore, carrying out integration by parts in the representations of Fourier coefficients 1 ak = ak (f ) = π

π −π

1 f (t) cos ktdt, bk = bk (f ) = π

π f (t) sin ktdt,

(4.3)

rπ )dt. 2

(4.4)

−π

we get ∞

a0 (f ) 1 f (x) = + 2 πk r k=1

π f (r) (x − t) cos(kt − −π

Since f (r) ∈ L0 for any f ∈ A(r) , we have Ar ⊆ A¯r . To prove the imbedding A¯r ⊆ Ar , we use the following two classical results:

114

Classes of Periodic Functions

Chapter 3

Theorem 4.1. If a function f ∈ C has a bounded variation on a period, then the Fourier series S[f ] converges uniformly to f (x). Theorem 4.2. Every Fourier series S[f ] can be integrated within any limits, no matter whether it is convergent or not. This means that the sum of the series of integrals of terms of a Fourier series is always equal to the integral of a function f (·). The proof of Theorem 4.1 is given in almost all manuals on Fourier series. The proof of Theorem 4.2 can also be found in textbooks, but we present it here because the relations obtained in the course of the proof are used in what follows. Let x a0 (f ) F (x) = [f (t) − ]dt. 2 0

The function F (x) is 2π-periodic and absolutely continuous, and, hence, it has a bounded variation on a period. Therefore, according to Theorem 4.1, the Fourier series of this function converges to it uniformly, i.e., ∞

a0 (F ) (ak (F ) cos kx + bk (F ) sin kx). + F (x) = 2 k=1

However, 1 ak (F ) = π

2π F (t) cos ktdt 0

1 1 sin kt 2π = [F (t) ]0 − π k kπ

2π a0 (f ) [f (t) − ] sin ktdt 2 0

=−

1 kπ

2π f (t) sin ktdt = −

bk (f ) . k

0

Similarly, we obtain bk (F ) = ak (f )/k. Thus, ∞

a0 (F ) 1 F (x) = + (ak (f ) sin kx − bk (f ) cos kx). 2 k k=1

Section 4

Classes of Differentiable Functions

By setting x = 0, we find that a0 (F )/2 = we get F (x) =

∞

115

bk (f )/k. Hence, for any x ∈ R,

k=1

∞ 1 (ak (f ) sin kx + bk (f )(1 − cos kx)). k k=1

Theorem 4.2 is proved. Let us proceed with proving Proposition 4.1. We take the function ϕ(·) from decomposition (4.1) as the function f (·) in Theorem 4.2. We get x F (x) =

∞

a0 (F ) 1 ϕ(t)dt = + (ak (ϕ) sin kx − bk (ϕ) cos kx) 2 k k=1

0

π

∞

a0 (F ) 1 = + 2 kπ k=1

−π

π dt. ϕ(x − t) cos kt − 2

(4.5)

The series in this expression converge to F (x) uniformly. By comparing relations (4.1) and (4.5), we conclude that, for r = 1, the function f (x) defined by (4.1) is absolutely continuous, i.e., A¯1 ⊆ A1 . If r > 1, then, by virtue of the uniform convergence of the series in (4.5), the function f (x) in (4.1) can be differentiated r − 1 times. Moreover, f (r−1) ∈ A1 , which proves the imbedding A¯r ⊆ Ar for all r ∈ N. 4.3. Thus, we have Ar = A¯r . This implies that, on the set Ar (or A¯r ), the operator of differentiation Dr of order r ∈ N can also be defined as an operator that associated every function f ∈ Ar with a function ϕ ∈ L0 . This approach allows one to extend the operation of differentiation to fractional values of r, r > 0, as follows: The series in (4.1) has a sense for any r > 0 and ϕ ∈ L0 . For fixed r > 0, we denote the set of functions f ∈ L(0, 2π) for which S[f ] coincides with the right-hand side of (4.1) by Jrr and call it the set of rth periodic integrals of functions ϕ ∈ L0 . On the set Jrr , we define Drr as an operator that associates a function f ∈ Jrr with a function ϕ ∈ L0 by the relation ∞

a0 (f ) 1 S[f ] = + 2 πk r k=0

π ϕ(x − t) · cos(kt − −π

rπ )dt. 2

In this way, we can define the operator of differentiation Drr for any r > 0.

(4.6)

116

Classes of Periodic Functions

Chapter 3

In this case, the function ϕ(·) is called the rth Weyl derivative of the function f (·) and the following notation is used: ϕ(·) = Drr (f ) = frr (·). 4.4. Besides functions f (·), the set Jrr contains all functions that differ from f (·) on an arbitrary set of measure zero. Therefore, generally speaking, we have Jrr ⊃ Ar for r ∈ N. In what follows, if the series in (4.6) is convergent, we take its sum as a function f (·) regarded as an element of the set Jrr . Then for r ∈ N, we obviously have Jrr = Ar , Drr = Dr , and, hence, frr (·) = f r (·). 4.5. Classes of functions differentiable in the sense of Weyl can be defined by analogy with the classes of functions differentiable in the ordinary sense. If f ∈ Jrr and, moreover, frr ∈ N, then we say that a function f belongs to the class Jrr N. If N is a specific class of functions, then we set Jrr N = Wrr N. For example, Wrr Hω is the class of functions f (·) for which frr ∈ Hω , Wrr SM is the class of functions f (·) for which the inequality |frr (·)| ≤ 1 holds almost everywhere, etc. The last class is also denoted by Wrr . In the notation Drr , Wrr , Jrr , and, frr , the subscript r is omitted if this does not lead to misunderstanding. It is clear that, for r ∈ N, the classes Wrr N coincide with the classes W r N introduced in Subsection 4.1. Therefore, if f ∈ Jrr for r ≥ 1, then this function is at least absolutely continuous. However, if r ∈ (0, 1), then the fact that a function f (·) belongs to Jrr does not guarantee even its ordinary continuity. For more details concerning functions f ∈ Jrr , see Sections 7 and 8.

5.

Conjugate Functions and Their Classes 5.1. Let f ∈ L(0, 2π) and let, as usual, ∞

S[f ] =

a0 (f ) (ak (f ) cos kx + bk (f ) sin kx) + 2

(5.1)

k=1

be its Fourier series. Consider the power series ∞

a0 (f ) (ak (f ) + ibk (f ))z k + 2

(5.2)

k=1

on the unit circle z = eix , the coefficients ak (f ) and bk (f ) of which are the coefficients of series (5.1). Then series (5.1) is the real part of series (5.2). The

Section 5

Conjugate Functions and Their Classes

series ˜ ]= S[f

∞

(ak (f ) sin kx − bk (f ) cos kx)

117

(5.3)

k=1

is the imaginary part of series (5.2); it is called a series conjugate to series (5.1). ˜ ]. Therefore, the series conjugate to the Note that there is no free term in S[f ˜ series S[f ] is the series S[f ] without free term. ˜ ] is called trigonometriThe function f˜ ∈ L(0, 2π) for which S[f˜] = S[f cally conjugate, or simply conjugate, to f (·). It can be shown that the functions f (·) and f˜(·) are connected by the equality 1 f˜(x) = − 2π

π f (x + t) cot −π

t dt, 2

(5.4)

in which the integral is understood in the sense of its principal value, i.e., π −π

t f (x + t) cot dt = lim ( ε→0 2

−ε π t + )f (x + t) cot dt. 2

−π

(5.5)

ε

5.2. The problem of the existence of the function f˜(·) (or, which is the same, ˜ of the limit in (5.5)), the correlation between the series S[f ], S[t], and S[f˜], and other problems close to this topic are well described in the fundamental books of Bari [1] and Zygmund [5, 6]. Let us present some statements from these books which will be necessary below. Theorem 5.1. If the limit in (5.5) is finite, then f˜(x) exists and the series ˜ ] converges to f˜(x). S[f Theorem 5.2. Suppose that f ∈ Hω and 1

ω(t) dt = A < ∞. t

(5.6)

0

˜ ] converge uniformly to f (x) and f˜(x), respecThen the series S[f ] and S[f tively. Moreover, the function f (x) is continuous and the equality ˜ ] S[f˜] = S[f holds. Equality (5.7) remains valid if f (·) and f˜(·) belong to L(0, 2π).

(5.7)

118

Classes of Periodic Functions

Chapter 3

The function f˜(·) conjugate to a continuous function f (·) is not necessarily continuous. However, if the majorant ω(t) of its modulus of continuity ω(f, t) satisfies condition (5.6), then, according to Theorem 5.2, the function f˜(·) is continuous. Condition (5.6) is, clearly, satisfied if ω(t) = Ktα , 0 < α < 1. In this case, a stronger statement is valid for the function f˜(·). Theorem 5.3 (Privalov [1]). If f ∈ KH α , 0 < α < 1, i.e., if |f (x) − f (x )| ≤ K|x − x |α

∀x, x ∈ R,

(5.8)

where K is a fixed constant, then f˜(x) also satisfies condition (5.8), probably, with a different constant K. This statement fails for α = 0 and α = 1; if α = 1, then ω(f˜, t) = O(t ln 1/t) as t → 0. 5.3. Classes of conjugate functions are introduced according to the following ˜ denotes the class principle: If M is a class of 2π-periodic functions, then M ˜ ˜ ω is the of functions f (·) conjugate to functions from the class M. Thus, H r H is the set of set of functions conjugate to functions from the class Hω , W ω ˜α functions conjugate to functions from the class W r Hω , etc. In particular, H is the class of functions conjugate to functions f ∈ H α . Note that Theorem 5.3 ˜ α , 0 < α < 1, then this function also belongs to the class implies that if f ∈ H KH α although the constant K is not necessarily equal to one in this case. In the ˜ ⊂ M is not true general case, as follows from Subsection 5.2, the imbedding M even if we use a homothet KM with an arbitrary coefficient K instead of M. 5.4. If f ∈ Jrr , then the Fourier series of this function has the form (4.6). In this case, if f˜ ∈ L(0, 2π), then, by virtue of equalities (5.7) and (5.3), we have π ∞ r+1 1 ˜ ]= S[f˜] = S[f ϕ(x − t) cos(kt − π)dt, ϕ ∈ L0 . πk r 2 k=1

(5.9)

−π

Hence, J˜rr is the set of functions the Fourier series of which have the form r (5.9). This means, in particular, that the classes W r N coincide with the classes of functions whose Fourier series have the form (5.9) with ϕ ∈ N. Therefore, if 0 ˜ rr = Dr ˜r df r we introduce an operator D r+1 that maps Jr = Jr+1 in L according df r r to relation (5.9), then the classes W r N = Wr+1 N can also be characterized as df

r r = fr+1 ∈ N. sets of functions f (·) for which Dr+1

Section 6

6.

Weyl–Nagy Classes

119

Weyl–Nagy Classes

6.1. One can conclude from the results in Subsections 4.3 and 5.4 that it is likely to be helpful to consider the sets Jβr of functions f (·) such that, for fixed r > 1 and β ∈ (−∞, ∞), ∞

a0 (f ) 1 + S[f ] = 2 πk r k=1

π ϕ(x − t) cos(kt − −π

βπ )dt 2

(6.1)

π ∞ rπ (β − r)π 1 a0 (f ) ϕ(x − t) cos(kt − + cos )dt = r 2 2 πk 2 k=1

−π

π ∞ r+1 (β − r)π 1 ϕ(x − t) cos(kt − π)dt, (6.2) + sin 2 πk r 2 k=1

−π

where ϕ ∈ L0 . Moreover, it is expedient to introduce an operator Dβr that acts from Jβr to L0 according to equality (6.1). It is clear that, for β = r + 2k, k ∈ Z, the set Jβr coincides with the set r Jr in Subsection 4.3; for β = r + 1 + 2k, functions that form the sets Jβr and r Jr+1 in Subsection 5.4 may differ only by constant terms. For other values of β, functions from the set Jβr are linear combinations of a pair of conjugate functions r , respectively. belonging to Jrr and Jr+1 The sets Jβr were first considered by Nagy. This is why we call the function ϕ(·) in decomposition (6.1) the Weyl–Nagy (r, β)-derivative; in what follows, it is sometimes denoted by fβr (·) so that fβr = Dβr f and it is easy to verify that S[fβr ]

=

∞ k=1

k −r (ak (f ) cos(kx +

βπ βπ ) + bk (f ) sin(kx + )). 2 2

(6.3)

6.2. The set Jβr can be completely described by using equality (6.3). Namely, this set consists of functions f (·) for which the series in (6.3) is the Fourier series of some summable function. To verify this it suffices to substitute decomposition (6.3) for ϕ(·) in equality (6.1) and perform elementary transformations. Thus, for any r > 0 and β ∈ (−∞, ∞), we have Jβr = {f (·) : Dβr f = fβr (·) ∈ L0 }.

(6.4)

120

Classes of Periodic Functions

Chapter 3

As in Subsection 4.4, if the series in (6.1) is convergent, we take its sum as a function f (·) regarded as an element of the set Jβr . 6.3. The classes of Weyl–Nagy differentiable functions are introduced in the already familiar way. If f ∈ Jβr and fβr ∈ N, then we say that f (·) belongs to the class Jβr N. As in Subsection 4.5, if N is a specific class of functions, we set Jβr N = Wβr N. It is clear that Wβr N = Wrr N for β = r + 2k, k ∈ Z, and Wβr N = r N=W r Wr+1 r N for β = r + 1 + 2k.

7.

Classes Lψ βN

7.1. By considering successively more and more general operators of differentiation (Dr , Drr , and then Dβr ), we can classify wider and wider classes of functions. Furthermore, there is a succession in the sense that we have Wrr N = W r N for r ∈ N and Wβr N = Wrr N for β = r. Moreover, there appears a possibility to indicate, for a given function f (·), a narrower set in which this function is contained; this, in turn, allows one to use its distinctive features more completely, e.g., for studying the problem of its approximation. For the same purpose, we consider a further generalization of the operation of differentiation. Formally, it is reduced to the replacement of the factor k −r in equalities (6.1) and (6.3) by a factor ψ(k), where ψ(k), k = 1, 2, . . . , generally speaking, is an arbitrary function of a natural variable. On the basis of this operation, we introduce classes of periodic functions which may coincide with the classes Wβr N (and, hence, with the classes Wrr N and W r N as well) for fixed values of their determining parameters and, at the same time, allow one to take into account the properties of functions that cannot be described in terms of known classes of functions. Let f ∈ L(0, 2π) and let S[f ] be the Fourier series of this function. Furthermore, let ψ(k) be an arbitrary function of a natural variable and let β be a fixed real number, β ∈ (−∞, ∞). Assume that the series ∞ k=1

1 βπ βπ (ak (f ) cos(kx + ) + bk (f ) sin(kx + )) ψ(k) 2 2

(7.1)

is the Fourier series of some function from L(0, 2π). We denote this function by fβψ (·) and call it the (ψ, β)-derivative of a function f (·). The set of functions satisfying this condition is denoted by Lψ β.

Classes Lψ βN

Section 7

121

If f ∈ Lψ β , then, by using Theorem 4.2 and performing elementary transformations, we obtain ∞

π

k=1

−π

a0 (f ) ψ(k) + 2 π

fβψ (x − t) cos(kt −

∞

βπ Ak (f ; x) = S[f ]. (7.2) )dt = 2 k=0

On the other hand, it is easy to prove that every function f (·) with the Fourier series of the form π ∞ βπ a0 (f ) ψ(k) ϕ(x − t) cos(kt − + )dt, (7.3) S[f ] = 2 π 2 k=1

−π

where ϕ ∈ L0 , belongs to Lψ β . For this purpose, it suffices to show that the Fourier series of the function ϕ(·) has the form (7.1). However, if equality (7.3) holds for S[f ], then, for any k = 1, 2, . . . , ak (f ) = ψ(k)(ak (ϕ) cos

βπ βπ − bk (ϕ) sin ), 2 2 (7.4)

βπ βπ bk (f ) = ψ(k)(ak (ϕ) sin + bk (ϕ) cos ). 2 2 Hence, ak (ϕ) =

1 βπ βπ (ak (f ) cos + bk (f ) sin ), ψ(k) 2 2

(7.5) 1 βπ βπ bk (ϕ) = (bk (f ) cos − ak (f ) sin ). ψ(k) 2 2 By comparing the relations obtained with expression (7.1), we can conclude that the Fourier series of a function ϕ(·) is indeed determined by this expression. ¯ψ Thus, the set Lψ β coincides with the set Lβ of functions from L(0, 2π) whose Fourier series have the form (7.3) with ϕ ∈ L0 . In this connection, if the Fourier series of a function f ∈ L(0, 2π) has the form (7.3) and ϕ ∈ L0 , then it is natural to call the function ϕ(·) the (ψ, β)- derivative of f. In this case, ψ 0 the operator that maps Lψ β in L according to relation (7.3) is denoted by Dβ so that Dβψ f = fβψ . ψ 7.2. It is clear that if ψ(k) = k −r , r > 0, then Lψ β = Jβ . If the series

in (7.2) is convergent and we take its sum as an element of the set Lψ β , then, for r β = r ∈ N, we have Lψ β =A .

122

Classes of Periodic Functions

Chapter 3

ψ 7.3. Let N be a subset of functions from L(0, 2π). If f ∈ Lψ β and fβ ∈

N, then we say that the function f (·) belongs to the class Lψ β N. Thus, every function ψ, every number β ∈ R, and every set N are associated with a class ψ −r r Lψ β N. In particular, for ψ(k) = k , we have Lβ N = Wβ N. 7.4. The function f (·) is called a convolution of two functions h(·) and g(·) from L(0, 2π) if it can be represented in the form 1 f (x) = π

π h(x − t)g(t)dt, −π

which is symbolically denoted by f = h ∗ g. In this case, the function g(t) is called the kernel of a convolution. First, let us verify that the function f (x) exists for almost all x. The general case can easily be reduced to the case where functions h(·) and g(·) are nonnegative. However, in this case, the function h(x−t)g(t), as a product of measurable functions, is measurable on the plane (x, t); moreover, since it is nonnegative, the order of integration makes no matter. Therefore, π −π

1 f (x)dx = π

π π ( h(x − t)g(t)dt)dx

−π −π

1 = π

π

−π

π g(t)( −π

1 h(x − t)dx)dt = π

π

−π

π g(t)dt

h(t)dt.

(7.6)

−π

This implies that f ∈ L(−π, π) and, hence, this function is finite for all x ∈ (−π, π). The function f (x) is obviously periodic. Let us determine its Fourier coefficients under the assumption that ∞

a0 (ak cos kx + bk sin kx), S[h] = + 2

(7.7)

k=1 ∞

a (ak cos kx + bk sin kx). S[g] = 0 + 2 k=1

(7.7 )

Classes Lψ βN

Section 7

123

It follows from (7.6) that the product h(x − t)g(t) is summable in the square −π ≤ x, t ≤ π. Therefore, by changing the order of integration, we get 1 ak (f ) = 2 π 1 = 2 π

π π h(x − t)g(t) cos kxdtdx −π −π

π

π g(t)

−π

h(x) cos k(x + t)dxdt = ak ak − bk · bk , k = 0, 1, . . . .

−π

Similarly, we can find that bk (f ) = ak bk + ak bk , k = 1, 2, . . . . Thus, the following theorem is true: Theorem 7.1. If h ∈ L(0, 2π) and g ∈ L(0, 2π), then the convolution f = h ∗ g of these functions also belongs to L(0, 2π). If, in this case, equalities (7.7) and (7.7 ) hold, then ∞

S[f ] =

a0 a0 ((ak ak − bk bk ) cos kx + (ak bk + ak bk ) sin kx). + 2

(7.8)

k=1

7.5. Consider a function ψ(k), k ∈ N, and a number β ∈ R and assume that the series ∞ βπ ψ(k) cos(kx − ) (7.9) 2 k=1

is the Fourier series of a function Dψ,β (x) summable on (0, 2π). Consider a convolution π 1 f (x) = ϕ(x − t)Dψ,β (t)dt, (7.10) π −π

where ϕ(·) is a function from S[f ] =

L0 .

In view of equalities (7.8) and (7.4), we get

π ∞ ψ(k) k=1

π

−π

ϕ(x − t) cos(kt −

βπ )dt. 2

(7.11)

By comparing equalities (7.3) and (7.11), we establish the following statement: Proposition 7.1. If series (7.9) is the Fourier series of a function Dψ,β ∈ L(0, 2π), then elements of the set Lψ β differ only by a free term from functions representable by convolution (7.10), in which ϕ(·) coincides with fβψ (·) almost everywhere.

124

Classes of Periodic Functions

Chapter 3

7.6. The representation of functions in the form (7.10) is often preferable to equalities of the form (7.3). Moreover, in problems of the theory of approximation, the presence of a free term does not play a principal role. Indeed, if we succeed in approximating a function f (x) by a function F (f, x), then F (f, x) − a0 (f )/2 would approximate the function f (x) − a0 (f )/2. Therefore, it is important to know in what cases the conditions of Proposition 7.1 are satisfied, or, in other words, under what conditions series (7.9) is the Fourier series of a function from L(0, 2π). This is one of central problems in the theory of Fourier series. Many deep results were obtained in this field, a part of which is presented in manuals on trigonometric series, e.g., in the books of Bari [1] and Zygmund [5, 6]. Some general results in this direction follow from Chapter 1. We restrict ourselves to the formulation of several facts necessary for what follows. Theorem 7.2. Let a function ψ(k), k ∈ N, be such that lim ψ(k) = 0,

(7.12)

Δ2 ψ(k) = ψ(k − 1) − 2ψ(k) + ψ(k + 1) ≥ 0.

(7.13)

k→∞

Then the series

∞

ψ(k) cos kx

(7.14)

k=1

converges everywhere except, probably, the point x = 0 to a function f ∈ L(0, 2π) and is the Fourier series of this function. Condition (7.13) is essential because one can construct a series of the form (7.14) with coefficients tending to zero monotonically such that its sum does not belong to L(0, 2π). Theorem 7.3. Let ψ(k), k ∈ N, tend to zero monotonically as k → ∞. Then the sum g(x) of the series ∞

ψ(k) sin kx

(7.15)

k=1

is integrable if and only if

∞ ψ(k) k=1

k

< ∞.

(7.16)

If this condition is satisfied, series (7.15) is the Fourier series of the function g(x) and series (7.14) is the Fourier series of some function f ∈ L(0, 2π).

Classes Lψ βN

Section 7

125

7.7. Denote the set of pairs of functions (ψ, β) for which series (7.9) is the Fourier series of some function from L(0, 2π) by F0 and the set of functions ψ(k) satisfying condition (7.16) and tending to zero monotonically as k → ∞ by F1 . If ψ ∈ F1 , then series (7.9), as a linear combination of series (7.14) and (7.15), is the Fourier series of some function from L(0, 2π), i.e., (ψ, β) ∈ F0 for any ψ ∈ F1 and β ∈ R. In view of Proposition 7.1, this yields the following assertion: Proposition 7.2. Let ψ ∈ F1 . Then, for any β ∈ R, elements f (·) of the set Lψ β can be represented as a0 (f ) 1 + f (x) = 2 π

π ϕ(x − t)Dψ,β (t)dt,

(7.17)

−π

where ϕ ∈ L0 , ϕ(·) coincides with fβψ (·) almost everywhere, and the sign of equality in (7.17) means equality of two functions from L(0, 2π), i.e., equality for almost all x. The function ψ(k) = k −r , k ∈ N, belongs to F1 for any r > 0. Therefore, functions from the sets Jβr can be represented in the form (7.17), i.e., a0 (f ) 1 f (x) = + 2 π where Dβr (t) =

π ϕ(x − t)Dβr (t)dt, ϕ ∈ L0 ,

(7.18)

−π

∞

k −r cos(kt − βπ/2).

(7.19)

k=1

For β = r + 1 + 2k, k ∈ Z, this relation gives a representation of functions from J˜rr (see Subsection 6.1) and, for β = r + 2k, it gives a representation of functions from Jrr . If, in addition, r are natural numbers, then equality (7.18) gives a representation of functions from the sets Ar and A˜r , respectively. 7.8. If we consider the statements of Propositions 7.1 and 7.2 in view of the definition of classes Lψ β N, we can find that, for (ψ, β) ∈ F0 (ψ ∈ F1 ), these classes are the classes of convolutions of functions ϕ ∈ L0 (or ϕ ∈ L(0, 2π) ) with kernels Dψ,β (·). In particular, the classes Wβr N are the classes of convolutions with kernels Dβr (·). Note that, for β = r and β = r + 1, the functions Dβr (·) are sometimes called Bernoulli kernels (or Bernoulli functions).

126

8.

Classes of Periodic Functions

Chapter 3

Classes Cβψ N ψ 8.1. Denote the subset of continuous functions f ∈ Lψ β by Cβ and the class

of functions f ∈ Cβψ such that fβψ ∈ N by Cβψ N. To establish conditions under which the classes Cβψ N are classes of convolutions, we need the following additional information about convolutions: Proposition 8.1. If h ∈ Lp , 1 < p < ∞, and g ∈ Lp (1/p + 1/p = 1), then convolution (7.6) is continuous on the entire axis and f C ≤

1 hp gp . π

(8.1)

Proof. The proof of this statement is based on the Hölder inequality: If h ∈ Lp , 1 ≤ p ≤ ∞, and g ∈ Lp , then π |h(t)g(t)|dt ≤ hp gp .

(8.2)

−π

By using this inequality, we find that, for any x and t, 1 |f (x + t) − f (x)| = | π

π (h(x + t − τ ) − h(x − τ ))g(τ )dτ |

−π

1 ≤ ( π

π |h(x + t − τ ) − h(x − τ )|p dτ )1/p gp

−π

1 = ( π

π |h(t − τ ) − h(τ )|p dτ )1/p gp .

−π

In view of Proposition 3.1, this implies that f (x) is continuous for all x. Inequality (8.1) also follows from (8.2) because the norm in the space Lp does not depend on a shift of an argument or a change of its sign. Proposition 8.2. If at least one of functions h(·) and g(·) belongs to the space M = L∞ , then convolution (7.6) is continuous; moreover, f C ≤

1 hM g1 if h ∈ M, π

(8.3)

Classes Cβψ N

Section 8 and

1 h1 gM if g ∈ M. π Indeed, assume, e.g., that h ∈ M. Then f C ≤

1 |f (x + t) − f (x)| ≤ π

127

(8.3 )

π |h(τ )||g(x + t − τ ) − g(x − τ )|dτ −π

1 ≤ hM π

π |g(t − τ ) − g(τ )|dτ −π

To prove the continuity of the function f (x), it remains to use Proposition 3.1. Inequality (8.3) is obvious. The case of g ∈ M can be examined similarly. By combining Propositions 8.1, 8.2, and 7.1 and using the completeness of a trigonometric system in the space C, we establish the following statement: Proposition 8.3. If (ψ, β) ∈ F0 , Dψ,β ∈ Lp , 1 ≤ p ≤ ∞, and N ⊂ Lp , 1/p+1/p = 1, then the set Cβψ N consists of functions which can be represented by the equality a0 (f ) 1 f (x) = + 2 π

π

fβψ (x − t)Dψ,β (t)dt

−π

at any point x ∈ R. In particular, if N ⊂ M, then this is true for any (ψ, β) ∈ F0 . One can easily conclude from Proposition 8.3, for example, that the classes Wβr N consist of continuous functions if r is an arbitrary nonnegative number and N ⊂ M or r > 1 and N is an arbitrary set from L(0, 2π). 8.2. Consider the sets Cβψ in the case where the function ψ(k) is chosen so that it decreases to zero faster than any power of k, i.e., lim k r ψ(k) = 0 ∀r ∈ R.

k→∞

(8.4)

Then series (7.2) can be differentiated arbitrarily many times and, as a result, we obtain uniformly convergent series. This means that, under condition (8.4), the sets Cβψ and, hence, the classes Cβψ N consists of infinitely differentiable functions.

128

Classes of Periodic Functions

Chapter 3

The functions ψ(k) = ψα,r (k) = exp(−αk r ) with arbitrary positive α and r can serve as an example of a function for which (8.4) is satisfied. 8.3. The operators Dβψ also enable us to classify the sets A2π of 2π-periodic functions analytic on the real axis. If f ∈ A2π , then we can choose the layer −a ≤ y ≤ b ( a > 0, b > 0 ) in which the function f (z) = f (x + iy) is regular. There is a close connection between numbers a and b and Fourier coefficients of such function. This connection is established by the following statement: Theorem 8.1. Let f (z) be an analytic 2π-periodic function regular in a closed layer −c ≤ y ≤ d ( c > 0, d > 0 ) and let ∞

ck eikx

(8.5)

k=−∞

be the Fourier series of the function f (x) in the complex form, i.e., c0 = ck =

a0 (f ) , 2

ak (f ) − ibk (f ) ak (f ) + ibk (f ) , c−k = , k = 1, 2, . . . . 2 2

(8.6)

Then there exists a constant K independent of k and such that |c−k | ≤ Ke−kd , |ck | ≤ Ke−kc , k = 0, 1, . . . .

(8.7)

On the other hand, if conditions (8.7) are satisfied for a function f ∈ L(0, 2π) with the Fourier series (8.5), then the series ∞

ck eikz

(8.8)

k=−∞

converges uniformly and absolutely inside the layer −c < y < d and represents a 2π-periodic regular function therein. Proof. The proof of the second part of Theorem 8.1 is obvious. In order to prove its first part, we integrate the function f (z)eikz along the boundary of the

Classes Cβψ N

Section 8

129

rectangle with vertices at the points (0, 0), (2π, 0), (2π, d), and (0, d). By taking into account the periodicity of this function and using the Cauchy theorem, we get 2π ikx

f (x)e

−kd

2π f (x + id)eikx dx.

dx = e

0

(8.9)

0

However,

2π f (x)eikx dx = 2πc−k .

(8.10)

0

Consequently, e−kd |c−k | ≤ 2π

2π

|f (x + id)|dx ≤ Ke−kd .

0

The inequalities for |ck | can be obtained similarly. Corollary 8.1. The values a and b that determine the layer of regularity of a function f (z) can be obtained from the relations a = lim inf ln |ck |−1/k , b = lim inf ln |c−k |−1/k . k→∞

k→∞

(8.11)

Indeed, if (8.11) is satisfied, then, for any ε > 0 and all sufficiently large k, we have |c−k | < e−k(b−ε) , |ck | < e−k(a−ε) . By virtue of the second part of Theorem 8.1, this implies that f (z) is regular inside a layer −(a − ε) < y < (b − ε), and since ε is arbitrary, f (z) is regular in the layer −a < y < b as well. It remains to show that this is the largest layer of regularity of f (z). If this were not true, then, by virtue of the first part of the theorem, we could set d > b in the inequalities |c−k | < Ke−kd , k = 0, 1, . . . , or c > a in the inequalities |ck | < Ke−kc , k = 0, 1 . . . . In the first case, the inequality lim inf ln |c−k |−1/k > b, k→∞

would hold and, in the second case, the inequality lim inf ln |ck |−1/k > a k→∞

130

Classes of Periodic Functions

Chapter 3

would be valid. However, both these inequalities contradict condition (8.11). 8.4. It follows from the statements proved above and equalities (8.6) and (7.4) that if the function ψ(k) satisfies the condition |ψ(k)| ≤ Ke−αk , k ∈ N, α > 0,

(8.12)

then the sets Cβψ consist of analytic functions which can be regularly extended into the layer |y| < α. Hence, in this case, the classes Cβψ N are the classes of functions f (x) which can be regularly extended into the layer |y| < α and whose (ψ, β)-derivatives belong to the class N. If the function ψ(k) is chosen so that lim inf ln |ψ(k)|−1/k = ∞, k→∞

(8.13)

then, by virtue of equalities (8.6) and (7.4) and Corollary 8.1, the sets Cβψ and, hence, the classes Cβψ N consist of functions regular on the whole complex plane, i.e., of entire functions. As an example of a function ψ(k) satisfying condition (8.12), we use again the function ψα,r (k) from Subsection 8.2 with α > 0 and r = 1. If r > 1, this function also satisfies condition (8.13), 1 r αk = ∞, k→∞ k

lim ln(exp(−αk r ))−1/k = lim

k→∞

ψ

i.e., in this case, the classes Cβ α,r N are classes of entire functions.

9.

Classes Lψ ¯N β ψ 9.1. In the definition of the sets Lψ β and classes Lβ N, the parameter β can

and classes take exactly one fixed value from R1 . Consider now the sets Lψ β¯ N which are determined by two functions of a natural variable, namely, by the Lψ β¯ df ¯ = βk , k ∈ N. The idea function ψ(k), as before, and the function β¯ = β(k) of definition of this classes is to replace the number β in (7.1) by numbers βk . Thus, a function f ∈ L(0, 2π) for which ∞

a0 (f ) (ak (f ) cos kx + bk (f ) sin kx), + S[f ] = 2 k=1

(9.1)

Classes Lψ N β¯

131

1 βk π βk π (ak (f ) cos(kx + ) + bk (f ) sin(kx + )) ψ(k) 2 2

(9.2)

Section 9 belongs to the set Lψ if the series β¯ ∞ k=1

is the Fourier series of some function fβ¯ψ from L(0, 2π). If f ∈ Lψ and fβ¯ψ ∈ N, we say that f (·) belongs to the class Lψ N. Clearly, β¯ β¯ ψ ψ = Lψ if βk = β for all k ∈ N, then Lψ β and, hence, Lβ¯ N = Lβ N. β¯

9.2. Many facts presented in Sections 7 and 8 for βk = β can be almost automatically extended to the general case with replacing β by β¯ or β¯k in the notation. Here, we only present several facts necessary for what follows. ¯ ψ¯ of functions from Proposition 9.1. The set Lψ coincides with the set L β¯ β L(0, 2π) whose Fourier series have the form ∞

a0 (f ) ψ(k) S[f ] = + 2 π k=1

π ϕ(x − t) cos(kt − −π

βk π )dt, 2

(9.3)

where ϕ ∈ L0 . , one can easily obtain an analog of relation (7.2): Indeed, for f ∈ Lψ β¯ ∞

π

k=1

−π

a0 (f ) ψ(k) + 2 π

=

∞

fβ¯ψ (x − t) cos(kt − βk π/2)dt

Ak (f ; x) = S[f ].

(9.4)

k=0

¯ ψ¯ . At the same time, if equality (9.3) holds for f (·), then ⊂L Hence, Lψ β¯ β ak (f ) = ψ(k)(ak (ϕ) cos βk π/2 − bk (ϕ) sin βk π/2), (9.5) bk (f ) = ψ(k)(ak (ϕ) sin βk π/2 + bk (ϕ) cos βk π/2).

132

Classes of Periodic Functions

Chapter 3

Therefore, ak (ϕ) =

1 (ak (f ) cos βk π/2 + bk (f ) sin βk π/2), ψ(k) (9.6)

1 bk (ϕ) = (bk (f ) cos βk π/2 − ak (f ) sin βk π/2). ψ(k) This implies that the series S[ϕ] has the form (9.2) and, hence, f ∈ Lψ , i.e., β¯ ψ ψ ¯ ¯. Lβ¯ = L β ¯ If f (·) satisfies equality (9.3), then the function ϕ(·) is called a (ψ, β)in L0 according derivative of the function f (·) and the operator that maps Lψ β¯ to formula (9.3) is denoted by Dβψ¯ so that Dβψ¯ f = fβ¯ψ . 9.3. In the case where the series ∞

ψ(k) cos(kx − βk π/2)

(9.7)

k=1

is the Fourier series of a function Dψ,β¯(x), by virtue of Theorem 7.1 and equalities (9.5), we can easily conclude that the following representation is valid for at almost all points x : elements of the set Lψ β¯ a0 (f ) 1 f (x) = + 2 π

π ϕ(x − t)Dψ,β¯(t)dt,

(9.8)

−π

where ϕ(·) is a function from L0 which coincides with a function fβ¯ψ (·) almost everywhere. It has been established above that, by virtue of Theorems 7.2 and 7.3, the fact that a function ψ(k) belongs to the set F1 guarantees the validity of equality (7.17) for any f ∈ Lψ β because Dψ,β (·) is a linear combination of series (7.14) and (7.15). A function Dψ,β¯(·) has the form Dψ,β¯(x) =

∞ k=1

ψ(k) cos βk π/2 cos kx +

∞

ψ(k) sin βk π/2 sin kx

(9.9)

k=1

In order to apply Theorems 7.2 and 7.3, we must now require that the functions df df ψ1 (k) = ψ(k) cos βk π/2 and ψ2 (k) = ψ(k) sin βk π/2 should belong to the set F1 .

¯ Order Relation for (ψ, β)-Derivatives

Section 10

133

9.4. Denote the subset of continuous functions from Lψ by Cβψ¯ and the β¯ class of functions f ∈ Cβψ¯ for which fβ¯ψ ∈ N by Cβψ¯ N. In the case where series (9.7) is the Fourier series of a summable function Dψ,β¯(x), we say that the ¯ belongs to the set F1,1 . In this case, clearly, Proposition 8.3 can be pair (ψ, β) reformulated as follows: ¯ ∈ F1,1 , D ¯ ∈ Lp , 1 ≤ p ≤ ∞, and N ⊂ Lp , Proposition 9.2. If (ψ, β) ψ,β 1/p + 1/p = 1, then the set Cβψ¯ N consists of functions that can be represented at any point x ∈ R by the equality a0 (f ) 1 f (x) = + 2 π

π

fβ¯ψ (x − t)Dψ,β¯(t)dt.

−π

¯ ∈ F1,1 . In particular, if N ⊂ M, then this is valid for all (ψ, β) 9.5. By virtue of equalities (9.5), condition (8.4) guarantees that functions f ∈ Cβψ¯ N are infinitely differentiable for such ψ. If inequalities (8.12) hold, then Cβψ¯ N are classes of analytic functions that can be extended regularly into the layer |y| < α; under condition (8.13), functions f ∈ Cβψ¯ N are regular on the entire complex plain.

¯ Order Relation for (ψ, β)-Derivatives

10.

10.1. If a function f (·) is r times differentiable (r ∈ N ), then it is absolutely continuous and has absolutely continuous derivatives f (ri ) (·) of all lower orders ri < r. Therefore, it is also natural to indicate analogs of “lower” derivatives for functions from sets Lψ . For this purpose, we give the following definiβ¯ tion: Assume that ψ1 = ψ1 (k), β¯1 = β1 (k), ψ2 = ψ2 (k), and β¯2 = β2 (k), k ∈ N, are arbitrary sequences of real numbers. The pair (ψ1 , β¯1 ) L-precedes L

2 1 ⊆ Lψ . In this case, we write (ψ1 , β¯1 ) ≤ (ψ2 , β¯2 ). If the pair (ψ2 , β¯2 ) if Lψ β¯ β¯ 2

Lβψ¯2 2

⊂

1 Lψ , β¯1

1

L then we write (ψ1 , β¯1 ) < (ψ2 , β¯2 ).

Thus, larger sets correspond to “lower” pairs (an operator Dβψ¯ with “lower” indices has a larger domain of definition).

134

Classes of Periodic Functions

Chapter 3 L

2 If a function f (·) belongs to Lψ and (ψ1 , β¯1 ) ≤ (ψ2 , β¯2 ), then it β¯ 2

has a derivative

fβψ¯1 1

belonging to

ψ /ψ Lβ¯2−β¯1 , 2 1

df

where ψ2 /ψ1 = (ψ2 (k)/ψ1 (k)),

df β¯2 − β¯1 = (β2 (k) − β1 (k)), k ∈ N ; furthermore, ψ /ψ

S[(fβ¯ψ1 )β¯2−β¯1 ] = S[fβ¯ψ2 ]. 1

2

1

(10.1)

2

1 Indeed, the imbedding Lβψ¯2 ⊆ Lψ implies that the derivative fβψ¯1 (·) exists and β¯1 2 1 the series

∞ k=1

1 (ak (f ) cos(kx+βi (k)π/2)+bk (f ) sin(kx+βi (k)π/2)), i = 1, 2 (10.2) ψi (k)

are the Fourier series of the function fβψ¯1 (x) and fβψ¯2 (x), respectively. For i = 1 2 1, we have S[fβ¯ψ1 ] = 1

∞

(ak (fβ¯ψ1 ) cos kx + bk (fβ¯ψ1 ) sin kx), 1

k=1

1

(10.3)

where 1 (ak (f ) cos β1 (k)π/2 + bk (f ) sin β1 (k)π/2), ψ1 (k)

ak (fβ¯1 )ψ1 = ψ1

bk (fβ¯1 )

1 = (bk (f ) cos β1 (k)π/2 − ak (f ) sin β1 (k)π/2). ψ1 (k)

(10.4)

Consider the series ∞ ψ1 (k) k=1

ψ2 (k)

(ak (fβ¯ψ1 ) cos(kx + (β2 (k) − β1 (k))π/2) 1

+ bk (fβ¯ψ1 ) sin(kx + (β2 (k) − β1 (k))π/2)). (10.5) 1

By using equalities (10.4), we can check that this series coincides with S[fβψ¯2 ] 2 and, at the same time, it is the Fourier series of the (ψ2 /ψ1 , β¯2 − β¯1 )- derivative of the function fβψ¯1 . 1

L

It follows from (10.1), in particular, that if (ψ1 , β¯1 ) ≤ (ψ2 , β¯2 ), then ψ /ψ

fβ¯ψ1 ∈ Lβ¯2−β¯1 N 1

2

1

2 ∀f ∈ Lψ N, N ⊂ L0 . β¯ 2

(10.6)

¯ Order Relation for (ψ, β)-Derivatives

Section 10

135

¯ derivatives. It is clear that In this way, we establish an order relation for (ψ, β)the pair for which ψ(k) ≡ 1 and β(k) ≡ 0 has the “lowest” order because Lψ = L in this case. β¯ 10.2. In examining the sets Cβψ¯ N, we find that the required continuity of “lower” derivatives is closely connected with the properties of the set containing ¯ (ψ, β)-derivatives. In this connection, we introduce one more notion. As before, let ψ1 (k), β1 (k), ψ2 (k), and β2 (k) be arbitrary sequences of real numbers. We say that the pair (ψ1 , β¯1 ) CN-precedes the pair (ψ2 , β¯2 ), CN

(ψ1 , β¯1 ) ≤ (ψ2 , β¯2 ) if the inclusion f ∈ Cβψ¯2 N yields fβψ¯1 ∈ C. 2

CN If f ∈ Cβψ¯2 N and (ψ1 , β¯1 ) ≤ (ψ2 , β¯2 ), then, 2 automatically, fβψ¯1 ∈ L0 ). Hence, equality (10.1) 1

1

by definition, fβψ¯1 ∈ C (and, 1

is also valid in this case. This

yields the following analog of relation (10.6): ψ /ψ

fβ¯ψ1 ∈ Cβ22−β11 N 1

∀f ∈ Cβψ¯ 2 N, N ⊂ L0 . 2

(10.7)

10.3. It is clear that there exist pairs (ψ2 , β¯2 ) and (ψ1 , β¯1 ) for which the L

CN

order relations ≤ or ≤ are meaningless. For example, let ψ1 (k) ≡ ψ2 (k) ≡ k −2 , β1 (k) ≡ 0, and β2 (k) ≡ 1. By virtue of Theorems 7.2 and 7.3, the function ∞ cos kx g(x) = k 2 ln k k=1

has a (ψ1 , β¯1 )-derivative because it is obviously continuous. Hence, it belongs to Lβψ¯1 and does not belong to Lβψ¯2 . Similarly, we conclude that the function 1

2

∞ sin kx h(x) = k 2 ln k k=1

belongs to

2 Lψ β¯2

and does not belong to Lβψ¯1 . This means that neither the relation 1

L

(ψ1 , β¯1 ) ≤ (ψ2 , β¯2 ) nor the inverse one holds. This fact could be a shortage if our aim were ordering pairs of sequences. However, in the case under consideration, it only illustrates the variety in the sets Lψ . β¯ 10.4. After all, we can give fairly simple conditions that guarantee Lpreceding and CN-preceding of such pairs.

136

Classes of Periodic Functions

Chapter 3

Theorem 10.1. Let ψi (k) and βi (k), i = 1, 2, k ∈ N, be arbitrary sequences of real numbers. If the pair (ψ2 /ψ1 , β¯2 − β¯1 ) belongs to the set F1,1 , i.e., the series ∞ ψ2 (k) (10.8) cos(kx + (β2 (k) − β1 (k))π/2) ψ1 (k) k=1

is the Fourier series of some summable function Dψ,β¯(x), ψ(k) = ψ2 (k)/ψ1 (k), β¯ = β2 (k) − β1 (k), k ∈ N, then L

(ψ1 , β¯1 ) ≤(ψ2 , β¯2 ).

(10.9)

If, in addition, N ⊂ M, then CN

(10.9 )

(ψ1 , β¯1 ) ≤ (ψ2 , β¯2 ).

2 ⊆ Lβψ¯1 , i.e., that, for Proof. To prove relation (10.9) we must show that Lψ β¯ 2

all f ∈ Lβψ¯2 , the series

1

2

∞ k=1

1 (ak (f ) cos(kx + β1 (k)π/2) + bk (f ) sin(kx − β1 (k)π/2)) (10.10) ψ1 (k)

is the Fourier series of a summable function fβψ¯1 (x). For this purpose, we con1 sider the function 1 J (x) = π

π

fβ¯ψ2 (x − t)Dψ,β¯(t)dt. 2

(10.11)

−π

Since fβψ¯2 (·) and Dψ,β¯(·) are summable, the function J (x) is also summable 2 by virtue of Theorem 7.1. By writing the Fourier series for this function and taking equalities (10.4) into account, we find that this series coincides with series (10.10). This proves relation (10.9). If f ∈ Cβψ¯2 N and N ⊂ M, then fβψ¯2 is essentially bounded. Hence, accord2 2 ing to Proposition 8.2, J (x) is continuous. In this case, since S[J ] coincides with series (10.10), expression (10.10) is the Fourier series of a continuous function. This proves relation (10.9 ). 10.5. If ψ2 (k)/ψ1 (k) ≡ k −r , r > 0, k ∈ N, and β1 (k) and β2 (k) take df

fixed values β1 (k) ≡ β1 and β2 (k) ≡ β2 for any k ∈ N so that β2 − β1 = γ,

¯ ψ-Integrals of Periodic Functions

Section 11

137

then, according to Subsection 7.7, series (10.8) is always the Fourier series of a summable function Dγr (·). This means that, in this case, relations (10.9) and (10.9 ) are always valid. In particular, for ψ1 (k) = k −r1 and ψ2 (k) = k −r2 , r1 , r2 > 0, Theorem 10.1 implies that if r2 > r1 , then the pair (ψ1 , β1 ) Lprecedes and CN-precedes ( N ⊂ M ) the pair (ψ2 , β2 ) for any β1 and β2 . Note that the examples in Subsection 10.3 show that, for r1 = r2 , these facts do not take place in the general case.

11.

¯ ψ-Integrals of Periodic Functions

¯ 11.1. The function ϕ in equality (9.3) is called (ψ, β)-derivative of the ¯ function f. Hence, it is natural to call the function f (ψ, β)-integral of ϕ. Then ¯ the set Lψ is the set of (ψ, β)-integrals of all functions ϕ ∈ L0 . Using this β¯ notation, we give following definitions: Definition 11.1. Let L denote the space of integrable 2π-periodic functions f ∈ L, and let ∞

∞

k=1

k=0

a0 (ak cos kx + bk sin kx) ≡ Ak (f ; x) S[f ] = + 2

(11.1)

be the Fourier series of a function f. Further, let ψ¯ = (ψ1 , ψ2 ) be a pair of arbitrary fixed systems of numbers ψ1 (k) and ψ2 (k), k = 1, 2, . . . . Consider the following series: A0 +

∞

(ψ1 (k)Ak (f ; x) + ψ2 (k)A˜k (f ; x)),

(11.2)

k=1

where A0 is a certain number and A˜k (f ; x) = ak sin kx − bk cos kx. If series ¯ is the Fourier series of a certain (11.2), for a given function f (·) and a pair ψ, function F ∈ L, then we say that F is the integral of the function f generated ¯ or simply, the ψ-integral ¯ by the pair ψ, of the function f, and denote this as ¯ ψ follows: F (·) = J (f ; ·). ¯ ¯ We denote the set of ψ-integrals of all functions f ∈ L by Lψ . If N is a ¯ ¯ subset of L, then Lψ N denotes the set of ψ-integrals of functions f ∈ N.

Definition 11.2. Let f ∈ L, let (11.1) be its Fourier series, and let a pair ¯ ψ = (ψ1 , ψ2 ) satisfy the following condition: ψ¯2 (k) = ψ12 (k) + ψ22 (k) = 0, k ∈ N.

(11.3)

138

Classes of Periodic Functions

If the series

Chapter 3

∞ ψ1 (k) ψ2 (k) ˜ A A (f ; x) − (f ; x) k k ψ¯2 (k) ψ¯2 (k)

(11.4)

k=1

¯ is the Fourier series of some function ϕ ∈ L, then we say that ϕ is the ψ¯ ¯ ψ ψ derivative of the function f and write ϕ(·) = D (f ; ·) = f (·). We denote the ¯ ¯ ψ¯ . If f ∈ L ¯ ψ¯ and subset of functions f ∈ L for which ψ-derivatives exist by L ¯ ¯ ψ¯ N. f ψ ∈ N, where N ⊂ L, then we set f ∈ L ¯ Remark 11.1. It is clear from the definition of ψ-integral that if a function ¯ has a ψ-integral F (·), any function F1 (·) that differs from F (·) on any set of ¯ ¯ measure zero is also its ψ-integral. On the other hand, any ψ-integral F (·) of a ¯ given function f (·) is a ψ-integral of any function f1 (·) that differs from f (·) on an arbitrary set of measure zero. This ambiguity is not essential for the present investigation and can be removed by the standard method, namely, by the identification of functions that are equivalent with respect to the Lebesgue measure. ¯ This remark obviously remains true for ψ-derivatives. In view of this, in the case of convergence of series (11.2) and/or (11.4) to summable functions, we always ¯ ¯ regard the functions F (·) = J ψ (f ; ·) and/or ϕ(·) = Dψ (f ; ·), respectively, as the sums of the corresponding series. ¯ ¯ 11.2. The relationship between ψ-integrals and ψ-derivatives is given by the following statement: Proposition 11.1. If f ∈ L, series (11.1) is its Fourier series, and condi¯ ¯ tion (11.3) is satisfied, then the function J ψ (f ; x) has a ψ-derivative and the following equality is true: ¯

¯

Dψ (J ψ (f ; ·)) = f (·) − ¯

a0 . 2

(11.5) ¯

¯ ψ and series (11.1) is its Fourier series, then the function Dψ (f ; x) has If f ∈ L ¯ a ψ-integral and ¯ ¯ J ψ (Dψ (f ; ·)) = f (·) + A0 , (11.6) where A0 is a certain constant. This proposition is proved by simple verification in view of the fact that if U is the operator of trigonometric conjugation, then U U (Ak (f ; x)) = −Ak (f ; x).

¯ ψ-Integrals of Periodic Functions

Section 11

139

Remark 11.2. In view of equality (11.5), it is natural to say that the func¯ ¯ tion f (x) − a0 /2 is the ψ-derivative of a function F (x) = J ψ (f ; x) regardless of whether condition (11.3) is satisfied or not. Thus, we always assume that df ¯ ¯ ¯ Dψ (J ψ (f ; x)) = f (x) − a0 /2, i.e., F ψ (·) = f (·) − a0 /2. Relations (11.5) and (11.6) yield the following statement: Proposition 11.2. For any pair ψ¯ satisfying condition (11.3) the following equality holds: ¯ ψ¯ = Lψ¯ . L (11.7) ¯ integrals of all funcThus, if condition (11.3) is satisfied, then the set of ψ¯ tions f ∈ L coincides with the set of functions having ψ-derivatives. Hence, in ¯ order to verify that the given function f ∈ L belongs to Lψ , it suffices to show ¯ψ. that f ∈ L 11.3. If we represent series (11.1) in the complex form, then series (11.2) can be rewritten as a0 μk ck eikx , ck = (ak − ibk )/2, c−k = (ak + ibk )/2, (11.8) + 2 |k|≥1

where μk =

ψ1 (k) − iψ2 (k),

k ≥ 1,

ψ1 (|k|) + iψ2 (|k|), k ≤ −1,

and series (11.4) can be rewritten in the form λk ck eikx , |k|≥1

where λk =

¯ (ψ1 (k) + iψ2 (k))/ψ(k),

k ≥ 1,

¯ (ψ1 (|k|) − iψ2 (|k|))/ψ(|k|), k ≤ −1, ¯ ψ(k) = (ψ12 (k) + ψ22 (k))1/2 . ¯

¯

Thus, the operators J ψ and Dψ are multipliers, and μk λk = 1, |k| ≥ 1.

140

Classes of Periodic Functions

Chapter 3

11.4. Definition 11.3. If a pair ψ¯ = (ψ1 , ψ2 ) is such that the series ∞

(ψ1 (k) cos kx + ψ2 (k) sin kx) =

1 μk eikx 2

(11.9)

|k|≥1

k=1

is the Fourier series of a certain function Ψ(x), then we write ψ¯ ∈ L. ¯ Proposition 11.3. If ψ¯ ∈ L, then, for every function f ∈ L, its ψ-integral exists and, almost everywhere, 1 J (f ; x) = A0 + π ψ¯

π f (x − t)Ψ(t)dt.

(11.10)

−π

¯

Thus, the elements of the set Lψ are functions representable by A0 + f ∗ Ψ, f ∈ L. In particular, the following statement is true: ¯ Proposition 11.4. Suppose that ψ¯ ∈ L. Then, for any function f ∈ Lψ , the following equality is valid almost everywhere:

a0 1 f (x) = + 2 π

π

¯

f ψ (x − t)Ψ(t)dt = −π

a0 ¯ + (f ψ ∗ Ψ)(x), 2

(11.11)

where a0 is the free term of the Fourier series of a function f (·). Equality (11.10) is proved by simple verification, and equality (11.11) follows from (11.10) in view of Remark 11.2. ¯ ¯ 11.5. The notion of ψ-derivatives coincides with the notion of (ψ, β)derivatives assumed in Subsection 3.9 in the sense that the fact of the existence of one of such derivatives yields the existence of the other one with the corresponding values of the determining parameters. In order to prove this, it suffices to show by using relations (11.4) and (9.2) that the system ⎧ ψ1 (k) 1 π ⎪ ⎪ = cos βk , ⎪ ⎪ 2 ¯ ψ(k) 2 ⎨ ψ (k) ψ¯2 (k) = 0, ψ(k) = 0, (11.12) ⎪ ⎪ 1 (k) π ψ ⎪ 2 ⎪ = sin βk , ⎩ ¯2 ψ(k) 2 ψ (k)

¯ ψ-Integrals of Periodic Functions

Section 11

141

is solvable with respect to ψ1 (k) and ψ2 (k) if the pair ψ(k), β(k) is given, and with respect to ψ(k) and β(k) if ψ1 (k) and ψ2 (k) are given. If ψ(k) and β(k) are given, we set π π ψ1 (k) = ψ(k) cos βk , ψ2 (k) = ψ(k) sin βk . 2 2

(11.13)

It is clear that such values satisfy system (11.12). If the pair ψ1 (k), ψ2 (k) is given, then the required values are determined by the equalities ¯ ψ(k) = ψ(k), cos βk

π ¯ = ψ1 (k)/ψ(k), 2

π ¯ = ψ2 (k)/ψ(k). 2 Thus, the following statement is true: sin βk

(11.14)

¯ ¯ Proposition 11.5. Any (ψ, β)-derivative of a function f ∈ L is also a ψderivative if the components ψ1 (k) and ψ2 (k) are chosen according to equali¯ ¯ ties (11.13). Moreover, any ψ-derivative is a (ψ, β)-derivative if the parameters ψ(k) and β(k) are determined by relations (11.14). In both cases, the following equalities hold: ¯ ψ¯ = Lψ¯ , L ¯ ψ¯ N = Lψ¯ N ∀N ⊂ L0 . L (11.15) β β where L0 = {ϕ ∈ L :

π

ϕ(t)dt = 0, i.e., ϕ⊥1}.

−π

¯ Note that relations (11.13) imply that a (ψ, β)-derivative is also a ψderivative if π ψ1 (k) = ψ(k) cos β , 2

π ψ2 (k) = ψ(k) sin β . 2

(11.13 )

(r)

The known rth (r > 0) Weyl derivative fr (·) of a function f (·) is a (ψ, β)derivative under the condition that ψ(k) = k −r and β = r. Consequently, by ¯ (r) virtue of (11.13 ), fr (·) coincides with f ψ (·) if we set π ψ1 (k) = k −r cos r , 2

π ψ2 (k) = k −r sin r . 2

11.6. If, in condition (11.3), we set ϕ¯ = (ϕ1 , ϕ2 ), where ϕ1 (k) = ψ1 (k)/ψ¯2 (k),

ϕ2 (k) = ψ2 (k)/ψ¯2 (k),

(11.16)

142

Classes of Periodic Functions

Chapter 3

then the values of ψ1 (k) and ψ2 (k) are determined by the equalities ¯ ψ1 (k) = ϕ1 (k)/Φ(k),

¯ ψ2 (k) = −ϕ2 (k)/Φ(k),

¯ 2 (k) = ϕ2 (k) + ϕ2 (k). Φ 1 2

(11.17)

In this case, for f ∈ L0 , series (11.4) has the form ∞

(ϕ1 (k)Ak (f ; x) + ϕ2 (k)A˜k (f ; x)),

(11.18)

k=1

and series (11.2) has the form ∞ ϕ1 (k) ϕ2 (k) ( ¯ 2 Ak (f ; x) − ¯ 2 A˜k (f ; x)). Φ (k) Φ (k)

(11.18 )

k=1

¯

This implies that if condition (11.3) is satisfied, then the integral J ψ (f ; ·) can be ¯ regarded on the set L0 as the derivative Dϕ¯ (f ; ·), and the derivative Dψ (f ; ·) can be regarded as the integral J ϕ¯ (f ; ·). In other words, generally speaking, it ¯ suffices to consider only one of these notions, e.g., the notion of ψ-integral, or to interchange the names of these notions. However, it turns out to be convenient to separate these notions, and the principal reason for this is the traditional conjecture that the properties of a function become “better” after its integration, and become “worse” after its differentiation. In view of this fact, in the present paper, we assume that ψ1 (k) and ψ2 (k) are infinitely small and, consequently, ϕ1 (k) and ϕ2 (k) in (11.16) are infinitely ¯ large sequences. Hence, as a rule, the fact of the existence of a ψ-integral for a ¯ function f ∈ L can easily be established. At the same time, the existence of a ψderivative of a given function imposes substantial requirements on this function: The faster the functions ψ1 (k) and ψ2 (k) decrease, the lower the probability for ¯ ¯ ψ¯ a given function to have a ψ-derivative or, which is the same, to hit the set L ¯ and, hence, according to Proposition 11.2, the set Lψ , as well. ¯ ¯ Different pairs ψ¯ and ψ¯ correspond to different sets Lψ and Lψ . One can use Subsection 3.10 and take into consideration Proposition 11.5 for establishing the relationship between such sets. ¯ 11.7 Do all functions f ∈ L have at least one ψ-derivative? The positive answer to this question is obvious if we do not require that lim ψ1 (k) = lim ψ2 (k) = 0.

k→∞

k→∞

(11.19)

¯ ψ-Integrals of Periodic Functions

Section 11

143 ¯

Indeed, if we set, e.g., ψ1 (k) = ψ2 (k) = 1∀k ∈ N, we get f ψ (·) = f (·). Therefore, it is interesting to consider the case where condition (11.19) is satisfied. In this case, the answer is also positive but nontrivial. This follows from Salem’s statement [1] (see, e.g., N. K. Bari [1, p.243]), for the formulation of which we need the following standard definitions: Definition 11.4. A sequence of real numbers λ = λ(k), k = 1, 2, . . . , is called convex (convex downwards) if Δ2 λ(k) = λ(k) − 2λ(k + 1) + λ(k + 2) ≥ 0.

(11.20)

If δ 2 λ(k) ≤ 0, then the sequence λ(k) is called concave (convex upwards). The set of convex downwards sequences λ(k) for which lim λ(k) = 0,

k→∞

(11.21)

is denoted by M. Proposition 11.6 (Salem[1]). Let f (·) be any function from the set C (or L ) and let ∞ a0 + S[f ] = (ak cos kx + bk sin kx) 2 k=1

be its Fourier series. Then one can always indicate a concave sequence λ = λ(k) such that its terms tend to infinity (λ(k) ↑ ∞) without decreasing and the series ∞

a0 λ(k)(ak cos kx + bk sin kx) λ(0) + 2 k=1

is the Fourier series of a continuous (or, respectively, summable) function F (·). We can assume that, in this statement, we always have λ(k) ≥ 1. Therefore, for a pair ψ¯ = (ψ1 , ψ2 ) with ψ1 (k) ≡ 1/λ(k) and ψ2 (k) ≡ 0, taking into account that ψ1 , ψ2 ∈ M and, for any function f ∈ L, according to (11.4) ¯

S[f ψ ] =

∞ k=1

1 Ak (f ; x), ψ1 (k)

we obtain the following statement from Proposition 11.6:

144

Classes of Periodic Functions

Chapter 3

¯ Proposition 11.7. Every function f ∈ C (or f ∈ L ) has at least one ψ¯ ψ ¯ derivative f (·) which belongs to C (or L ). In this case, the pair ψ = (ψ1 , ψ2 ) can be chosen so that ψ1 , ψ2 ∈ M. ¯ This implies that every function f ∈ C (or f ∈ L ) has an infinite set of ψderivatives, the determining components of which ψ1 and ψ2 belong to M, and the derivatives themselves are continuous (or summable, respectively). Moreover, as is well known, for ψ ∈ M, the series ∞

ψ(k) cos kt,

(11.22)

k=1

is always the Fourier series of a summable function Ψ(t). Then, by virtue of Proposition 11.3, we arrive at the following statement: Proposition 11.8. Every function f ∈ C (or f ∈ L ) is representable as the convolution π 1 ϕ(x − t)Ψ(t)dt, (11.23) f (x) = A0 + π −π

where ϕ ∈ C (or ϕ ∈ L ) and Ψ(t) is a function whose Fourier series has the form (11.22). In addition, ψ = ψ(k) belongs to M. Thus, the following equalities hold: ∪

¯

Lψ = L,

ψ1 ,ψ2 ∈M

¯

∪

¯

¯

C ψ = C, (C ψ = Lψ ∩ C)

ψ1 ,ψ2 ∈M

(11.24)

and, moreover, ¯

∪ Lψ∗ = L,

ψ∈M

¯

¯

¯

∪ C ψ∗ = C, (C ψ∗ = Lψ∗ ∩ C),

ψ∈M

(11.24 )

¯

where Lψ∗ are the sets of functions representable by equality (11.23). 11.8. Let F be the set of all trigonometric polynomials, i.e., functions f (·) representable by the equalities a0 f (x) = (ak cos kx + bk sin kx), + 2 n

k=1

where n may be any positive integer. It is clear that, for any pair ψ¯ = (ψ1 , ψ2 ) ¯ satisfying condition (11.3), every function f ∈ F has a ψ-derivative. This

Section 11

¯ ψ-Integrals of Periodic Functions

145

¯

implies, in particular, that the set Lψ is always nonempty. It is also clear that ¯ ∩ψ¯ Lψ = F if ψ¯ runs through the set of pairs satisfying condition (11.3). Moreover, the following stronger statement is also true: Proposition 11.9. The following equality is true: ∩

¯

Lψ = F.

(11.25)

ψ1 ,ψ2 ∈M

¯

Proof. If f ∈ F, then, as indicated above, f ∈ Lψ for any pair satisfying condition (11.3). Therefore, it remains to prove that a function f (·) having any ¯ ψ-derivative for ψ1 , ψ2 ∈ M must belong to F. This fact follows from the statement presented below. ¯ F, then one can choose a pair ψ¯ = Proposition 11.10. If f ∈ L but f ∈ ¯ Lψ¯ , i.e., f ψ¯ (·) does not exist. (ψ1 , ψ2 ) such that ψ1 , ψ2 ∈ M and f ∈ Proof. Let a0 , a1 , . . . , b1 , b2 , . . . be the Fourier coefficients of a function f (·). We set a(x) = sup |ak |, b(x) = sup |bk |, x ≥ 1. k>x

k>x

The functions a(x) and b(x) are piecewise constant and nonincreasing. Moreover, since the Fourier coefficients tend to zero as k → ∞, we have lim a(x) = lim b(x) = 0.

x→∞

x→∞

¯ F is equivalent to the fact that at least one of these functions is The condition f ∈ always positive. For definiteness, assume that a(x) > 0∀x ≥ 1. Denote by kj , j = 1, 2, . . . , the points, arranged in ascending order, at which the function a(x) changes its value. It is clear that a(kj ) = |akj |. We set zj = (kj , |akj |) and construct a function l(x) in the following way: We rotate the ray l1 which originates at z1 in the direction opposite to the positive direction of the y-coordinate axis, counterclockwise up to the moment when this ray meets one of the points zj , j > 1. Denote this point by zj1 . In the case where several points lie on the ray, we denote by zj1 the point with the greatest abscissa. On the interval [1, kj1 ], we define l(x) so that its graph coincides with the straight line connecting the points z1 and zk1 . Further, we rotate the ray l2 that originates at the point zj1 and whose direction coincides with the ray l1 in the final position again counterclockwise up to the moment when it meets one of the remaining points zj , j > j1 . We denote this point by zj2 . If several such points lie on the

146

Classes of Periodic Functions

Chapter 3

ray, then we denote by zj2 the point with the greatest abscissa. On the interval [kj1 , kj2 ], we define l(x) so that the graph of this function coincides with the straight line connecting the points zj1 and zj2 . By repeating this procedure, we construct the function l(x) for all x ≥ 1. This function possesses the following properties: (a) l(x) is convex downwards and lim l(x) = 0; x→∞

(b) l(kjs ) = a(kjs ) = |akjs |, s = 1, 2, . . . . Therefore, if we set ψ1 (k) = l(k), then, by virtue of property (a), we conclude that ψ1 ∈ M, and, by virtue of property (b), we have ψ1 (kjs ) = |akjs |, s = 1, 2, . . . .

(11.26)

Consider the pair ψ¯ = (ψ1 , ψ2 ), where ψ2 (k) = 0, k = 1, 2, . . . . Then series (11.4) that corresponds to this pair and the function f (·) takes the form ∞ k=2

1 Ak (f ; x). ψ1 (k)

By virtue of (11.26), the coefficients of this series do not tend to zero. Hence, it cannot be the Fourier series of any function from L. Consequently, the function ¯ f (·) has no ψ-derivative with such parameters. Proposition 11.10 and, hence, Proposition 11.9 are thus proved. Note that equalities (11.24) show that, in the case where a pair ψ¯ = (ψ1 , ψ2 ) runs through the set M × M, the sets L and/or C are decomposed into subsets ¯ ¯ (classes) Lψ and/or C ψ . Equality (11.25) means that only trigonometric polynomials remain indistinguishable under such a classification. Therefore, in the course of investigation of various problems of the theory of ¯ approximation for functions from the sets Lψ , the main attention should be given to the case where the sequences ψ1 (k) and ψ2 (k), k = 1, 2, . . . , are chosen from the set M. In this case, the considerable mathematical apparatus has already been ¯ created. This enables one to obtain the results for the classes Lψ N practically with the same degree of completeness as for the classes of functions defined by derivatives in the Weyl sense. Some special properties of convex functions are one of the most important elements of this apparatus. Numerous properties of this type will be established in subsequent sections.

Section 12

12.

Sets M0 , M∞ , and MC

147

Sets M0 , M∞ , and MC

12.1. Without loss of generality, we assume that the sequences ψ(k) from the set M are restrictions of certain positive continuous convex downwards functions ψ(t) of continuous argument t ≥ 1 that vanish at infinity to the set of natural numbers. As above, we denote the set of such functions by M. Thus, we have M = {ψ(t) : ψ(t) > 0, ψ(t1 ) − 2ψ((t1 + t2 )/2) + ψ(t2 ) ≥ 0 ∀t1 , t2 ∈ [1, ∞),

lim ψ(t) = 0}.

t→∞

The set M is very inhomogeneous in the rate of convergence of its elements to zero as t → ∞ because the functions ψ(t) can decrease either arbitrarily slowly or arbitrarily rapidly. Moreover, it turns out that the form of results on ¯ the approximation of functions from the sets Lψ , their substance, and methods for their derivation significantly depend on this rate. Therefore, these arises the necessity of decomposing the set M into subsets of functions ψ ∈ M with the same, in a certain sense, character of convergence to zero. As a characteristic on the basis of which it is convenient to perform such a decomposition, we use a pair of functions η(t) = η(ψ; t) and μ(t) = μ(ψ; t) defined as follows: Let ψ ∈ M. Then we denote by η(t) = η(ψ; t) the function connected with ψ by the equality 1 ψ(η(t)) = ψ(t), 2

t ≥ 1.

(12.1)

By virtue of the strict monotonicity of ψ, the function η(t) from (12.1) is uniquely defined for all t ≥ 1 : 1 η(t) = η(ψ; t) = ψ −1 ( ψ(t)). 2

(12.2)

The function μ(t) is defined by the equality μ(t) = μ(ψ; t) =

t . η(t) − t

(12.3)

As follows from (12.1), the quantity η(t) − t is the length of the segment [t, η(t)] on which the value of the function ψ reduces exactly to half. In this connection, the function μ(ψ, t) was called the modulus of half-decay of the function ψ.

148

Classes of Periodic Functions

Chapter 3

If ψ1 (t) = t−r , r > 0, then μ(ψ1 ; t) = (21/r − 1)−1 . If ψ2 (t) = 1/ ln(t + a), a > e, then μ(ψ2 ; t) = t/((t + a)2 + a − t), and if ψ3 (t) = e−t , then μ(ψ3 ; t) = t/ ln 2. These examples demonstrate that the quantity μ(ψ; t) can be bounded from above and below by certain positive numbers, can tend to zero as t → ∞, and can be unbounded above. On the basis of these features, we select the following subsets of the set M : M0 = {ψ ∈ M : 0 < μ(ψ; t) ≤ K

∀t ≥ 1},

(12.4)

M∞ = {ψ ∈ M : 0 < K ≤ μ(ψ; t) < ∞ ∀t ≥ 1},

(12.5)

MC = {ψ ∈ M : 0 < K1 ≤ μ(ψ; t) ≤ K2

(12.6)

∀t ≥ 1}.

Here and below, K, K1 , . . . are certain positive constants independent of the quantities that are parameters in the case under investigation (in the case considered, they are independent of the variable t). Further, we denote by M+ 0 the subset of functions ψ ∈ M0 for which the quantity μ(ψ; t) tends monotonically to zero as t → ∞ : M+ 0 = {ψ ∈ M : μ(ψ; t) ↓ 0}.

(12.7)

By M+ ∞ , we denote the subset of functions ψ ∈ M∞ for which μ(ψ; t) monotonically and infinitely increases as t → ∞ : M+ ∞ = {ψ ∈ M : μ(ψ; t) ↑ ∞}.

(12.8)

12.2. Note that the functions t−r , r > 0, t−r lnε (t + e) for ε ∈ R1 , etc. can be regarded as natural examples of functions from the set MC ; the set M+ 0 contains the functions lnε (t + e) for ε < 0; the functions exp(−αtr ) for any α > 0 and r > 0 belong to the set M+ ∞ . This can easily be verified with the use of the following statement: Theorem 12.1. A function ψ ∈ M belongs to M0 if and only if the quantity α(t) =

ψ(t) df , ψ (t) = ψ (t + 0), t|ψ (t)|

(12.9)

satisfies the condition α(t) ≥ K > 0 ∀t ≥ 1.

(12.10)

Sets M0 , M∞ , and MC

Section 12

149

A function ψ ∈ M belongs to M∞ if and only if α(t) ≤ K

∀t ≥ 1.

(12.11)

A function ψ ∈ M belongs to MC if and only if 0 < K1 ≤ α(t) ≤ K2

∀t ≥ 1.

(12.12)

If the function α(t) does not decrease and lim α(t) = ∞,

(12.13)

t→∞

then ψ ∈ M+ 0 . If α(t) does not increase and lim α(t) = 0,

(12.14)

t→∞

then ψ ∈ M+ ∞. Proof. In view of equality (12.9) with t ≥ 1, we get η(t) ψ(η(t))

dτ ≤ τ α(τ )

t

η(t) η(t) η(t) ψ(τ ) dτ |ψ (τ )|dτ = dτ ≤ ψ(t) τ α(τ ) τ α(τ ) t

t

t

or, with regard for (12.2), ψ(t) 2

η(t)

dτ ψ(t) ≤ ≤ ψ(t) τ α(τ ) 2

t

η(t)

dτ , τ α(τ )

t

i.e., for all t ≥ 1, we have 1 ≤ 2

η(t)

dτ ≤ 1. τ α(τ )

t

If condition (12.10) is satisfied, we obtain K ≤ 2

η(t) t

dτ η(t) = ln = ln(1 + 1/μ(ψ; t)). τ t

(12.15)

150

Classes of Periodic Functions

Chapter 3

Thus, in this case, we have μ(ψ; t) ≥ (eK/2 − 1)−1 , i.e., ψ ∈ M0 . Similarly, if condition (12.11) is satisfied, then relation (12.15) yields ln(1 + 1/μ(ψ; t)) ≤ K, which immediately implies that ψ ∈ M∞ . By analogy, one can verify that if condition (12.12) is satisfied, then the function ψ belongs to the set MC . If α(t) does not decrease and relation (12.13) is true, then, by virtue of inequality (12.15), we have α(t) ≤ 2 ln(1 + 1/μ(ψ; t))

(12.16)

for all t ≥ 1. In view of (12.13), this yields lim μ(ψ; t) = 0.

t→∞

(12.17)

Let us verify that the function μ(ψ; t) monotonically decreases. For this purpose, we consider the function 1/μ(t) = η(t)/t − 1 and conclude that this is possible if and only if tη (t) − η(t) ≥ 0. (12.18) According to (12.1) and (12.9), we have 1 ψ(t) = ψ(η(t)) = −η(t)ψ (η(t))α(η(t)). 2 Combining this relation with (12.9) and taking into account the fact that, by virtue of (12.2), ψ (t) η (t) = (12.19) 2ψ (η(t)) for every ψ ∈ M, we establish that, in the case under consideration, tη (t)α(t) =1 η(t)α(η(t)) or tη (t) = η(t)

α(η(t)) ≥ η(t), α(t)

(12.20)

Section 12

Sets M0 , M∞ , and MC

151

i.e., relation (12.18) is indeed true. Thus, we have proved that if α(t) does not decrease and condition (12.13) is satisfied, then ψ ∈ M+ 0 . By analogy, we verify that if α(t) does not increase and condition (12.14) is satisfied, then ψ ∈ M+ ∞. It remains to show that condition (12.10) is satisfied for all ψ ∈ M0 , condition (12.11) is satisfied for any ψ ∈ M∞ , and condition (12.12) is satisfied for any ψ ∈ MC . For every function ψ ∈ M, we denote by η¯(t) = η¯(ψ; t) the function inverse to η(t). By virtue of (12.19), for any t ≥ 1 we have η (t) > 1/2, which implies that the function η(t) is strictly monotonic. Therefore, on the set t ≥ η(1), the function η¯(t) is uniquely defined and, for every t ≥ η(1), the following relation is true: t ψ(t) = − ψ (τ )dτ ≥ |ψ (t)|(t − η¯(t)) (12.21) η¯(t)

or

ψ(t) t − η¯(t) ≥ . t|ψ (t)| t We now assume that a function ψ ∈ M is such that μ(ψ; t) =

t ≤K η(t) − t

∀t ≥ 1

(12.22)

(12.23)

(i.e., ψ ∈ M0 ). Then, by setting z = η¯(t), we obtain t η(z) z = =1+ ≤K +1 t − η¯(t) η(z) − z η(z) − z or

t − η¯(t) 1 ≥ . t K +1 Substituting this estimate into (12.22), we conclude that if relation (12.23) is true, then ψ(t) ≥ K1 > 0 t|ψ (t)|

for all t ≥ η(1). It is clear that the same inequality is also true for t ∈ [1, η(1)]. This proves that condition (12.10) is satisfied if ψ ∈ M0 and, furthermore, the left-hand side of relation (12.12) is true if ψ ∈ MC . For every function ψ ∈ M and any t ≥ 1, we have 1 |ψ (η(t))|(η(t) − t) ≤ ψ(t) = − 2

η(t) ψ (τ )dτ ≤ |ψ (t)|(η(t) − t). t

(12.24)

152

Classes of Periodic Functions

Chapter 3

Hence, ψ(t) η(t) − t ≤2 . t|ψ (t)| t If ψ ∈ M is such that μ(ψ; t) =

t ≥K η(t) − t

∀t ≥ 1,

i.e., ψ ∈ M∞ , then ψ(t) 2 ≤ . t|ψ (t)| K Thus, if ψ ∈ M∞ , then condition (12.11) is satisfied, and if ψ ∈ MC , then the right-hand side of relation (12.12) is true. Theorem 12.1 is proved. Representing equality (12.9) in the form ψ (t) 1 =− ψ(t) tα(t) and integrating the last relation from 1 to t, t > 1, we get t ψ(t) = ψ(1) exp(−

dτ ). τ α(τ )

1

Analyzing this equality, we arrive at the following statement: Corollary 12.1. If ψ ∈ M0 , then there exists r1 > 0 such that, for all t ≥ 1, ψ(t) ≥ Kt−r1 . If ψ ∈ M∞ , then there exists r2 > 0 such that, for all t ≥ 1, ψ(t) ≤ Kt−r2 . If ψ ∈ MC , then there exist r1 , r2 > 0 such that, for all t ≥ 1, K1 t−r1 ≤ ψ(t) ≤ K2 t−r2 .

Set F

Section 13

13.

153

Set F

13.1. The further investigation of functions ψ ∈ M is connected with the behavior of the quantity η (t) = η(ψ; t), for which, as mentioned above, equality (12.19) is true. According to this equality, 2η (t) is the number that indicates by how many times the value of ψ (τ ) changes as τ runs through the segment [t, η(t)]. We have mentioned above that, for all ψ ∈ M, 1 η (t) ≥ , 2

t ≥ 1.

(13.1)

The functions ψ(t) = t−r , r > 0, and ψ(t) = ln−ε (t + e), ε > 0, demonstrate that, for various functions ψ ∈ M, this quantity can be bounded from above as well as unbounded. For this reason, we set F = {ψ ∈ M : η (ψ; t) ≤ K}.

(13.2)

The set F has certain specific features. First, we formulate the following statement: Theorem 13.1. The following inclusion is true: MC ∪ M+ ∞ ⊆ F.

(13.3)

Proof. Parallel with relations (12.21) and (12.24), we also consider their analog for the segment [η(t), η(η(t))], namely 1 |ψ (η(η(t)))|(η(η(t)) − η(t)) ≤ ψ(t) ≤ |ψ (η(t))|(η(η(t)) − η(t)). 4

(13.4)

Combining relations (12.21) and (13.4), for all ψ ∈ M and arbitrary t ≥ η(1) we obtain η (t) =

ψ (t) (t − η¯(t)) η(η(t)) − η(t) · · 2ψ (η(t)) η(η(t)) − η(t) t − η¯(t)

≤2

η(η(t)) − η(t) η(η(t)) − η(t) t η(t) =2 · · . t − η¯(t) η(t) t − η¯(t) t

(13.5)

If ψ ∈ MC , then it follows from (12.6) that each fraction in the last relation is bounded from above. Therefore, for every ψ ∈ MC and any t ≥ η(1), the quantity η (t) is also bounded. It is clear that this is also true for t ∈ [1, η(1)]. Thus, MC ⊂ F.

154

Classes of Periodic Functions

Chapter 3

If ψ ∈ M+ ∞ , then, by virtue of (12.8), the following equality is true: η(t) = t(1 + γ(t)),

(13.6)

where γ(t) is a function monotonically tending to zero as t → ∞. Therefore, η (t) ≤ 1 + γ(t) and, hence, M+ ∞ ⊂ F. 13.2. Now we establish several criteria for a function ψ ∈ M to belong to the set F. Theorem 13.2. In order that a function ψ ∈ M belong to F, it is necessary and sufficient that, for all t ≥ 1, the following relation be true: K1 |ψ (t)|(η(t) − t) ≤ ψ(t) ≤ K2 |ψ (η(t))|(η(t) − t), η(t) = η(ψ, t). (13.7) Indeed, if ψ ∈ F, then (13.7) follows from the (12.24), (13.1), (12.19) and (13.2). In this case, we have |ψ (η(t))| ≤ |ψ (t)| ≤ 2K|ψ (η(t))|.

(13.8)

On the other hand, if relation (13.7) is true, then, dividing it by |ψ (η(t))|(η(t) − t), we obtain η (t) ≤ K2 /2K1 , i.e., we indeed have ψ ∈ F. 13.3. Remark 13.1. Setting λ(t) = ψ(t)/|ψ (t)|, and using (13.7), we obtain K1 (η(t) − t) ≤ λ(t) ≤ K2 (η(t) − t) ∀ψ ∈ F. Theorem 13.3. In order that a function ψ ∈ M belong to F, it is necessary and sufficient that, for all t ≥ 1, the following inequality be true: η(η(t)) − η(t) ≤ K, η(t) − t

η(t) = η(ψ; t).

(13.9)

Proof. Let ψ ∈ F and let K0 > 1/2 be such that η (t) ≤ K0 for any t ≥ 1. Then η(t) η(η(t)) − η(t) = η (τ )dτ ≤ K0 (η(t) − t), t

(13.10)

Set F

Section 13

155

which yields inequality (13.9) for K = K0 . Now let inequality (13.9) be true. Then, by virtue of (12.21) and (13.4), for all t ≥ η(1) we get η (t) =

ψ (t) η(η(t)) − η(t) η(t) − t ≤2 · ≤ 2K 2 . 2ψ (η(t)) t − η¯(t) η(t) − t

Thus, for t ≥ η(1), the quantity η (t) is bounded. It is clear that this quantity is also bounded for t ∈ [1, η(1)]. 13.4. Remark 13.2. Since, for every ψ ∈ M, the estimate η (t) ≥ 1/2 is true, estimating integral (13.10) from below we conclude that, for all t ≥ 1, η(η(t)) − η(t) 1 ≥ ∀ψ ∈ M. η(t) − t 2

(13.11)

Thus, according to (13.9) and (13.11), for all t ≥ 1 we have 2(η(t) − t) ≤ η(η(t)) − η(t) ≤ K(η(t) − t) ∀ψ ∈ F.

(13.12)

Theorem 13.4. In order that, for a given function ψ ∈ M and all t ≥ 1, the relation ∞ ψ(τ ) K1 ψ(t) ≤ (13.13) dτ ≤ K2 ψ(t), η(t) = η(ψ; t), τ −t η(t)

be true, it is necessary and sufficient that ψ belong to F. If ψ ∈ F, then, for all t ≥ 1, we have η(t) t

ψ(t) − ψ(τ ) dτ ≤ Kψ(t), τ −t ∞

(13.14)

ψ(τ ) dτ ≤ K. τ

(13.15)

1

Proof. Since ∞

ψ(τ ) dτ > τ −t

η(t)

η(η(t))

η(t)

=

ψ(τ ) ψ(t) dτ > τ −t 4

η(η(t))

dτ τ −t

η(t)

ψ(t) η(η(t)) − η(t) ln(1 + ), 4 η(t) − t

(13.16)

156

Classes of Periodic Functions

Chapter 3

the upper bound for the integral in (13.13) is possible only if condition (13.9) is satisfied. Thus, the inclusion ψ ∈ F is necessary for inequality (13.13) to be satisfied. If ψ ∈ F, then, by virtue of (12.24) and (13.9), we have ∞

ψ(τ ) dτ ≤ 2 τ −t

η(t)

∞

|ψ (τ )|

η(τ ) − τ dτ ≤ Kψ(t), τ − η¯(τ )

η(t)

and the lower bound in (13.13) follows from (13.16) and (13.11). 13.5. In view of inequality (13.7), we obtain estimate (13.8): η(t) t

ψ(t) − ψ(τ ) dτ ≤ |ψ (t)|(η(t) − t) ≤ Kψ(t). τ −t

To prove inequality (13.15), we first note that, for any ψ ∈ F and t ≥ η(t), t η(t) − t =

.

η (τ )dτ ≤ K(t − η¯(t)) < Kt.

(13.17)

η¯(t)

This yields for all t ≥ 1 μ(ψ; t) =

t ≥ K1 > 0 ∀ψ ∈ F. η(t) − t

(13.18)

By using relation (12.24) and estimate (13.18), we obtain ∞

ψ(τ ) dτ ≤ 2 τ

1

∞

|ψ (τ )|

η(τ ) − τ dτ ≤ K. τ

1

Theorem 13.4 is proved.

14.

Two Counterexamples

14.1. Relation (13.18) means the validity of the inclusion F ⊆ M∞ . Thus, by virtue of Theorem 13.1, we have MC ∪ M+ ∞ ⊆ F ⊆ M∞ .

(14.1)

In fact, both inclusions in this relation are strict due to the following statement:

Section 14

Two Counterexamples

157

Theorem 14.1. The set M∞ \ F is nonempty. In particular, it contains functions ψ for which η(ψ; t) − t < 2. (14.2) The set F \ (MC ∪ M+ ∞ ) is also nonempty and also contains functions ψ satisfying condition (14.2). Proof. Let us prove the first part of Theorem 14.1. For this purpose, it suffices to indicate a function ψ ∗ ∈ M satisfying condition (14.2) for which ψ(1) = 1 and the quantity η (t) or, by virtue of Theorem 13.3, the quantity R(ψ; t) =

η(η(t)) − η(t) η(t) − t

(14.3)

is unbounded on the set t ≥ 1. Setting ψ(t) = 2x, we get η(t) − t = ψ −1 (x) − ψ −1 (2x), R(ψ; t) =

ψ −1 (x/2) − ψ −1 (x) 1 , x ∈ (0, ]. −1 −1 ψ (x) − ψ (2x) 2

(14.4) (14.5)

Hence, to prove the required statement, it suffices to indicate a function g(x) convex downwards on the interval (0, 1] and such that lim g(x) = ∞,

(14.6)

g(x) − g(2x) < 2,

(14.7)

x→0

and the quantity G(x) =

g(x/2) − g(x) g(x) − g(2x)

(14.8)

is unbounded on (0, 1/2). In this case, ψ ∗ = g −1 (t) is the required function. Indeed, g −1 (t), as the function inverse to a function convex downwards, is also convex downwards for all t ≥ 1 and, by virtue of (14.6), vanishes at infinity, i.e., ψ ∗ ∈ M. Relation (14.2) follows from (14.7) (and, hence, ψ ∗ ∈ M∞ ) . Since R(ψ ∗ ; t) = G(x) for x = ψ ∗ (t)/2, the unboundedness of g(x) in the neighborhood of zero would yield the unboundedness of R(ψ ∗ ; t) at infinity. 14.2. Let ak , k = 0, 1, . . . , be an arbitrary sequence of real numbers monotonically tending to zero, let a0 = 1, and let, for any k ∈ N, the following conditions be satisfied: ak+1 ≤ ak /2 < ak < 2ak < ak−1 .

(14.9)

158

Classes of Periodic Functions

Chapter 3

Further, let ck , k = 0, 1, . . . , be an arbitrary nondecreasing sequence of positive numbers and let ϕ(t) be a function defined by the conditions ϕ(t) = ck , ak+1 ≤ t ≤ ak , k = 0, 1, . . . . We set

(14.10)

1 ϕ(t)dt.

g(x) = 1 +

(14.11)

x

It is easy to see that, for any k = 1, 2, . . . , we have g(ak ) = 1 +

k−1

ci hi , hi = ai − ai+1 ,

(14.12)

i=0

and, with regard for conditions (14.9), g(2ak ) = g(ak ) − ck−1 ak ,

(14.13)

g(ak /2) = g(ak ) + ck ak /2.

(14.14)

Therefore, for k ∈ N, we get G(ak ) =

g(ak /2) − g(ak ) ck . = g(ak ) − g(2ak ) 2ck−1

(14.15)

14.3. Let us verify that the numbers ak and ck can be chosen so that g(x) satisfies conditions (14.9), (14.6), (14.7), and ck = ∞. k→∞ 2ck−1 lim

(14.16)

It is clear that this will prove the required statement. According to the results presented below, these numbers can be chosen in 2 2 numerous ways. Let, e.g., ak = 2−k and ck = 2k , k = 0, 1, . . . . Then 2

and

hk = ak − ak+1 = 2−k (1 − 2−(2k+1) ),

(14.17)

1 2 2 ak = 2−(k +1) > 2−(k+1) = ak+1 , 2

(14.18)

2ak = 2−k

2 +1

2

< 2−(k−1) = ak−1 ,

(14.19)

Section 14

Two Counterexamples

159

i.e., the sequence ak thus chosen satisfies conditions (14.9). At the same time, ck = lim 22(k−1) = ∞. lim k→∞ 2ck−1 k→∞ It remains to verify conditions (14.6) and (14.7). Assume that x ∈ [ak+1 , ak ] for a certain k ≥ 1. Then, by virtue of (14.11), (14.12), and (14.17), we have g(x) ≥ g(ak ) > k, which immediately yields equality (14.6). Furthermore, in this case, according to (14.19), we have 2x ≤ 2ak ≤ ak−1 . Taking (14.12) and (14.17) into account, we get g(x) − g(2x) ≤ g(ak+1 ) − g(ak−1 ) = ck−1 hk−1 + ck hk < 2. 14.4. The first part of the theorem is proved. To prove the second part of the theorem, we also use the function g(x) constructed above. However, in this case, we set ak = 2−N k and ck = 2N k , k = 0, 1, . . . , where N is any number greater then 4. In this case, we have hk = ak − ak+1 = 2−N k (1 − 2−N ),

(14.20)

1 ak = 2−(N k+1) > 2−N (k+1) = ak+1 , 2 and

2ak = 2−N k+1 < 2−N (k−1) = ak−1 ,

i.e., for the values chosen, conditions (14.9) are satisfied and hence, relations (14.12)–(14.14) remain valid. Let us show that, in this case, the quantity G(x) defined by relation (14.8) is bounded on the interval (0, 1/2). Let x be an arbitrary point from (0, 1/2) or, more specifically, let x ∈ [ak+1 , ak ], where k ≥ 1. Then, in view of (14.9) and (14.12)–(14.14), we obtain G(x) = ≤

g(x/2) − g(x) g(x) − g(2x) g(ak+2 ) − g(ak ) ck+1 hk+1 ck+1 hk+1 + ck hk =1+ . = g(ak ) − g(ak−1 ) ck hk ck hk

By using this relation and (14.20), we conclude that G(x) ≤ 2 for any x ∈ (0, 1/2). Therefore, for the function ψ ∗ (t) inverse to g(x), which obviously belongs to M, the quantity R(ψ ∗ ; t) is bounded for all t ≥ 1. This, in turn, means that ψ ∗ ∈ F by virtue of Theorem 13.3. Since, by virtue of (14.20), g(x) − g(2x) ≤ g(ak+1 ) − g(ak−1 ) = ck−1 hk−1 + ck hk ≤ 2,

160

Classes of Periodic Functions

Chapter 3

the function ψ ∗ satisfies condition (14.2). To complete the proof of the theorem, it remains to show that the quantity μ(ψ ∗ ; t) is not monotone or, which is the same, that the function f (x) = g(x)/g(2x) is not monotone on (0, 1/2). Note that the function f (x) is continuously differentiable everywhere in (0, 1/2) except the points ak and ak /2, k = 1, 2, . . . . Therefore, it suffices to indicate the segments of continuous differentiability of the function f (x) on which the expression ϕ(x) = g (x)g(2x) − 2g(x)g (2x) has opposite signs. Since ak /4 = 2−N k−2 ≥ 2−N (k+1) = ak+1 , in this case we have g (x) = g (2x) = −ck . Therefore, the equality ϕ(x) = −ck (g(2x) − 2g(x)) is true. Since g(x) decreases, we have ϕ(x) > 0, which completes the proof of the theorem.

Function ηa (t) and Sets Defined by It

15.

15.1. Let us replace the constant 1/2 in equality (12.1) by an arbitrary constant a ∈ (0, 1) and denote the function η defined by the relation obtained by ηa : (15.1) ψ(ηa (t)) = aψ(t), t ≥ 1. This definition is correct in the sense that, for any ψ ∈ M and all t ≥ 1, the value of ηa (t) is uniquely defined by the equality ηa (t) = ηa (ψ; t) = ψ −1 (aψ(t))

(15.2)

η1/2 (t) = η(t).

(15.3)

and Further, by setting μa (t) = μa (ψ; t) =

t , ηa (t) − t

(15.4)

we can introduce analogs of the set M0 , M∞ , MC , etc., namely, (a)

M0 = {ψ ∈ M : 0 < μa (ψ; t) ≤ K

∀t ≥ 1},

M(a) ∞ = {ψ ∈ M : 0 < K ≤ μa (ψ; t) < ∞ ∀t ≥ 1}, (a)

MC = {ψ ∈ M : 0 < K1 ≤ μa (ψ; t) ≤ K2 (a)+

M0

∀t ≥ 1},

(15.5)

= {ψ ∈ M : μa (ψ; t) ↓ 0}, M(a)+ = {ψ ∈ M : μa (ψ; t) ↑ ∞}, ∞

Section 15

Function ηa (t) and Sets Defined by It

161

and an analog of the set F : Fa = {ψ ∈ M : ηa (ψ; t) ≤ K}.

(15.6)

It is easy to see that Theorems 12.1–13.4 remain valid and their proof, in fact, does not change if we insert the index a in the corresponding places of their formulation and proof. After similar changes, Theorem 14.1 also remains valid. Its proof remains the same if we replace 1/2 in the corresponding places by a number a. In this direction, the following statement is more informative: Theorem 15.1. Let a1 and a2 be arbitrary numbers from the interval (0, 1). Then Fa1 = Fa2 . (15.7) Thus, for a ∈ (0, 1), all sets Fa are equal to one another and, in particular, Fa = F1/2 = F

∀a ∈ (0, 1).

(15.8)

Proof. For example, let ψ ∈ Fa1 . This means that there exists a constant K1 for which a1 ψ (t) ηa 1 (t) = (15.9) ≤ K1 ∀t ≥ 1. ψ (ηa1 (t)) We must prove that, in this case, there exists a constant K2 such that, for every t ≥ 1, a2 ψ (t) ηa 2 = (15.10) ≤ K2 . ψ (ηa2 (t)) If a1 < a2 , then ηa2 (t) < ηa1 (t) and, hence, |ψ (ηa1 (t))| ≤ |ψ (ηa2 (t))|. a2 Therefore, relation (15.10) follows from (15.9) for K2 = K1 . If a2 < a1 , we a1 set t = t0 , t1 = ηa1 (t0 ), t2 = ηa1 (t1 ), . . . , tk+1 = ηa1 (tk ), . . . and denote by n any natural number such that tn ≥ ηa2 (t). On each segment [tk , tk+1 ], k = 0, 1, . . . , in view of (15.9), the value of the derivative ψ (·) increases at most by K1 /a1 times. Therefore, |ψ (tn )| ≥ (K1 /a1 )−n |ψ (t)|. However, |ψ (tn )| ≤ |ψ (ηa2 (t))|. Therefore, K1 n ηa 2 ≤ a2 ( ) , a1 which proves Theorem 15.1. 15.2. Theorem 15.2. In order that, for a given function ψ ∈ M, any a1 , a2 ∈ (0, 1), and all t ≥ 1, the relation K1 ≤

ηa1 (t) − t ≤ K2 , ηa2 (t) − t

ηai (t) = ηai (ψ; t), i = 1, 2,

(15.11)

162

Classes of Periodic Functions

Chapter 3

where the constants K1 and K2 , generally speaking, may depend on the values of a1 and a2 , be true, it is necessary and sufficient that ψ ∈ F. Proof. If relation (15.11) holds, then, in particular, K1 ≤

η1/4 (t) − t ≤ K2 . η1/2 (t) − t

However, η(η(t)) = η1/2 (η1/2 (t)) = η1/4 (t). Therefore, R(ψ, t) =

η1/4 (t) − t η(η(t)) − η(t) = − 1. η(t) − t η1/2 (t) − t

Consequently, R(ψ; t) ≤ K2 − 1. By virtue of Theorem 13.3, this means that ψ ∈ F. The necessity of the conditions of the theorem is proved. If ψ ∈ F, then ψ ∈ Fa1 and ψ ∈ Fa2 by virtue of Theorem 15.1. Therefore, in view of an analog of relation (13.7) for the sets Fa1 and Fa2 , we conclude that, for any t ≥ 1, K1 |ψ (t)|(ηai (t) − t) ≤ ψ(t) ≤ K2 |ψ (t)|(ηai (t) − t),

i = 1, 2,

which yields relation (15.11).

16.

Sets B and M0

16.1. Let c be a certain number that satisfies the condition c > 1. Denote by Bc the set of functions ψ(t) monotonically decreasing for all t ≥ 1 for which there exists a constant K such that, for all t ≥ 1, the following inequality holds: ψ(t) ≤ K. ψ(ct)

(16.1)

If c and c1 are arbitrary numbers greater than the unity, then Bc = Bc1 . Indeed, let ψ ∈ Bc . If c1 ≤ c, then the inequality ψ(t) ≤ K1 ψ(c1 t)

(16.2)

Sets B and M0

Section 16

163

follows from inequality (16.1) for K1 = K because the function ψ decreases. If c1 > c, then, denoting by n any number for which cn ≥ c1 , we get ψ(t) ψ(t) ψ(cn−1 t) ψ(c2 t) ψ(ct) ≤ · . . . ≤ K n. ψ(c1 t) ψ(cn t) ψ(cn−1 t) ψ(c2 t) ψ(ct) df

Thus, all sets Bc with c > 1 coincide and are equal, e.g., to B2 = B. Theorem 16.1. In order that a function ψ ∈ M belong to M0 , it is necessary and sufficient that this function belong to the set B, i.e., M0 = M ∩ B.

(16.3)

Proof. It follows from the definition of the set M0 that if ψ ∈ M0 , then one can find ε > 0 such that, for every t ≥ 1, we have η(ψ; t) − t ≥ε t or η(ψ; t) ≥ (1 + ε)t. Therefore, if ψ ∈ M0 , then ψ(t) ≤ 2, ψ((1 + ε)t) i.e., ψ ∈ Bc for c = (1 + ε) and, hence, ψ ∈ M ∩ B. Now let ψ ∈ M ∩ B. By virtue of (12.24), for every ψ ∈ M, we have η(ψ; t) − t ≥

1 ψ(t) . 2 |ψ (t)|

(16.4)

If, in addition, ψ ∈ B, then there exists a constant K such that, for all t ≥ 1, K≥

ψ(t) ψ(t) − ψ(2t) |ψ (ξ)|t |ψ (2t)| = +1= +1≥ t + 1. ψ(2t) ψ(2t) ψ(2t) ψ(2t)

For all t ≥ 2, this yields t ψ(t) ≥ . |ψ (t)| 2(K − 1) Thus, we can indicate ε > 0 such that, for all t ≥ 1, ψ(t) ≥ εt. |ψ (t)| Substituting this estimate in (16.4), we establish that ψ ∈ M0 .

4. INTEGRAL REPRESENTATIONS OF DEVIATIONS OF POLYNOMIALS GENERATED BY LINEAR PROCESSES OF SUMMATION OF FOURIER SERIES

1.

First Integral Representation (n)

(n)

1.1. Let Λ = λk , k, n = 0, 1, . . . , λ0 = 1, be an arbitrary rectangular numerical matrix. We associate every function f ∈ L whose Fourier series has the form ∞ S[f ] = Ak (f ; x), (1.1) k=0

with the following sequence of series: ∞

Un (f ; x; Λ) =

a0 (n) λk Ak (f ; x), n = 1, 2, . . . . + 2

(1.2)

k=1

In this chapter, we obtain integral representations for the quantities δn (f ; x; Λ) = f (x) − Un (f ; x; Λ)

(1.3)

¯ in the case where f (·) is a ψ-integral of some function ϕ ∈ L. Hence, by virtue of (3.11.2), the following equality is true: S[δn (f ; x; Λ)] =

∞

(n) (1 − λk )(ψ1 (k)Ak (ϕ; x) + ψ2 (k)A˜k (ϕ; x)).

(1.4)

k=1

The first integral representation of the quantity δn (f ; x; Λ) is established in the following statement: 165

166

Integral Representations of Deviations of Polynomials

Chapter 4

Proposition 1.1. Suppose that the series ∞

(ψ1 (k) cos kx + ψ2 (k) sin kx)

(1.5)

(n)

(1.6)

k=1

and

∞

λk (ψ1 (k) cos kx + ψ2 (k) sin kx)

k=1 df (n) (n) are Fourier series, i.e., pairs ψ¯ = (ψ1 , ψ2 ) and λψ¯ = (λk ψ1 (k), λk ψ2 (k)) belong to L. Then, for every function ϕ ∈ L, the following equality holds almost everywhere: π 1 δn (f ; x; Λ) = ϕ(x − t)Kn (t)dt, (1.7) π −π

¯

where f (·) = J ψ (ϕ; ·) and ¯ Λ; t) = Kn (t) = Kn (ψ;

∞

(n)

(1 − λk )(ψ1 (k) cos kt + ψ2 (k) sin kt).

(1.8)

k=1

To prove relation (1.7), it suffices to write the Fourier series for the convolution ϕ ∗ Kn and verify that it coincides with the right-hand side of (1.4). ¯ 1.2. In the general case, if f ∈ Lψ and the pairs ψ¯ and λψ¯ belong to L, in view of Proposition 1.1, equality (1.7) is valid almost everywhere. If, in ¯ ¯ addition, f (·) belongs to C ψ = Lψ ∩ C, Kn (·) belongs to Lp , 1 ≤ p ≤ ∞, and ϕ(·) belongs to Lp , then, in view of Proposition 3.8.1, both sides in (1.7) are continuous functions. Under these conditions, equality (1.7) is valid at any point x. In particular, this is always true if ψ¯ ∈ L, ϕ ∈ M, and Λ is a triangular matrix. (n)

1.3. If Λ = λk , n = 0, 1, . . . , k = 0, 1, . . . , n − 1, is a triangular matrix (n) such that λk ≡ 1, then Un (f ; x; Λ) coincides with the partial sum Sn (f ; x) of the (n − 1)th order of series (1.1). In this case, we set δn (f ; x; Λ) = ρn (f ; x), so that, by definition ρn (f ; x) = f (x) − Sn−1 (f ; x) ∀f ∈ L, and, therefore, Proposition 1.1 takes the following form:

(1.9)

Section 2

Second Integral Representation

167

¯ Proposition 1.1. If ψ¯ ∈ L, then, for any function f ∈ Lψ , the following equality holds almost everywhere:

1 ρn (f ; x) = π here

∞

¯ t) = Fn (ψ;

π

¯ ¯ t)dt, f ψ (x − t)Fn (ψ;

(1.10)

−π

(ψ1 (k) cos kt + ψ2 (k) sin kt).

(1.11)

k=n ¯

¯

¯ ·) ∈ Lp , 1 ≤ p ≤ ∞, and f ψ ∈ Lp , then If, in addition, f ∈ C ψ , Fn (ψ; ¯ equality (1.10) is valid at any point x. In particular, this is true if f ψ ∈ M. ¯ t) in (1.11) is orthogonal to any 1.4. If ψ¯ ∈ L, then the function Fn (ψ; trigonometric polynomial tn−1 (·) of the (n − 1)th degree: π ¯ τ )dτ = 0. tn−1 (τ )Fn (ψ;

(1.12)

−π ¯

Therefore, we can replace f ψ (x − t) in (1.10) by ¯

Δ(x − t) = f ψ (x − t) − tn−1 (x − t),

(1.13)

where tn−1 (·) is an arbitrary polynomial of the (n − 1)th degree. In this case, equality (1.10) has the form 1 ρn (f ; x) = π

π ¯ t)dt. Δ(x − t)Fn (ψ;

(1.14)

−π

It is clear that Proposition 1.1 is valid for it.

2.

Second Integral Representation

2.1. It is often convenient to represent the quantity δn (f ; x; Δ) in another form, namely, as a convolution determined by integration over the entire number axis. Let, as above, M be the set of essentially bounded functions from L : M = {ϕ : ϕ ∈ L, ϕM = esssup|ϕ(·)| < ∞}. Let us prove the following statement:

(2.1)

168

Integral Representations of Deviations of Polynomials

Chapter 4

Lemma 2.1. Suppose that τ1 (v) and τ2 (v) are functions continuous for all v ≥ 0, and their transformations 1 τˆ1+ (t) = π

∞

1 τ1 (v) cos vtdv, τˆ2− (t) = π

0

∞ τ2 (v) sin vtdv

(2.2)

0

are absolutely summable on the entire number axis (ˆ τ1+ , τˆ2− ∈ L(R1 )) : ∞

∞ |ˆ τ1+ (t)|dt < ∞;

−∞

|ˆ τ2− (t)|dt < ∞.

(2.3)

−∞

For any function ϕ ∈ M, consider the convolution ∞ Φτ (x) = (ϕ ∗ τˆ)(x) =

ϕ(x − t)ˆ τ (t)dt,

(2.4)

−∞

where τˆ(t) = τˆ1+ (t) + τˆ2− (t),

(2.5)

and the integral is regarded as the limit of integrals over extending symmetric intervals, i.e., ∞ A = lim . (2.6) −∞

A→∞ −A

This convolution is a continuous 2π-periodic function, ϕ ∗ τ ∈ C, and S [ϕ ∗ τˆ] =

∞

(τ1 (k)Ak (ϕ; x) + τ2 (k)A˜k (ϕ, x)),

(2.7)

k=0

where Ak (ϕ; x) = ak (ϕ) cos kx + bk (ϕ) sin kx, A˜k (ϕ; x) = U Ak (ϕ; x), and ak (ϕ) and bk (ϕ) are the Fourier coefficients of the function ϕ(·). Proof. The inclusion ϕ ∗ τˆ ∈ C is a simple consequence of the fact that ϕ ∈ M and relations (2.3) implying that τˆ ∈ L(R1 ). Therefore, it remains to prove the validity of (2.7). To this end, we calculate the Fourier coefficients αk and βk of function (2.4). The product ϕ(x − t) · τˆ(t) is absolutely integrable

Section 2

Second Integral Representation

169

in the layer x ∈ [−π, π], t ∈ R1 . Therefore, by using the Fubini theorem on the change of the order of integration, we have 1 αk = π

∞

π ϕ(x − t) cos kxdxdt

τˆ(t) −∞

−π

∞

∞ τˆ(t) cos ktdt − bk

= ak −∞

1 βk = π

τˆ(t) sin ktdt,

(2.8)

−∞

∞

π τˆ(t)

−∞

ϕ(x − t) sin kxdxdt

−π

∞

∞ τˆ(t) sin ktdt + bk

= ak −∞

τˆ(t) cos ktdt.

(2.8 )

−∞

Below, we use the following fact from the theory of Fourier integrals: Proposition 2.1. Suppose that γ(v) is a function continuous for all v ≥ 0 and such that its transformations ∞ γˆ+ (t) =

∞ γ(v) cos vtdv, γˆ− (t) =

0

γ(v) sin vtdv 0

are absolutely summable on R1 . Then, at any point v ∈ [0, ∞), we have ∞

∞ γˆ+ (t) cos vtdt =

−∞

and

γˆ− (t) sin vtdt = γ(v)

(2.9)

−∞

∞

∞ γˆ+ (t) sin vtdt =

−∞

γˆ− (t) cos vtdt = 0.

(2.9 )

−∞

In view of equalities (2.5), (2.9), and (2.9 ), we have ∞

∞ τˆ(t) cos ktdt =

−∞

(ˆ τ1+ (t) + τˆ2− (t)) cos ktdt = τ1 (k), −∞

(2.10)

170

Integral Representations of Deviations of Polynomials

Chapter 4

∞ τˆ(t) sin ktdt = τ2 (k).

(2.10 )

−∞

Therefore, returning to relations (2.8) and (2.8 ), we find αk = ak τ1 (k) − bk τ2 (k), βk = ak τ2 (k) + bk τ1 (k). Hence, S[ϕ ∗ τˆ] =

∞

(ak τ1 (k) − bk τ2 (k)) cos kx + (ak τ2 (k) + bk τ1 (k)) sin kx

k=1

=

∞

τ1 (k)Ak (ϕ; x) + τ2 (k)A˜k (f ; x).

k=0

Lemma 2.1 is proved. In what follows, we assume that the systems of numbers ψ1 (k), ψ2 (k), and continuous functions ψ1 (v), ψ2 (v), set of positive integers. By setting τ1 (v) = (1 − λn (v))ψ1 (v), τ2 (v) = (1 − λn (v))ψ2 (v), and (n) λn (k) = λn (k) = λk in Lemma 2.1, we obtain the following statement:

(n) λk , k = 1, 2, . . . , are restrictions of certain and λn (v) of a continuous variable v to the

Theorem 2.1. Suppose that the functions τ1 (v) = (1 − λn (v))ψ1 (v)

and τ2 (v) = (1 − λn (v))ψ2 (v),

λn (k) = λn (k) = λk

(n)

(2.11)

¯ satisfy all conditions of Lemma 2.1. If a function f (·) is a ψ-integral (ψ¯ = (ψ1 , ψ2 )) of some ϕ ∈ M, then the integral ∞ ϕ(x − t)ˆ τn (t)dt,

Φn (f ; τˆn ; x) =

(2.12)

−∞

where 1 τˆn (t) = π

∞ ((1 − λn (v))ψ1 (v) cos vt + (1 − λn (v))ψ2 (v) sin vt)dv, (2.13) 0

Section 2

Second Integral Representation

171

is a continuous 2π-periodic function. Moreover, S[Φn (f ; τˆn ; x)] =

∞

(n)

(1 − λk )(ψ1 (k)Ak (ϕ; x) + ψ2 (k)A˜k (ϕ; x))

k=1

= S[δn (f ; x; Λ)].

(2.14)

Now consider convolution (2.4) in the case where ϕ ∈ L. It is clear that, in this case, conditions (2.3) do not even guarantee the existence of integrals (2.4) at any point x ∈ R1 . However, these conditions are sufficient for the convolution Φτ (x) to exist almost everywhere and to be a summable (and, obviously, 2πperiodic) function, because π

π ∞ |Φτ (x)|dx ≤

−π

∞ |ϕ(x − t)ˆ τ (t)|dtdx = ϕ1

−π −∞

|ˆ τ (t)|dt.

(2.15)

−∞

The last equality is valid because the order of integration of nonnegative functions is not essential. It follows from (2.15) that the product ϕ(x − t)ˆ τ (t) is absolutely integrable in the layer x ∈ [−π, π], t ∈ R1 . Therefore, evaluating Fourier coefficients of the function Φτ (x), in view of Proposition 2.1, we arrive at the following statements: Lemma 2.1. Suppose that the functions τi (v), i = 1, 2, satisfy all conditions of Lemma 2.1. Then, for every ϕ ∈ L, convolution (2.4) represents a function from L for which equality (2.7) holds. Theorem 2.1. Suppose that the functions τ1 (v) and τ2 (v) in (2.11) satisfy ¯ all conditions of Lemma 2.1. Then, for any function f (·) that is a ψ-integral of some function ϕ from L, function (2.12) belongs to L and equality (2.14) holds. ¯

2.2. If f ∈ Lψ M and the conditions of Theorem 2.1 are satisfied, then, for any n ∈ N, the functions Φn (f ; τn ; x) belong to C. Therefore, the left-hand side of (2.14) is the Fourier series of a continuous function. Hence, if f (·) and Un (f ; x; Λ) are also continuous, then, by virtue of (2.14), at any point x ∈ R1 , the following equality holds: ∞ f (x) − Un (f ; x; Λ) =

¯

f ψ (x − t)ˆ τn (t)dt.

(2.16)

−∞ ¯

¯

Consequently, by setting C ψ M = Lψ M ∩ C, we obtain from Theorem 2.1 the following statement:

172

Integral Representations of Deviations of Polynomials

Chapter 4

¯

Theorem 2.2. Suppose that f ∈ C ψ M and the functions τi (·), i = 1, 2, determined by equalities (2.11) satisfy all conditions of Lemma 2.1. Then, at any point x ∈ R1 , equality (2.16) is valid if Un (f ; ·; Λ) ∈ C (which is always true if Λ is a triangular matrix). Theorem 2.1 yields the following consequence: Theorem 2.2. Suppose that f ∈ Lψ and τî (·), i = 1, 2, satisfy all conditions of Lemma 2.1. Then equality (2.16) holds almost everywhere on R1 . ¯

2.3. Consider one more fact which follows from Lemma 2.1 . Namely, the following statement is true: Proposition 2.2. Suppose that the functions ψ1 (v) and ψ2 (v) are continuous for all v ≥ 0 and their transformations 1 ψˆ1+ (t) = π

∞

1 ψ1 (v) cos vtdv, ψˆ2− (t) = π

0

∞ ψ2 (v) sin vtdv

(2.17)

0

are absolutely integrable on

R1 .

Consider the convolution ∞

ˆ Φψ (x) = (ϕ ∗ ψ)(x) =

ˆ ϕ(x − t)ψ(t)dt,

(2.18)

−∞

where ˆ = ψˆ1+ (t) + ψˆ2− (t) ψ(t)

(2.19)

and the integral is regarded in the sense of (2.6). This convolution is a function from L and S[Φψ ] =

∞

(ψ1 (k)Ak (ϕ; x) + ψ2 (k)Ak (ϕ; x)).

(2.20)

k=0

This implies, in particular, that if the function ψi (v) satisfies the conditions of this proposition, then the following equality holds: ψ¯

∞ ˆ ϕ(x − t)ψ(t)dt + A0 ,

J (ϕ; x) = Φψ (x) =

(2.21)

−∞ ¯ ¯ where J ψ (ϕ; x) is the ψ-integral of the function ϕ(·) and A0 is a certain constant.

Section 3

3.

Representation of Deviations of Fourier Sums

173

Representation of Deviations of Fourier Sums ¯ ¯ on Sets C ψ M and Lψ

3.1. First, note that, for the validity of representation (2.18) for a given sequence Un (f ; x; Λ) with the functions τ1 (v) and τ2 (v) chosen according to (2.11), the principal condition is the continuity of these functions for v ≥ 0 and the fact that their transformations (2.2) belong to L(R1 ). It is clear that such functions are not unique. One may use this fact and choose τ1 (v) and τ2 (v) so that the right-hand of (2.18) has the most convenient form for investigation. In particular, in order to obtain the integral representation for the deviations of the Fourier sums a0 Sn−1 (f ; x) = (ak cos kx + bk sin kx) + 2 n−1 k=1

¯

from f ∈ C ψ M, it suffices to take any continuous functions τ1 (v) and τ2 (v) satisfying (2.3) and such that 0, k < n, τi (k) = (3.1) ψi (k), k ≥ n, i = 1, 2. Following this reasoning, we obtain integral representations of the quantities ρn (f ; x) = f (x) − Sn−1 (f ; x)

(3.2)

¯ ¯ for the cases where f ∈ C ψ M and f ∈ Lψ and the pairs ψ¯ = (ψ1 , ψ2 ) are such that ψ1 ∈ M (or −ψ1 ∈ M ) and ψ2 ∈ M (or −ψ2 ∈ M ) and satisfy the following condition: ∞ |ψ2 (k)|/k < ∞. (3.3) k=1

Note that the conditions imposed below on the functions ψ1 and ψ2 are nonsymmetric. This is explained by well-known fact (see Theorem 3.7.3) that condition (3.3) with ψ2 ∈ M, is necessary (and, obviously, sufficient) for the series ∞ ∞ ψ2 (k) sin kt to be a Fourier series, while the series ψ1 (k) cos kt is alk=1

k=1

ways a Fourier series for ψ1 ∈ M. In what follows, as above, we denote by M the set of continuous positive functions ψ(·) which are convex downwards for all v ≥ 1 and lim ψ(v) = 0.

v→∞

(3.4)

174

Integral Representations of Deviations of Polynomials

Chapter 4

By M we denote the subset of functions ψ(·) from M that satisfy, in addition, the following condition: ∞ |ψ(t)| dt < ∞. (3.5) t 1

The notation ±ϕ ∈ A means that either ϕ ∈ A or −ϕ ∈ A. It is clear that if ±ψ2 ∈ M , then the values of |ψ2 (k)| satisfy relation (3.3).

3.2. First, we present the following auxiliary statement: Lemma 3.1. Let ±ψ1 ∈ M, ±ψ2 ∈ M , and let n be an arbitrary positive integer, c ∈ [0, n), and

τi (c; v) =

⎧ ⎪ 0, ⎪ ⎪ ⎪ ⎪ ⎪ ⎨v − c

0 ≤ v ≤ c,

ψi (n), c ≤ v ≤ n, ⎪ n−c ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ψ (v), v ≥ n, i = 1, 2. i

(3.6)

Then the transformations 1 τˆ1+ (c; t) = π

∞ τ1 (c; v) cos vtdv

(3.7)

τ2 (c; v) sin vtdv

(3.7 )

0

and 1 τˆ2− (c; t) = π

∞ 0

are absolutely integrable on R1 : ∞

∞ |ˆ τ1+ (c; t)|dt < ∞,

−∞

|ˆ τ2− (c; t)|dt < ∞. −∞

(3.8)

Section 3

Representation of Deviations of Fourier Sums

175

Proof. Considering equality (3.6), we get 1 τˆ1+ (c; t) = π

∞ τ1 (c; v) cos vtdv 0

ψ1 (n) = π

n c

v−c 1 cos vtdv + n−c π

∞ ψ1 (v) cos vtdv n

df

= J1 (ψ1 , t, c)0 + J2 (ψ1 , n; t)0 .

(3.9)

By integrating by parts, we find J1 (ψ1 , t, c)0 =

=

ψ1 (n) sin nt 2 sin(n + c)t sin((n − c)t/2) ] [ − π t (n − c)t2

ψ1 (n) (n − c)t − sin(n − c)t 1 − cos(n − c)t sin nt + cos nt]. [ 2 π (n − c)t (n − c)t2

(3.10)

(3.11)

It follows from (3.10) that J1 (ψ1 ; t; c)0 = O(1)ψ1 (n), t → 0,

(3.12)

where O(1) is a quantity uniformly bounded in t. This implies that the function J1 (ψ1 ; t; c)0 is summable in the neighborhood of origin. Let us show that the function J2 (ψ1 ; t)0 is also of this sort. For any t ∈ R1 , we have 1 J2 (ψ1 ; n; t)0 = π

∞

ψ1 (n) sin nt 1 ψ1 (v) cos vtdt = − − πt πt

n df

=−

ψ1 (n) sin nt 1 − J3 (ψ1 ; t)0 . πt π

∞

ψ (v) sin vtdv

n

(3.13)

The first term on the right-hand side in (3.13) is bounded in the neighborhood of origin. Therefore, it suffices to prove that the function J3 (ψ1 ; t)0 is summable near the point t = 0. Now, for definiteness, we consider that the function ψ1 (v) is positive, i.e., that ψ1 ∈ M. In this case, let −ψ1 (n), 0 < v ≤ n, ϕn (v) = (3.14) −ψ1 (v), v ≥ n.

176

Integral Representations of Deviations of Polynomials

Chapter 4

For any fixed n, the function ϕn (v) does not increase, and, hence, df

∞ ϕn (v) sin vtdv ≥ 0 ∀t ≥ 0.

Φn (t) =

(3.15)

0

However,

2ψ1 (n) nt (3.16) sin2 − tJ3 (ψ; t)0 . t 2 Thus, if J3 (ψ1 ; t)0 ≥ 0 at some point t > 0, then, by virtue of (3.15) and (3.16), Φn (t) = −

J3 (ϕ; t)0 ≤ −

2ψ (n) nt sin2 < −n2 ψ (n)/2. 2 t 2

(3.17)

For every a > 0, let us decompose the interval (0, a) into the sets e+ and e− , by setting e+ = {t : t ∈ (0, a), J3 (ψ1 ; t) ≥ 0}, e− = (0, a) \ e+ . Then a

a |J3 (ψ1 ; t)0 |dt = 2

−a

|J3 (ψ1 ; t)0 |dt 0

a

J3 (ψ1 ; t)0 dt −

= 2(2 e+

J3 (ψ1 ; t)0 dt).

(3.18)

0

Estimate (3.17) is valid on the set e+ , thus, 4 J3 (ψ1 ; t)0 dt ≤ 2a|ψ (n)|n2 .

(3.19)

e+

By changing the order of integration, we find a

∞ J3 (ψ1 ; t)0 dt| = 2|

2|

ψ1 (v)

n

0

∞ ≤2

av

sin t dtdv| t

0

|ψ1 (v)| · (

π + siav)dv < 2πψ1 (n), 2

(3.20)

sin t dt t

(3.21)

n

where

∞ six = − x

is the integral sine.

Section 3

Representation of Deviations of Fourier Sums

177

By combining relations (3.18)–(3.20), we get a

|J3 (ψ1 ; t)0 |dt ≤ 2πψ1 (n) + 2a|ψ1 (n)|n2

(3.22)

−a

for any ψ1 ∈ M and for any a > 0. Thus, the function J2 (ψ1 ; t)0 (as well as J1 (ψ1 ; t; c)0 ) is summable in any neighborhood of origin. Consequently, by virtue of (3.9), the function τˆ1+ (c; t) is also of this sort and it is continuous for all |t| > 0. Therefore, in order to prove the first relation of (3.8), it suffices to show the summability of τ1 (c; t) for large values of |t|. 3.3. It follows from relations (3.9), (3.10), and (3.13) that τˆ1+ (c; t) = − Let us show that

2ψ1 (n) 1 sin(n + c)t sin((n − c)t/2) − J3 (ψ1 ; t)0 . (3.23) 2 π(n − c)t π J3 (ψ1 ; t)0 = O(1)t−2 , t → ∞.

(3.24)

For this purpose, we consider the function ∞ Φ(x) = Φβ (ϕ; x) =

ϕ(v) sin(vt +

βπ )dt, x > 0, t > 0, 2

(3.25)

x

where ϕ(·) is nonnegative and nonincreasing function for all v ≥ 1 and β is any real number. The function Φ(x) is continuous for every fixed t and β. On every interval between the consecutive zeros vk and vk+1 of the function sin(vt + βπ/2), the function Φ(x) has one simple zero xk . Indeed, it is obvious that this function is continuous and existence of zeros xk is a consequence of the Leibniz theorem on alternating series: v ∞ i+1 sign Φ(vk ) = sign ϕ(v) sin(vt + βπ/2)dv i=k vi vk+1

= sign

ϕ(v) sin(vt + βπ/2)dv vk

= sign sin(ξt + βπ/2), ξ ∈ (vk , vk+1 ).

(3.26)

178

Integral Representations of Deviations of Polynomials

Chapter 4

The fact that the zero xk is unique on the interval (vk , vk+1 ) follows from the equality sign Φ (x) = −sign sin(xt + βπ/2). For zeros xk , the following condition is satisfied: (kπ − βπ/2)/t ≤ xk ≤ ((k + 1)π − βπ/2)/t.

(3.27)

Thus, assuming that xk is the zero closest from the right to the point n, we have n ≤ xk ≤ n + 2π/t.

(3.28)

In view of this, by setting ϕ(v) = ψ1 (v) and β = 0 in (3.25), we find 1 |J3 (ψ1 ; t)0 | = | t

∞

1 ψ1 (v) sin vtdv| = | Φ0 (ψ1 ; n)| t

n

1 =| t

xk

ψ1 (v) sin vtdv|

1 ≤| t

n+2π/t

n

≤

ψ1 (v)dv|

n

2π|ψ1 (n)|/t2 .

(3.29)

This proves relation (3.24). Combining (3.23) and (3.24), we get τ1+ (c; t) = O(1)t−2 , t → ∞,

(3.30)

which yields the summability of the function τ1+ (c; t) for large values of |t|. 3.4. Let us prove the second relation in (3.8). Relation (3.6) yields ψ2 (n) τˆ2− (c; t) = π

n c

v−c 1 sin vtdv + n−c π

∞ ψ2 (v) sin vtdv n

df

= J1 (ψ2 ; t; c)1 + J2 (ψ2 ; n; t)1 .

(3.31)

Since J1 (ψ2 ; t; c)1 = =−

ψ2 (n) cos nt 2 cos(n + c)t sin((n − c)t/2) [− + ] π t (n − c)t2

ψ2 (n) (n − c)t − sin(n − c)t 1 − cos(n − c)t cos nt + sin nt], [ 2 π (n − c)t (n − c)t2

(3.32) (3.33)

Section 3

Representation of Deviations of Fourier Sums

179

relation (3.32) yields J1 (ψ2 ; t; c)1 = O(1)ψ2 (n), t → 0.

(3.34)

Setting ϕ(v) = ψ2 (v) and β = 0 in (3.25), by virtue of (3.27) and (3.28) we obtain 1 |Φ0 (ψ2 ; n)| π

|J2 (ψ2 ; n; t)1 | =

1 = | π

xk

1 ψ2 (v)dv| < π

n+2π/t

n

|ψ2 (v)|dv.

(3.35)

n

Therefore, for any a > 0, in view of (3.5) we get a −a

2 |J2 (ψ2 ; n; t)1 |dt ≤ π

a

n+2π/t

|ψ2 (v)|dvdt n

0

2 = (t π

n+2π/t

a

|ψ2 (v)|dv

|a0

+ 2π

n

|ψ2 (n + 2π/t)| dt) t

0

∞ < 4(|ψ2 (n)| + π

|ψ2 (t + n)| dt) t

2π/a

= O(1)(ψ2 (n) + 1).

(3.36)

Relations (3.34) and (3.35) yield the summability of the function τˆ2− (c; t) in any neighborhood of origin. Let us verify its summability for large values of |t|. For any t ∈ R, we have 1 J2 (ψ2 ; n; t)1 = π

∞ ψ2 (v) sin vtdv n

ψ2 (n) cos nt 1 = + πt πt

∞

ψ2 (v) cos vtdv

n df

=

ψ2 (n) cos nt 1 + J3 (ψ2 ; t)1 . πt π

(3.37)

180

Integral Representations of Deviations of Polynomials

Chapter 4

Thus, in view of (3.31) and (3.32), τˆ2− (c; t) =

2ψ2 (n) 1 + J3 (ψ2 ; t)1 . 2 π(n − c)t π

(3.38)

It is now necessary to obtain J3 (ψ2 ; t)1 = O(1)t−2 , t → ∞.

(3.39)

Setting ϕ(v) = ψ2 (v) and β = 1 in (3.25), we get 1 |J3 (ψ2 ; t)1 | = | Φ1 (ψ2 ; n)|. t

(3.40)

Therefore, following the derivation of estimate (3.29), we obtain |J3 (ψ2 ; t)1 | ≤ 2π|ψ1 (n)|/t2 .

(3.41)

Relation (3.39) is established. By comparing (3.38) and (3.41), one can verify the summability of τˆ2− (c; t) for large values of |t|. Therefore, relations (3.8) are proved. 3.5. In Lemma 3.1, let c = n − 1. Then, according to (3.6), ⎧ 0, 0 ≤ v ≤ n − 1, ⎪ ⎪ ⎨ τi (n − 1, v) = (v − n + 1)ψi (n), n − 1 ≤ v ≤ n, ⎪ ⎪ ⎩ ψi (v), v ≥ n, i = 1, 2 and, hence, τi (n − 1, k) =

0,

k < n,

ψi (k), k ≥ n, i = 1, 2.

(3.42)

(3.42 )

Thus, condition (3.1) is satisfied, and, consequently, the left-hand side in (2.18) is the quantity ρn (f ; x) = f (x) − Sn−1 (f ; x). Therefore, by virtue of Theorems 2.2 and 2.2 and Lemma 3.1, we arrive at the following statement: ¯

Theorem 3.1. If f ∈ C ψ M, ±ψ1 ∈ M, and ±ψ2 ∈ M , then, at any point x, the following equality holds: ∞

¯

f ψ (x − t)ˆ τn (t)dt,

ρn (f ; x) = −∞

(3.43)

Section 3

Representation of Deviations of Fourier Sums

181

here 1 τˆn (t) = π

∞ (τ1 (n − 1, v) cos vt + τ2 (n − 1, v) sin vt)dv,

(3.44)

0

and τi (n − 1, v), i = 1, 2, are determined by formula (3.42). ¯ However, if f ∈ Lψ and, as above, ±ψ1 ∈ M and ±ψ2 ∈ M , then equality (3.43) holds almost everywhere. Let Fn be the set of trigonometric polynomials of degree ≤ n. If tn−1 (·) ∈ Fn−1 , then, taking into account equalities (3.42 ) and (2.7), we have ∞ tn−1 (x − t)ˆ τn (t)dt = 0.

(3.45)

−∞

Therefore, equality (3.43) can be rewritten as ∞

¯

(f ψ (x − t) − tn−1 (x − t))ˆ τn (t)dt,

ρn (f ; x) =

(3.46)

−∞

where tn−1 (·) is any function from Fn−1 . In particular, the following equality holds: ∞ ¯ ¯ ρn (f ; x) = (f ψ (x − t) − f ψ (x))ˆ τn (t)dt. (3.46 ) −∞

By virtue of (3.42), 1 π

∞ τ1 (n − 1, v) cos vtdv 0

ψ1 (n) = π

n

1 (v − n + 1) cos vtdv + π

n−1

∞ ψ1 (v) cos vtdv n

df

= J1 (n, t)0 + J2 (ψ1 , n, t)0 .

(3.47)

Taking the first integral, we find J1 (n, t)0 =

ψ1 (n) t − sin t 1 − cos t sin nt + cos nt). ( π t2 t2

(3.48)

182

Integral Representations of Deviations of Polynomials

Chapter 4

Similarly, we have 1 π

∞ τ2 (n − 1, v) sin vtdv = J1 (n, t)1 + J2 (ψ2 , n, t)1 ,

(3.49)

ψ2 (n) t − sin t 1 − cos t cos nt + sin nt) (− π t2 t2

(3.50)

0

where J1 (n, t)1 = and 1 J2 (ψ2 , n, t)1 = π

∞ ψ2 (v) sin vtdv.

(3.51)

n

3.6. The following lemma will be useful: Lemma 3.2. Suppose that ϕ(·) is an arbitrary summable 2π-periodic function ( ϕ ∈ L ). Then ∞ t − sin t ϕ(t) dt = 0, (3.52) t2 −∞

∞ −∞

1 − cos t 1 ϕ(t) dt = t2 2

π ϕ(t)dt,

(3.53)

−π

where the integrals are understood in the sense of their principal values. Proof. Let γ(t) = (t − sin t)/t2 , then ∞

−π π a ϕ(t)γ(t)dt = lim [ ϕ(t)γ(t)dt + ϕ(t)γ(t)dt + ϕ(t)γ(t)dt]. a→∞

−∞

−a

−π

π

It is clear that the quantity a can be chosen in the form a = (2p + 1)π, p = 1, 2, . . . . Therefore, a ϕ(t)γ(t)dt =

p

(2k+1)π

π

ϕ(t)γ(t)dt =

k=1(2k−1)π

π

π =

ϕ(t) π

p

k=1−π

p t + 2kπ − sin t k=1

(t + 2kπ)2

dt.

ϕ(t)γ(t + 2kπ)dt

Section 3

Representation of Deviations of Fourier Sums

183

By analogy, we have −π π p t − 2kπ − sin t ϕ(t)γ(t)dt = dt. (t − 2kπ)2 −π k=1

−a

Thus, ∞

π ϕ(t)γ(t)dt = lim

p→∞ −π

−∞

1 1 1 + ) ϕ(t)[ + ( t t + 2kπ t − 2kπ p

k=1

1 1 1 + ( + ))]dt. t2 (t + 2kπ)2 (t − 2kπ)2 p

− sin t(

k=1

The multiplier γ(t, p) of the function ϕ(t) under the integral sign on the righthand side of this equality is uniformly bounded in t and p. Therefore, taking the limit and using the well-known decompositions ∞

1 1 1 1 ( = + + ) 2 tan(t/2) t t + 2kπ t − 2kπ

(3.54)

k=1

and

∞

1 1 1 1 ( + ), = 2+ 2 t (t + 2kπ)2 (t − 2kπ)2 4 sin (t/2) k=1

we get

(3.55)

π lim

p→∞ −π

ϕ(t)γ(t, p)dt = 0,

which is equivalent to relation (3.52). Equality (3.53) is established by analogy. 3.7. In view of equalities (3.48), (3.50), (3.52), and (3.53), for every ϕ ∈ L we obtain ∞ π ψ1 (n) ϕ(t)J1 (n, t)0 dt = ϕ(t) cos ntdt, (3.56) 2π −∞ ∞

ϕ(t)J1 (n, t)1 dt = ∞

In particular, if we set ϕ(t) = (3.56) and (3.57) yield

ψ2 (n) 2π

−π π

ϕ(t) sin ntdt.

(3.57)

−π

¯ f ψ (x − t) − tn−1 (x − t)

for all x, then equalities

184

Integral Representations of Deviations of Polynomials ∞

Chapter 4

¯

(f ψ (x − t) − tn−1 (x − t))(J1 (n, t)0 + J1 (n, t)1 )dt −∞

1 = 2π

π

¯

(f ψ (x − t) − tn−1 (x − t)) · (ψ1 (n) cos nt + ψ2 (n) sin nt)dt. (3.58) −π

By analogy, we get ∞

¯

¯

(f ψ (x − t) − f ψ (x))(J1 (n, t)0 + J1 (n, t)1 )dt −∞

π =

¯

¯

(f ψ (x − t) − f ψ (x))(ψ1 (n) cos nt + ψ2 (n) sin nt)dt. (3.58 )

−π

Combining the statement of Theorem 3.1, equalities (3.46), (3.46 ), (3.47), (3.49), (3.58), and (3.58 ) and setting ¯

¯

δn (u) = δn (f ψ ; tn−1 , u) = f ψ (u) − tn−1 (u), ¯

δ0 (u) = δ0 (f ψ ; u) = f ψ (u) − f ψ (u + t), t ∈ R1 ,

(3.59) (3.59 )

we arrive at the following statement: ¯

Corollary 3.1. If f ∈ C ψ M, ±ψ1 ∈ M, and ±ψ2 ∈ M , then, at any point x ∈ R1 , the following equality holds: ∞ Δ(x − t)(J2 (ψ1 ; n; t)0 + J2 (ψ2 ; n; t)1 )dt

ρn (f ; x) = −∞

1 + 2π

π Δ(x − t)(ψ1 (n) cos nt + ψ2 (n) sin nt)dt, (3.60) −π

¯

where Δ(u) is either f ψ (u) or δn (u) from (3.57), or δ0 (u) from (3.57 ), 1 J2 (ψ1 ; n; t)0 = π

∞ ψ1 (v) cos vtdv, n

(3.61)

Section 3

Representation of Deviations of Fourier Sums 1 J2 (ψ2 ; n; t)1 = π

∞ ψ2 (v) sin vtdv.

185

(3.61 )

n ¯ Lψ

If f ∈ and, as above, ±ψ1 ∈ M and ±ψ2 ∈ M , then equality (3.60) is valid almost everywhere.

5. APPROXIMATION BY FOURIER SUMS IN SPACES C AND L1 In this chapter, we present results related to the estimation of the quantities ρn (f ; x)X = f (x) − Sn−1 (f ; x)X ,

(0.1)

where X denotes either C or L = L1 . If, in addition, X = C, then the ¯ ψ¯ and C ψ Hω0 defined as follows: functions f (·) are chosen from the classes C∞ ¯

¯

¯

0 ψ = {f : f ∈ C ψ , f ψ ∈ SM }, C∞ ψ¯

ψ¯

(0.2)

ψ¯

C Hω0 = {f : f ∈ C , f ∈ Hω0 }, 0 SM

where is the unit sphere in the space and orthogonal to a constant, i.e.,

M0

(0.3)

of functions essentially bounded π

0 SM

Hω0

= {ϕ : ϕM = ess sup |t| ≤ 1, t

ϕ(t)dt = 0},

(0.4)

−π

is the class of 2π-periodic functions ϕ(·) satisfying the condition |ϕ(t) − ϕ(t )| ≤ ω(|t − t |) ∀t, t ,

(0.5)

π ϕ(t)dt = 0, −π

and ω = ω(t) is a fixed modulus of continuity. ¯ ¯ If X = L1 , then functions f (·) in (0.1) belong to Lψ S10 and Lψ Hω01 : ¯

¯

¯

Lψ S10 = {f : f ∈ Lψ , f ψ ∈ S10 π = {ϕ : ϕ1 =

π |ϕ(t)|dt ≤ 1,

−π

187

ϕ(t)dt = 0}}, (0.6) −π

Approximation by Fourier Sums in Spaces C and L1

188 ¯

¯

Chapter 5

¯

Lψ Hω01 = {f : f ∈ Lψ , f ψ ∈ Hω01 π = {ϕ :

π |ϕ(x + t) − ϕ(x)|dx ≤ ω(t)}

ϕ(t)dt = 0}, (0.7)

−π

π

and ω = ω(t) is a fixed modulus of continuity. ¯

¯

ψ Note that, by virtue of Proposition 3.11.2, the sets C∞ and C ψ Hω0 consist ¯ 0 0 ¯ of ψ-integrals of all functions from SM and Hω , respectively. The sets Lψ S1 ¯ ¯ and Lψ Hω1 consist of ψ-integrals of functions from S1 and Hω1 , respectively. The quantities

En (N)X = sup ρn (f ; x)X , f ∈N

(0.8)

where N is one of the mentioned classes, are the main subject of our investigation aimed at obtaining asymptotic equalities for them, i.e., equalities of the form En (N)X = ϕ(n) + o(ϕ(n)),

(0.9)

where ϕ(n) = ϕ(N; n; X) is a certain function of a natural variable. If such an equality is obtained, we say that the Kolmogorov–Nikol’skii problem is solved in a given metric of the space X for a given class of functions N. Here, we find an analog of the well-known Lebesgue inequality with exact ¯ ¯ constant of the principal term for the sets C ψ C and Lψ : ρn (f ; x)C ≤

4 (ln n + K)En (f )C , ϕC = max |ϕ(x)|, x π2

(0.10),

where K is a certain constant and En (f )C is the quantity of the best approximation of a function f (·) by trigonometric polynomials of the (n − 1)th degree: En (f )C = inf f (·) − tn−1 (·)C . tn−1

(0.11)

The material is presented in ascending order of smoothness of functions. First, we ¯ consider the approximation of functions from the sets Lψ for ±ψ1 , ±ψ2 ∈ M0 and then for ±ψ1 , ±ψ2 ∈ F. Finally, we examine the case where elements of the ¯ sets Lψ are restrictions of entire functions to the real axis.

Simplest Extremal Problems in Space C

Section 1

1.

189

Simplest Extremal Problems in Space C

1.1. This section plays an auxiliary role. Here, we present a method for estimating integrals of the form b f (t)ϕ(t)dt,

(1.1)

a

where ϕ(·) is a fixed function and f (·) runs through a certain set of functions. We begin with the simplest but very important case. Assume that, on an arbitrary segment [a, b], a summable function ϕ(·) is given ( ϕ ∈ L(a, b) ) which is symmetric with respect to c = (a + b)/2, i.e., ϕ(t) = −ϕ(2c − t).

(1.2)

Denote the set of such functions ϕ(·) by Va,b . Further, let Hω [a, b] be the class of functions satisfying equality (0.5) for any t and t from [a, b] and let b Eω (ϕ) = Eω (ϕ; Hω [a, b]) =

sup f ∈Hω [a,b]

|

f (t)ϕ(t)dt|.

(1.3)

a

The problem is to determine the exact value of Eω (ϕ) or, at least, to find its upper and lower bounds unimprovable in a certain sense. As a rule, problems of this sort are solved in two steps: First, one establishes the required upper estimate, and then a function f ∗ (·) from a given class is indicated for which this estimate is unimprovable. In this case, the function f ∗ (·) is called extremal for a given problem. In the case under consideration, we establish the required upper estimate as follows: We decompose the integral over the interval (a, b) into two integrals over the intervals (a, c) and (c, b), respectively, and change the variables z = 2c − t in the second integral. By virtue of equality (1.2), we get b |

c f (t)ϕ(t)dt| = |

a

(f (t) − f (2c − t))ϕ(t)dt|. a

This implies that, for any f ∈ H[a, b], b |

c f (t)ϕ(t)dt| ≤

a

ω(2c − 2t)|ϕ(t)|dt. a

(1.4)

Approximation by Fourier Sums in Spaces C and L1

190

Consequently,

Chapter 5

c Eω (ϕ) ≤

ω(2c − 2t)|ϕ(t)|dt.

(1.5)

a

This is just the required upper estimate. Further, we set −ω(2c − 2t)/2, t ∈ [a, c], f ∗ (t) = ω(2t − 2c)/2, t ∈ [c, b].

(1.6)

For this function, we have b |

f ∗ (t)ϕ(t)dt| =

a

c

b ω(2c − 2t)|ϕ(t)|dt =

a

Therefore, in the case of

(1.7)

c

f∗

∈ Hω [a, b], equality (1.7) yields

c

b ω(2c − 2t)|ϕ(t)|dt =

Eω (ϕ) =

ω(2t − 2c)|ϕ(t)|dt.

a

ω(2t − 2c)|ϕ(t)|dt.

(1.8)

c

If ω = ω(t) is a convex function, then the inclusion f ∗ ∈ Hω [a, b] holds. Indeed, let t and t be arbitrary points from [a, b]. If both these points lie on the same side of the point c (e.g., c ≤ t < t ), then, by semiadditivity of a modulus of continuity (see Subsection 3.1.3), we have 1 |f ∗ (t ) − f ∗ (t )| = (ω(2t − 2c) − ω(2t − 2c)) 2 1 ≤ ω(2t − 2t ) ≤ ω(t − t ). 2

(1.9)

Consider the case where t < c < t . We set t − t = 2h and t0 = c − h. For definiteness, we assume that t ≤ t0 . Then, obviously, t ≤ c + h and t0 − t = c + h − t . Since the function f ∗ (·) increases on the segment [a, b], we have |f ∗ (t ) − f ∗ (t )| = (f ∗ (c + h) − f ∗ (t0 )) + (f ∗ (t0 ) − f ∗ (t )) + (f ∗ (t ) − f ∗ (c + h)) c−h c+h ∗ f (τ )dτ − f ∗ (τ )dτ. = ω(2h) + t

t

(1.10)

Simplest Extremal Problems in Space C

Section 1

191

By virtue of the convexity of the function ω(t), the derivative f ∗ (·) does not decrease on the interval (a, c) and does not increase on the interval (c, b). Therefore, c−h c+h ∗ f (τ )dτ − f ∗ (τ )dτ t

t

≤ f ∗ (c − h)(c − h − t ) − f ∗ (c + h)(c + h − t ) = 0. Thus, in this case, we also have |f ∗ (t ) − f ∗ (t )| ≤ ω(2h) = ω(t − t ). Hence, the following statement is true: Lemma 1.1. Suppose that ϕ ∈ Va,b and ω = ω(t) is an arbitrary convex modulus of continuity. Then b Eω (ϕ) =

sup f ∈Hω [a,b]

|

(a+b)/2

ω(a + b − 2t)|ϕ(t)|dt

f (t)ϕ(t)dt| = a

a

b ω(2t − a − b)|ϕ(t)|dt.

=

(1.11)

(a+b)/2

The upper bound in this equality is realized by any function of the form K ±f ∗ (·), where K is an arbitrary constant and f ∗ (·) is defined by equality (1.6). 1.2. In the case where ω(t) is not convex, the explicit value of Eω (ϕ) is unknown. Although the function f ∗ (t) in (1.6) still satisfies equality (1.7) in this case, it does not necessarily belong to the class Hω [a, b]. Nevertheless, it can be shown that, for any modulus of continuity, the function f∗ (t) = 23 f ∗ (t) belongs to the class Hω [a, b]. Indeed, if t and t lie on the same side of the point c, then inequality (1.9) always holds. However, if, e.g., t ∈ [a, c] and t ∈ [c, b], then, by virtue of the monotonicity of the function f ∗ (·), we have 1 |f∗ (t ) − f∗ (t )| = (ω(2t − 2c) + ω(2c − 2t )) 3 1 df = (ω(2t − c) + ω(2(t − t ) + 2(c − t ))) = Δ(ω). 3

Approximation by Fourier Sums in Spaces C and L1

192

Chapter 5

Thus,

1 1 Δ(ω) ≤ ω(2(t − t )) + ω(t − t ) ≤ ω(t − t ), 3 3 if t − c ≤ t − t ≤ 2(t − c), and 1 1 Δ(ω) ≤ ω(t − t ) + ω(2(t − t )) ≤ ω(t − t ), 3 3 if t − t ≥ 2(t − c), i.e., f∗ ∈ Hω [a, b]. If ϕ ∈ Va,b , then, obviously, b |

2 f∗ (t)ϕ(t)dt| = 3

a

c ω(2c − 2t)|ϕ(t)|dt. a

Comparing this equality with inequality (1.5), we obtain 2 3

c

c ω(2c − 2t)|ϕ(t)|dt ≤ Eω (ϕ) ≤

a

ω(2c − 2t)|ϕ(t)|dt.

(1.12)

a

By virtue of equality (1.2), we have

b

ϕ(t)dt = 0. Therefore, in the definition of

a

Eω (ϕ), the supremum of the integral in (1.3) can be only taken over the subset of functions f from Hω [a, b] that vanish at some point from [a, b], e.g., at the point t = c. This subset is compact in itself, which guarantees the existence of the value Eω (ϕ), whatever modulus of continuity ω(t) and function ϕ ∈ Va,b are taken. Thus, relation (1.12) and Lemma 1.1 yield the following assertion: Lemma 1.2. Suppose that ϕ ∈ Va,b and ω(t) is an arbitrary modulus of continuity. Then c Eω (ϕ) = θω

b ω(2c − 2t)|ϕ(t)|dt = θω

a

ω(2t − 2c)|ϕ(t)|dt,

(1.13)

c

where c = (a + b)/2, 2/3 ≤ θω ≤ 1, and θω = 1 for a convex function ω(t). 1.3. To illustrate applications of Lemmas 1.1 and 1.2, we establish some estimates necessary in what follows, namely, the estimates of upper bounds of nth Fourier coefficients of functions f (·) from the class Hω 1 an = an (f ) = π

π −π

1 f (t) cos ntdt, bn = bn (f ) = π

π f (t) sin ntdt. −π

Simplest Extremal Problems in Space C

Section 1

193

First, note that bn (f ) can be represented in the form 1 bn (f ) = π

π −π

π 1 f (t) cos(nt + )dt = 2 π

π f (t − π/2n) cos ntdt. −π

Since a shift in an argument does not remove a function from the class Hω , we have 1 sup |bn (f )| = sup | π f ∈Hω f ∈Hω

π

−π

f (t −

π ) cos ntdt| = sup |an (f )|, 2n f ∈Hω

(1.14)

i.e., in the class Hω , the upper bounds of the coefficients an (f ) and bn (f ) coincide. Therefore, it suffices to consider only one of them. Let us estimate, e.g., the value 1 en (ω) = sup |an (f )| = sup | π f ∈Hω f ∈Hω

π f (t) cos ntdt|, n ∈ N.

(1.15)

−π

Every function defined on the entire axis can be represented as a sum of its even and odd components, i.e., f (t) = f1 (t) + f2 (t), f1 (t) = (f (t) + f (−t))/2, f2 (t) = (f (t) − f (−t))/2. It is easy to see that if f ∈ Hω , then f1 ∈ Hω and f2 ∈ Hω . Since an (f2 ) = 0, we get 1 en (ω) = sup | f ∈Hω π

π f1 (t) cos ntdt|

−π

1 = sup | (0) π f ∈H ω

π

−π

2 f (t) cos ntdt| = sup | (0) π f ∈H ω

π f (t) cos ntdt|,

(1.16)

0

(0)

where Hω denotes the subset of even functions from the class Hω . For any fixed n, we denote by Hω,n the set of even 2π/n-periodic functions from Hω . Let us show that 2 sup | (0) π f ∈H ω

π 0

2 f (t) cos ntdt| = sup | f ∈Hω,n π

π f (t) cos ntdt|. 0

(1.17)

Approximation by Fourier Sums in Spaces C and L1

194

Chapter 5

Indeed, let f (·) be an arbitrary function from the class Hω . We set ti = iπ/n, i = 0, n, and take i0 satisfying the following condition: ti0+1 ti+1 df f (t) cos ntdt| = | f (t) cos ntdt| = M (f ). max | i

ti

(1.18)

ti0

Then 2 | π

π

2 f (t) cos ntdt| = | π

n−1

ti+1 2n f (t) cos ntdt| ≤ M (f ). π

(1.19)

i=0 t i

0

Let us show that, in the class Hω,n , there is a function ϕ0 (t) = ϕ0 (f ; t) for which this relation turns into an equality; this would prove equality (1.17). For this purpose, we set ϕ1 (t) = (f (t) − f (2ci − t))/2, t ∈ (ti0 , ti0 +1 ), ci = (2i + 1)π/2n,

ϕ2 (t) =

i = 0, n − 1, ϕ1 (t), t ∈ [ti0 , ti0 +1 ], ϕ1 (2ti0 +1 − t), t ∈ [ti0 +1 , ti0 +2 ],

(1.20)

and denote by ϕ0 (t) a 2π/n-periodic extension of the function ϕ2 (t). The function ϕ0 (t) is the required one. Indeed, for this function, relation (1.19) turns into an equality by construction; on the segment [ti0 , ti0 +1 ], the function ϕ0 (t) satisfies condition (0.5) because the function f (·) satisfies it; on the segment [ti0 , ti0 +2 ], which is the period of ϕ0 (t), this function satisfies condition (0.5) by virtue of its symmetry with respect to the straight line t = ti0 +1 . Equality (1.17) is proved. However, in this case, according to (1.16) and (1.17), we have

2 en (ω) = sup | f ∈Hω,n π

π 0

2n f (t) cos ntdt| = sup | π f ∈Hω,n

π/n f (t) cos ntdt|. (1.21) 0

It is clear that every function f (·) defined on [0, π/n] and satisfying condition (0.5) on this segment can be extended to a 2π/n-periodic even function for which condition (0.5) is also satisfied. Therefore, it follows from (1.21) that 2n en (ω) = | sup π f ∈Hω [0,π/n]

π/n f (t) cos ntdt|. 0

(1.22)

Simplest Extremal Problems in Space C

Section 1

195

To calculate the right-hand side of this equality, we use Lemma 1.2. In this case, a = 0, b = π/n, and c = π/2n. Hence, 2θω en (ω) = π

π/2 2t ω( ) sin tdt, n

(1.23)

0

where θω ∈ [2/3, 1], and θω = 1 for a convex modulus of continuity ω(t). This, in particular, yields 2 π en (ω) < ω( ). (1.23 ) π n

1.4. It is often sufficient to obtain the upper bound of an integral of type (1.1) which may not be exact on a given set of functions f (·). In this case of f ∈ Hω , the following simple reasoning is helpful for this purpose: Assume that the function ϕ(·) in (1.1) satisfies equality (1.2). Then, for any ξ ∈ [a, b], we have b sup f ∈Hω [a,b]

|

b f (t)ϕ(t)dt| =

sup f ∈Hω [a,b]

a

|

(f (t) − f (ξ))ϕ(t)dt| a

b |ϕ(t)|dt.

≤ ω(b − a)

(1.24)

a

This fact can be extended to a more general case. More precisely, the following lemma is true: Lemma 1.3. Let a function ϕ(t) be summable on J ( ϕ ∈ L(J ) ). If J = {x : a ≤ x ≤ b} and xk , k = 1, n, a ≤ x1 < x2 < . . . < xn ≤ b, is the set of points for which xk+1 ϕ(t)dt = 0, k = 1, n − 1, (1.25) xk

then, for any f ∈ Hω (J ),

Approximation by Fourier Sums in Spaces C and L1

196 b

x1 f (t)ϕ(t)dt| ≤ max |f (t)

|

Chapter 5

a≤t≤x1

a

|ϕ(t)|dt a

xn

b |ϕ(t)|dt + max |f (t)|

+ ω(Δ)

xn ≤t≤b

x1

|ϕ(t)|dt. (1.26)

xn

If J = {x : x ≥ a} and xk , k = 1, 2, . . . , a ≤ x1 < x2 < . . . , is the set of points for which (1.25) holds for any k ∈ N, then, for any f ∈ Hω (J ), ∞

x1 f (t)ϕ(t)dt| ≤ max |f (t)|

| a

a≤t≤x1

∞ |ϕ(t)|dt + ω(Δ)

a

|ϕ(t)|dt.

(1.26 )

x1

Here and in (1.26), Δ = sup(xk+1 − xk ). k

Proof. Since relation (1.25) holds for all k ∈ N, we decompose the integral over J into the sum of integrals over the intervals (a, x1 ), (xk , xk+1 ), k = 1, n − 1, and (xn , b) in the first case and over the intervals (a, x1 ), (xk , xk+1 ), k ∈ N, in the second case. Then, by analogy with the proof of estimate (1.24), we obtain relations (1.26) and (1.26 ). 1.5. The following obvious statements are helpful for determining the required set of points: Proposition 1.1. Suppose that a function ϕ(·) is summable on [a, b], vanishes at points tk ∈ [a, b], k = 1, n, a ≤ t1 < t2 < . . . < tn ≤ b, and preserves its sign almost everywhere on the intervals (tk , tk+1 ), so that sign αk = −sign αk+1 , where tk+1

αk =

ϕ(t)dt. tk

If the numbers |αk | do not increase, then, on every interval [tk , tk+1 ], the function x Φ1 (x) = ϕ(t)dt tn

Simplest Extremal Problems in Space C

Section 1

197

takes the value zero and changes its sign at some point xk . If the numbers |αk | do not decrease, the same is true for the function x Φ2 (x) = ϕ(t)dt. t1

Proposition 1.1.

Suppose that a function ϕ(·) is summable on the set t ≥ a. If this function vanishes at points tk , a ≤ t1 < t2 < . . . , sign αk = −sign αk+1 , k ∈ N, and the numbers |αk | form a nonincreasing sequence, then, on every interval [tk , tk+1 ], k ∈ N, the function ∞ Φ(x) = ϕ(t)dt x

vanishes at some point xk . It is clear that Lemma 1.3 can be applied to the sets of points xk constructed so because, surely, they satisfy relation (1.25). 1.6. Lemma 1.1 is a special case of the following more general statement known as Korneichuk–Stechkin lemma: Lemma 1.4 (Korneichuk–Stechkin). Let c be a given point from the segment [a, b], a < c < b, and let ψ(t) be a summable function positive (or negative) almost everywhere on (a, c) and negative (or positive, respectively) almost everywhere on (c, b). Moreover, b ψ(t)dt = 0. a

Further, let ω = ω(t) be an arbitrary modulus of continuity and Hω [a, b] = {f : |f (t) − f (t )| ≤ ω(|t − t |), t, t ∈ [a, b]}. Then b sup f ∈Hω [a,b]

|

c f (t)ψ(t)dt| ≤

a

|ψ(t)|ω(ρ(t) − t)dt a

b = c

|ψ(t)|ω(t − ρ−1 (t))dt,

(1.27)

198

Approximation by Fourier Sums in Spaces C and L1

Chapter 5

where ρ(x) is a function determined on [a, c] by the equalities x

ρ(x) ψ(t)dt = ψ(t)dt, a ≤ x ≤ c ≤ ρ(x) ≤ b,

a

a

and ρ−1 (x) is the function inverse to ρ(x). If ω(t) is a convex modulus of continuity, then (1.27) is an equality, and, moreover, the upper bound is realized by the functions of the form K ± f0 (x) from the class Hω [a, b], where K is an arbitrary constant and ⎧ c ⎪ ⎪ ⎪ ⎪ − ω (ρ(t) − t)dt, x ∈ [a, c], ⎪ ⎪ ⎨ f0 (x) = x x ⎪ ⎪ ⎪ ⎪ ω (t − ρ−1 (t))dt, x ∈ [c, b]. ⎪ ⎪ ⎩ c

Simplest Extremal Problems in Space L1

2.

2.1. In what follows, we shall need an analog of Lemma 1.3 in the integral metric. In this case, inequalities of type (1.24), (1.26), and (1.26 ) can be readily obtained as before, but their justification requires the following auxiliary statement: Lemma 2.1. Suppose that two strictly monotone and absolutely continuous functions Φ(x) and F (t) are defined on the segments a ≤ x ≤ b and ta ≤ t ≤ tb and their domains of values coincide. Furthermore, let t = ρ(x) be a mapping of the segment [a, b] onto [ta , tb ] defined as follows: Φ(x) = F (ρ(x)).

(2.1)

Then (i) the function t = ρ(x) is strictly monotone; (ii) if, in addition,

mes E(F (t) = 0) = 0,

then ρ(x) is also absolutely continuous.

(2.2)

Simplest Extremal Problems in Space L1

Section 2

199

The first assertion of Lemma 2.1 follows from equality (2.1), according to which ρ(x) = F −1 (Φ(x)). (2.1 ) To prove the second part, we use the following well-known facts: Condition (2.2) is necessary and sufficient for the function t = F −1 (y) to be absolutely continuous; the superposition of two absolutely continuous functions is absolutely continuous, provided that it is strictly monotone. In the case under consideration, the function ρ(x) is strictly monotone and, by virtue of (2.1), it can be represented as a superposition of absolutely continuous functions. Hence, it is absolutely continuous. 2.2. Let us formulate an analog of relation (1.24) in the metric of L1 . Lemma 2.2. Suppose that, on the segment [a, b], a point c, a < c < b, and a summable function ϕ(t) are given such that ϕ(t) > 0(ϕ(t) < 0) almost everywhere on (a, c) and ϕ(t) < 0(ϕ(t) > 0) almost everywhere on (c, b). Let also b ϕ(t)dt = 0. (2.3) a

Further, assume that Hω1 is the class of functions f ∈ L(0, 2π) such that π |f (x) − f (x + t)|dx ≤ ω(t),

(2.4)

−π

where ω = ω(t) is an arbitrary fixed modulus of continuity. Then df

b

Eω1 = a

π b f (x + t)ϕ(t)dt1 = | f (t + x)ϕ(t)dt|dx −π

c ≤

b |ϕ(t)|ω(ρ(t) − t)dt =

a

a

|ϕ(t)|ω(t − ρ−1 (t))dt,

(2.5)

a

where ρ(x) is a function defined on [a, c] by the equalities Φ(x) = Φ(ρ(x)), a ≤ x ≤ c ≤ ρ(x) ≤ b,

(2.6)

200

Approximation by Fourier Sums in Spaces C and L1

Chapter 5

x ϕ(t)dt;

Φ(x) = a

and ρ−1 (x) is the function inverse to ρ(x). Proof. It follows immediately from the assumptions imposed on the function ϕ(·) that the function Φ(x) defined by (2.6) is absolutely continuous on the entire segment [a, b], strictly decreasing (increasing) on [a, c], and strictly increasing (decreasing) on [c, b] and mes E(Φ (x) = 0) = 0. Therefore, denoting the points of the segment [c, b] by t and the function Φ(t) on this segment by F (t), we find that the functions Φ(x), x ∈ [a, c], and F (t), t ∈ [c, b], satisfy the conditions of Lemma 2.1. However, in this case, the function t = ρ(x) in (2.6) is strictly monotone (decreasing) and absolutely continuous and, furthermore, ρ(a) = b, ρ(c) = c.

(2.7)

Similarly, one can establish that the function x = ρ−1 (t) is strictly decreasing, absolutely continuous on [c, b], and ρ−1 (c) = c, ρ−1 (b) = a.

(2.7 )

By virtue of these facts, we can change the variables t = ρ(x) and x = ρ−1 (t) in integrals over corresponding intervals and, in view of (2.6), the following equalities hold almost everywhere: ϕ(t) = ϕ(ρ(t))ρ (t), t ∈ (a, c);

(2.8)

ϕ(ρ−1 (t))(ρ−1 (t)) = ϕ(t), t ∈ (c, b).

(2.8 )

For any f ∈ Hω1 , by setting t = ρ(x) and taking (2.7) and (2.8) into account, we get b

c f (x + t)ϕ(t)dt = −

c

f (ρ(t) + x)ϕ(ρ(t))ρ (t)dt

a

c =−

f (ρ(t) + x)ϕ(t)dt. a

Hence, for any f ∈ Hω1 , by virtue of (2.4), we have

Simplest Extremal Problems in Space L1

Section 2

201

π b π c | f (t + x)ϕ(t)dt|dx = | (f (t + x) − f (ρ(t) + x))ϕ(t)dt|dx −π

−π

a

a

c

π |ϕ(t)|

≤

|f (t + x) − f (ρ(t) + x)|dxdt

−π

a

c ≤

ω(ρ(t) − t)|ϕ(t)|dt.

(2.9)

a

Here, the change of the order of integration can easily be justified, e.g., by the same argument as in Subsection 3.7.4. If we set t = ρ−1 (z) in the last integral and use equalities (2.7 ) and (2.8 ), we obtain c

b ω(ρ(t) − t)|ϕ(t)|dt =

a

ω(t − ρ−1 (t))|ϕ(t)|dt.

(2.9 )

c

Combining relations (2.9) and (2.9 ), we obtain estimate (2.5). 2.3. Since ρ(t) − t ≤ b − a,

(2.10)

Lemma 2.2 yields the following statement: Corollary 2.1. Under the conditions of Lemma 2.2, the following inequality holds: b Eω1 (ϕ) = a

ω(b − a) f (t + x)ϕ(t)dt1 ≤ 2

b |ϕ(t)|dt.

(2.11)

a

It is clear that inequality (2.11) is completely analogous to inequality (1.24). 2.4. By using inequality (2.11), we can obtain the required analog of Lemma 1.3 following the reasoning that led us to Lemma 3.1 from estimate (1.24) with the only extra definition: If a function ϕ(·) satisfies the conditions of Lemma 2.2 c (ϕ ∈ V c ). Thus, the on a segment [a, b], then we say that it belongs to a set Va,b a,b following lemma is true:

Approximation by Fourier Sums in Spaces C and L1

202

Chapter 5

Lemma 2.3. Let ϕ ∈ L(J ). If J = [a, b] and xk , k = 1, n, a ≤ x1 < x2 < . . . < xn ≤ b, is a set of points for which ϕ ∈ Vxckk,xk+1 , k = 1, n, where ck is a fixed point from (xk , xk+1 ), then, for any f ∈ Hω1 , b

x1 f (t + x)ϕ(t)dt1 ≤

a

f (x + t)ϕ(t)dt a

ω(Δ) + 2

xn

b |ϕ(t)|dt +

x1

f (t + x)ϕ(t)dt1 . (2.12)

xn

If J = {x : x ≥ a} and xk , k = 1, 2, . . . , is a set of points for which ϕ ∈ Vxckk,xk+1 , k ∈ N, ck ∈ (xk , xk+1 ), then ∞

x1 f (t + x)ϕ(t)dt1 ≤

a

∞ |ϕ(t)|dt,

f (t + x)ϕ(t)dt1 + ω(Δ) a

(2.12 )

x1

where Δ = sup(xk+1 − xk ). 2.5. Let us estimate from above, by using Lemma 2.3, the supremum over the class Hω1 of the value 1 An (f ; x) = π

π f (t + x) cos ntdt, −π

which is the nth harmonic of the Fourier expansion of a function f (·). On every segment [xk , xk+1 ], xk = kπ/n, k ∈ Z, the function cos nt obviously belongs to the set Vxckk,xk+1 , ck = (xk + xk+1 )/2. Therefore, by virtue of (2.12), we have ω(Δ) sup An (f ; x)1 ≤ 2π f ∈Hω1

π −π

nω(Δ) | cos nt|dt = π

π/2n

cos ntdt =

2ω(Δ) . π

0

However, in this case, Δ = π/n and, consequently, sup An (f ; x) ≤

f ∈Hω1

2 !π " . ω π n

(2.13)

Section 3

3.

Approximations of Functions of Small Smoothness

203

Approximations of Functions of Small Smoothness by Fourier Sums

3.1. Here we obtain the results of approximation of the functions from the ¯ ¯ sets C ψ = Lψ ∩C when ψ1 , ψ2 ∈ M0 . We use, as above, the following notation: M is the set of functions ψ(v) convex downwards for all v ≥ 1 such that lim ψ(v) = 0;

v→∞

M0 is the subset of functions ψ ∈ M which satisfy the following condition: 0 < μ(ψ; t) =

1 t ≤ K < ∞, η(t) = η(ψ; t) = ψ −1 ( ψ(t)); η(t) − t 2

M is the subset of functions ψ(·) from M such that ∞

ψ(t) dt ≤ K < ∞. t

(3.1)

1

If A is some subset from M, then the notation ±ψ ∈ A means that either ψ ∈ A or −ψ ∈ A. N0 is the subset of functions f ∈ N which are orthogonal to the constant (f ⊥ 1) : π f (t)dt = 0. (3.2) −π ¯

ψ The first statement relates to approximations of the functions from the sets C∞ .

Theorem 3.1. Let ±ψ1 ∈ M0 and ±ψ2 ∈ M0 = M ∩ M0 . Then the quantity ¯

¯

¯

o ψ ψ ψ En (C∞ ) = sup{|f (x) − Sn−1 (f ; x)| : f ∈ C∞ }, C∞ = C ψ SM

does not depend on the value of x, and the following asymptotic equality holds as n → ∞ : ∞ 2 |ψ2 (t)| 4 ¯ ψ¯ ¯ En (C∞ ) = ln n + O(1)ψ(n), (3.3) dt + 2 ψ(n) π t π n

¯ where ψ(n) = (ψ12 (n) + ψ22 (n))1/2 and O(1) is a quantity uniformly bounded in n.

204

Approximation by Fourier Sums in Spaces C and L1

Chapter 5

3.2. Note some important special cases of this statement. If ψ ∈ MC (see Section 3.12), then there exist positive constants K1 and K2 such that K1 t|ψ (t)| ≤ |ψ(t)| ≤ K2 t|ψ (t)|.

(3.4)

Therefore, if ±ψ2 ∈ MC , then ∞

|ψ2 (t)| dt ≤ K2 | t

n

∞

ψ2 (t)dt| = K2 |ψ2 (n)|

(3.5)

n

and, hence, if ±ψ2 ∈ MC and ±ψ1 ∈ M0 , then equality (3.3) takes the form ¯

ψ )= En (C∞

4 ¯ ¯ ψ(n) ln n + O(1)ψ(n). π2

(3.4 )

Now let the functions ψ1 (·) and ψ2 (·) be chosen according to the following equalities: π π ψ1 (v) = ψ(v) cos β , ψ2 (v) = ψ(v) sin β (3.6) 2 2 under the condition that ψ ∈ MC and let β be an arbitrary positive number. ψ ψ¯ = Cβ,∞ = Lψ Then (see Section 3.11) C∞ β SM ∩ C and ±ψ1 , ±ψ2 ∈ MC . Therefore, according to (3.4 ), it follows from Theorem 3.1 that ψ )= En (Cβ,∞

4 ψ(n) ln n + O(1)ψ(n), ψ ∈ MC , β ∈ R1 . π2

(3.7)

If ψ(k) = k −r and β = r, r > 0, then (see Section 3.7) (ψ, β)-derivative fβψ (·) of the function f (·) coincides with Weyl (r, β)-derivative. In this case, ψ coincides with the class Wβr of functions f (·) whose ψ ∈ MC and the set Cβ,∞ (r, β)-derivatives are almost everywhere bounded by unity. Therefore, according to (3.7), we have

En (Wβr ) =

4 ln n + O(1)n−r , r > 0, β ∈ R1 . πnr

(3.8)

If r ∈ N and β = r, then relation (3.8) is well-known Kolmogorov equality, which laid the foundation of a whole direction in the theory of Fourier series and the theory of approximation of functions related to finding asymptotic equalities for upper bounds of deviations of approximating aggregates on given classes of functions. 3.3. The following statement is an analog of Theorem 3.1 for the classes ¯ C ψ Hω0 :

Section 3

Approximations of Functions of Small Smoothness

205

Theorem 3.2. Let ±ψ1 ∈ M0 and ±ψ2 ∈ M0 . Then the quantity ¯

¯

En (C ψ Hω0 ) = sup{|f (x) − Sn−1 (f ; x)| : f ∈ C ψ Hω0 }

(3.9)

does not depend on the value of x, and the following asymptotic equality holds as n → ∞ : En (C

ψ¯

Hω0 )

1 = θω ( | π

1

2t ω( ) n

0

∞ ψ2 (nv) sin vtdvdt| 1

2 ¯ + 2 ψ(n) ln n π

π/2 2t ¯ ω( ) sin tdt) + O(1)ψ(n)ω(1/n), (3.10) n 0

where θω ∈ [2/3, 1], and θω = 1 if ω(t) is a modulus of continuity convex upwards, and O(1) is a quantity uniformly bounded in n. Below we prove [see the proof of relation (5.4)] that if ±ψ2 ∈ M0 , then 1 | π

1

2t ω( ) n

0

∞

∞ ψ2 (nv) sin vtdvdt| = O(1)ω(1/n)

|ψ2 (t)| dt. t

(3.11)

n

1

Therefore, if ±ψ1 ∈ M0 and ±ψ2 ∈ MC , then, by virtue of relations (3.5) and (3.11), equality (3.10) yields En (C

ψ¯

Hω0 )

2θω ¯ ln n = 2 ψ(n) π

π/2 2t ¯ ω( ) sin tdt + O(1)ψ(n)ω(1/n). n

(3.10 )

0

If the functions ψ1 (·) and ψ2 (·) are chosen according to formulas (3.6), pro¯ vided that ψ ∈ MC and β ∈ R1 , then C ψ Hω0 = Cβψ Hω0 and ±ψ1 , ±ψ2 ∈ MC . Therefore, by virtue of equality (3.10 ), it follows from Theorem 3.2 that En (Cβψ Hω0 )

2θω = 2 |ψ(n)| ln n π

π/2 2t ω( ) sin tdt + O(1)|ψ(n)|ω(1/n). (3.10 ) n 0

3.4. An analog of the Lebesgue inequality (0.10) is established in the following theorem:

206

Approximation by Fourier Sums in Spaces C and L1

Chapter 5

Theorem 3.3. Let ±ψ1 ∈ M0 and ±ψ2 ∈ M0 . Then, for every function ¯ f ∈ C ψ C 0 and any n ∈ N, we have 2 f (x) − Sn−1 (f ; x)C ≤ ( π

∞

|ψ2 (t)| 4 ¯ ψ¯ ¯ ln n + O(1)ψ(n))E dt + 2 ψ(n) n (f ) t π

n ψ¯ ψ¯ ¯ = (En (C∞ ) + O(1)ψ(n))E n (f ),

(3.12) ¯

where O(1) is a quantity uniformly bounded in n ∈ N and f ∈ C ψ C 0 . ¯ ¯ For any function f ∈ C ψ C 0 and every n ∈ N, in the space C ψ C 0 one can ¯ ¯ find a function F (x) = F (f ; n; x) for which En (F ψ ) = En (f ψ ) and relation (3.12) turns into the equality. The second part of Theorem 3.3 shows, in particular, that inequality (3.12) is ¯ asymptotically exact on the entire space C ψ C 0 . This inequality is also asymp¯ ψ¯ is such one. totically exact on some important subsets of C ψ C 0 . The class C∞ 0 and considerIndeed, by taking into account that En (ϕ) ≤ 1 for every ϕ ∈ SM ψ¯ ing the upper bounds of both sides of relation (3.12) over the class C∞ , for every ±ψ1 ∈ M0 and ±ψ2 ∈ M0 , we get ψ¯ En (C∞ )

2 ≤ π

∞

|ψ2 (t)| 4 ¯ ¯ ln n + O(1)ψ(n). dt + 2 ψ(n) t π

(3.13)

n

According to (3.3), the strict inequality is impossible in this relation. Therefore, relation (3.13) is, in fact, the equality and, hence, inequality (3.12) is asymptotψ¯ . At the same time, inequality (3.12) can be strict ically exact on the class C∞ ¯ 0 ψ on the class C Hω . In this case, if ω(t) is a convex function, then, as is well known, we have 1 π sup En (ϕ) = ω( ). (3.14) 2 n 0 ϕ∈Hω By applying this equality to the right-hand side of (3.12), we obtain a quantity that ¯ differs in the general case from En (C ψ Hω0 ) obtained from equality (3.10). Note that, in equality (3.3) (as well as in equalities (3.10) and (3.12)), the first term on the right-hand side can be the principal one. For example, the function ψα (t) = (ln t lnα (ln t + e))−1 belongs to M0 for any α > 1 and, furthermore, we have ∞ ψα (t) dt) = 0. lim (ψα (n) ln n / n→∞ t n

Section 4

Auxiliary Statements

207

3.5. In connection with Theorem 3.3, we should also note the following fact: Let ε = εn , n ∈ N, be an arbitrary sequence of nonnegative numbers monotonically decreasing to zero. For every natural number n, we denote by Cn (ε) the set of continuous functions ϕ(·) that satisfy the inequality En (ϕ) ≤ ¯ ¯ εn , and by C ψ Cn0 (ε) the set of continuous functions whose ψ-derivatives belong 0 0 to Cn (ε) = Cn (ε) ∩ C . Then Theorem 3.3 yields the following statement: Theorem 3.3. Let ±ψ1 ∈ M0 and ±ψ2 ∈ M0 . Then, for any class and any n ∈ N, we have

¯ C ψ Cn0 (ε)

En (C

ψ¯

Cn0 (ε))

2 =( π

∞

|ψ2 (t)| 4 ¯ ln n + O(1))εn , dt + 2 ψ(n) t π

(3.15)

n

where O(1) is a quantity uniformly bounded in n. ¯

¯

¯

Indeed, if f ∈ C ψ Cn0 (ε), then f ψ (·) is continuous and En (f ψ ) ≤ εn . Therefore, by virtue of (3.12), the quantity ρn (f ; x)C does not exceed the right-hand side of (3.15). At the same time, for the function F (x) from Theorem 3.3 constructed by a function ϕ ∈ Cn0 (ε) such that En (ϕ) = εn , the value of ρn (F ; x)C coincides with the right-hand side of (3.15), which proves Theorem 3.3 .

4.

Auxiliary Statements

4.1. As a starting point of the proof of Theorems 3.1–3.3 , we use the following statement established in Corollary 4.3.1: ¯

If f ∈ C ψ M, ±ψ1 ∈ M and ±ψ2 ∈ M , then, at every point x ∈ R1 , the following equality holds: ∞ Δ(x, t)(J2 (ψ1 ; n; t)0 + J2 (ψ2 ; n; t)1 )dt

ρn (f ; x) = −∞

1 + 2π

π Δ(x, t)(ψ1 (n) cos nt + ψ2 (n) sin nt)dt, (4.1) −π

208

Approximation by Fourier Sums in Spaces C and L1 ¯

¯

Chapter 5 ¯

where Δ(x, t) is either f ψ (x − t), or f ψ (x − t) − f ψ (x), or f ψ (x − t) − tn−1 (x − t), where tn−1 (·) is an arbitrary polynomial from Tn−1 , and 1 J2 (ψ1 ; n; t)0 = π

∞ ψ1 (v) cos vtdv,

(4.2)

ψ2 (v) sin vtdv.

(4.2 )

n

1 J2 (ψ2 ; n; t)1 = π

∞ n

¯

If f ∈ Lψ and, as before, ±ψ1 ∈ M and ±ψ2 ∈ M , then equality (4.1) holds almost everywhere. 4.2. Now the problem is to simplify the right-hand sides of (4.1) without loss of their principal values. An intermediate result in this direction can be formulated as follows: Lemma 4.1. Let ±ψ1 ∈ M0 , ±ψ2 ∈ M0 , and let a be an arbitrary positive ¯ number. If f ∈ C ψ SM , then, for any n ∈ N, at every point x, we have

¯

f ψ (x − t)J2 (ψ2 ; n; t)1 dt

ρn (f ; x) = |t|≤a/n

¯ ψ(n) + π

¯

f ψ (x − t)

sin(nt − γn ) ¯ dt + O(1)ψ(n). (4.3) t

a/n≤|t|≤π/2 ¯

If f ∈ C ψ Hω , then, for any n ∈ N, at every point x, we have

¯

¯

(f ψ (x − t) − f ψ (x))J2 (ψ2 ; n; t)1 dt

ρn (f ; x) = |t|≤a/n

+

¯ ψ(n) π

¯

¯

(f ψ (x − t) − f ψ (x))

sin(nt − γn ) dt t

a/n≤|t|≤π/2

¯ + O(1)ψ(n)ω(1/n).

(4.4)

Section 4

Auxiliary Statements

209

¯

If f ∈ C ψ C, then, for any n ∈ N and tn−1 ∈ Tn−1 , at every point x one has ¯ (f ψ (x − t) − tn−1 (x − t))J2 (ψ2 ; n; t)1 dt ρn (f ; x) = |t|≤a/n

+

¯ ψ(n) π

¯

(f ψ (x − t) − tn−1 (x − t))

sin(nt − γn ) dt t

a/n≤|t|≤π/2 ψ ¯ (·) − tn−1 (·)C . + O(1)ψ(n)f

(4.5)

In equalities (4.3)–(4.5), ψ2 (n) ¯ ψ(n) = (ψ12 (n) + ψ22 (n))1/2 , γn = arctan ψ1 (n)

(4.6)

and the quantities O(1) depend neither on n ∈ N nor on the functions f (·) that belong to the corresponding classes under consideration. 4.3. We prove Lemma 4.1 in several steps. First, we denote by Rn = ¯ x) the second term on the right-hand side of (4.1) and note that the Rn (f ; ψ; following statement is true: Lemma 4.2. Let ψ1 (n) and ψ2 (n) be an arbitrary real numbers. If f ∈ ¯ C ψ SM , then, for every n ∈ N, we have ¯ x)C = O(1)ψ(n). ¯ Rn (f ; ψ;

(4.7)

¯

If f ∈ C ψ Hω , then, for every n ∈ N, we have ¯ x)C = O(1)ψ(n)ω(1/n). ¯ Rn (f ; ψ;

(4.7 )

¯

If f ∈ C ψ C, then, for any n ∈ N and any tn−1 ∈ Tn−1 , we have ψ¯ ¯ x)C = O(1)ψ(n)f ¯ (·) − tn−1 (·)C . Rn (f ; ψ;

(4.7 )

Indeed, for every function f ∈ C ψ C, we have 1 2π

π Δ(x, t)(ψ1 (n) cos nt + ψ2 (n) sin nt)dt −π

≤ O(1)Δ(x, t)M (|ψ1 (n)| + |ψ2 (n)|)

Approximation by Fourier Sums in Spaces C and L1

210

Chapter 5

which immediately yields (4.7 ). By setting tn−1 (·) ≡ 0 in (4.7 ), we obtain ¯ equality (4.7). For f ∈ C ψ Hω , we have ψ1 (n) Rn = 2π =

π

ψ2 (n) f (x − t) cos ntdt + 2π

π

−π

¯

f ψ (x − t) sin ntdt

ψ

−π

ψ1 (n) ψ2 (n) (αn cos nx + βn sin nx) + (αn sin nx − βn cos nx), (4.8) 2 2 ¯

¯

where αn and βn are the Fourier coefficients of the function f ψ (·). Since f ψ ∈ Hω , by virtue of (1.23 ), we get |αn |
0 and any n ∈ N, at every point x, the following equality holds: ψ1 (n) sin nt 1 Rn(1) = − Δ(x, t) (4.12) dt + bψ n (a; f ; x). π t |t|≥a/n ¯

¯

In this case, if f ψ ∈ SM , then Δ(x, t) = f ψ (x − t) and 1 bψ n (a; f ; x)C = O(1)|ψ1 (n)|.

(4.13)

Section 4

Auxiliary Statements

¯

¯

211

¯

If f ψ ∈ Hω , then Δ(x, t) = f ψ (x − t) − f ψ (x) and 1 bψ n (a; f ; x)C = O(1)|ψ1 (n)|ω(1/n).

¯

(4.14)

¯

If f ψ ∈ C, then Δ(x, t) = f ψ (x − t) − tn−1 (x − t), where tn−1 (·) is an arbitrary polynomial from Tn−1 , and ¯

ψ 1 bψ n (a; f ; x)C = O(1)|ψ1 (n)| f (·) − tn−1 (·)C .

(4.15)

Proof. Since ψ1 (n) sin nt 1 − J2 (ψ1 ; n; t)0 = − π t πt

∞

ψ1 (v) sin vtdv

n

=−

ψ1 (n) sin nt 1 − J3 (ψ1 ; t)0 , π t π

(4.16)

we have 2 bψ n (a; f ; x)

ψ1 (n) =− π

sin nt 1 dt − Δ(x, t) t π

|t|≤a/n

∞ Δ(x, t)J3 (ψ1 ; t)0 dt. (4.17) −∞

¯

If f ψ ∈ SM , then, for almost all x and t, we have |Δ(x, t)| ≤ 1. This yields ψ1 (n) sin nt |ψ1 (n)| | Δ(x, t) dt| ≤ 2a. (4.18) π t π |t|≤a/n ¯

If f ψ ∈ C, then, similarly, we get sin nt ψ1 (n) |ψ1 (n)| ¯ Δ(x, t) dt| ≤ 2af ψ (·) − tn−1 (·)C . | π t π

(4.19)

|t|≤a/n ¯

¯

¯

For f ψ ∈ Hω , we have |Δ(x, t)| = |f ψ (x − t) − f ψ (x)| ≤ ω(t). Therefore, ψ1 (n) sin nt | Δ(x, t) dt| π t |t|≤a/n

≤

2|ψ1 (n)| a ω( ) = O(1)|ψ1 (n)|ω(1/n). π n

(4.20)

Approximation by Fourier Sums in Spaces C and L1

212

Chapter 5

It remains to verify that estimates of the form (4.18)–(4.20) are also true for the second term on the right-hand side of equality (4.17). First, we consider the case ¯ where f ψ ∈ C. We have ∞ ∞ 2 ψ¯ 1 Δ(x, t)J3 (ψ1 ; t)0 dt| ≤ f (·) − tn−1 (·)C |J3 (ψ1 ; t)0 |dt. (4.21) | π π −∞

0

By representing the last integral by the sum of two integrals, e.g., integrals over intervals (0, 1/n) and (1/n, ∞), and using estimate (4.3.22) for the first integral and estimate (4.3.29) for the second one, we obtain ∞ |J3 (ψ1 ; t)0 |dt ≤ 2π(|ψ1 (n) + n|ψ1 (n)|). (4.22) 0

For any function ψ ∈ M0 , Theorem 3.12.1 yields the following relation:

Therefore,

|ψ(t)| ≤ Ktψ (t) ∀t ≥ 1.

(4.23)

|J3 (ψ1 ; t)0 |dt = O(1)|ψ1 (n)|

(4.24)

∞ 0 ¯

and, hence, by virtue of (4.21), for every f ∈ C ψ C, we get ∞ 1 ¯ | Δ(x, t)J3 (ψ1 ; t)0 dt| = O(1)|ψ1 (n)|f ψ (·) − tn−1 (·)C . π

(4.25)

−∞

It is clear that, by analogy, we can obtain an estimate of the form (4.18), namely, ∞ 1 ¯ ¯ f ψ (x − t)J3 (ψ1 ; t)0 dt| = O(1)|ψ1 (n)| ∀f ψ ∈ SM . (4.26) | π −∞

¯

Thus, it remains to show that, for every f ψ ∈ Hω , the following relation is true: 1 | π

∞ Δ(x, t)J3 (ψ1 ; t)0 dt| −∞

n =| π

∞ −∞

1 Δ(x, t) t

∞ 1

ψ1 (nv) sin vtdvdt| = O(1)|ψ1 (n)|ω(1/n). (4.26 )

Section 4

Auxiliary Statements

213

4.4. To prove the last relation, we use Lemma 1.3 or, in fact, estimate (1.26 ). For this purpose, we note that, on every interval ((k − 1)π, kπ), k ∈ N, the integral sine ∞ sin t si x = − dt, x > 0, t x

has a unique simple zero xk , which satisfies the equality xk = (k − 1/2)π + γk , 0 < γk < π/6.

(4.27)

This follows from the monotonicity of the function si x on the intervals ((k − 1)π, kπ) and from the equality sign si (k − 1/2)π = −sign si (k − 1/3)π,

(4.28)

which is established by a simple estimation of the values of the function si x represented in the following form: cos x si x = − + x

∞

cos t dt. t2

x

Let 1 J3 (ψ1 ; t) = J3 (ψ1 ; n; t) = t

∞

ψ1 (nv) sin vtdv.

(4.29)

1

We now need to establish the following statement: Proposition 4.1. On every interval (xk , xk+1 ), k = 0, 1, . . . , where xk are zeros of the integral sine enumerated in ascending order, the function ∞ J3 (ψ1 ; t)dt, t > 0, x > 0,

Φ3 (x) =

(4.30)

x

becomes zero with the change of its sign at a certain point x ¯k . Moreover, kπ + π/2 < x ¯k < (k + 1)π + 2π/3.

(4.31)

Approximation by Fourier Sums in Spaces C and L1

214

Chapter 5

Proof. We have ∞ ∞ Φ3 (x) = x

ψ1 (nv)

sin vt dvdt. t

(4.32)

1

In view of (4.3.29), the integrals in this representation converge uniformly. Hence, we can represent Φ3 (x) in the following form: ∞ Φ3 (x) =

ψ1 (nv)

∞

sin vt 1 dtdv = t x

x

1

∞

nv ψ1 ( )

∞

x

x

sin t dtdv. t

(4.33)

v

Therefore, 1 Φ3 (xk ) = xk where

x i+1

αi = xi

∞ xk

nt ψ1 ( ) xk

nt ψ1 ( ) xk

∞

∞ t

∞ sin τ 1 αi , dτ dt = τ xk

(4.34)

i=k

sin τ dτ dt = − τ

x i+1

ψ1 (

xi

t

nt )si tdt. xk

(4.35)

It is clear that sign αi = (−1)i .

(4.36)

Let us verify that, for all i = k, k + 1, . . . , |αi | > |αi+1 |.

(4.37)

Let, e.g., i be an even number. Then, αi > 0 and, in view of monotonicity of the function ψ1 (·), αi >

nxi+1 −|ψ1 ( )| xk

x i+1∞

xi

t

x i+2∞

αi

nxi+1 −|ψ1 ( )| xk

= −|ψ1 (

x i+2∞

xi

sin τ dτ dt τ

t

nxi+1 )|(cos xi − cos xi+2 ). xk

By virtue of (4.27), we get −(cos xi − cos xi+2 ) = (−1)i (sin γi − sin γi+2 ) > 0. In the case where i is even, inequality (4.37) is proved. By analogy, one can obtain it for odd i. By virtue of (4.35) and (4.37), series in (4.34) is an alternating one with decreasing (obviously, to zero) terms. Thus, sign Φ3 (xk ) = sign αk = (−1)k .

(4.38)

The function Φ3 (x) is continuous. Hence, by virtue of (4.38), on every interval (xk , xk−1 ) it has the required zero x ¯k . Moreover, according to (4.27), relation (4.31) holds. 4.5. Let us prove equality (4.26 ). Taking into account information from Proposition 4.1 about zeros of the function Φ3 (x) and using the second part of the Lemma 1.3, we get n | π

∞ 0

t 1 Δ(x, ) n t

∞

ψ1 (nv) sin vtdvdt|

1

∞ ≤ Kω(1/n) 0

n | t

∞

ψ1 (nv) sin vtdv|dt

1

∞ |J3 (ψ1 ; t)0 |dt.

= Kω(1/n)

(4.39)

0

It is clear that, since the function J3 (ψ1 ; t) is even, the same estimate also holds for the integral over the interval t < 0. Thus, in view of (4.24), we get estimate (4.26). Lemma 4.3 is proved.

Approximation by Fourier Sums in Spaces C and L1

216

Chapter 5

4.6. Lemma 4.4. Let ±ψ2 ∈ M0 . Then, for all a > 0 and for any n ∈ N, at every point x, the following equality holds: (2) Rn = Δ(x, t)J2 (ψ2 ; n; t)dt |t|≤a/n

+

ψ2 (n) π

Δ(x, t)

cos nt 2 dt + bψ n (a; f ; x). (4.40) t

|t|≥a/n ¯

¯

Moreover, if f ψ ∈ SM , then Δ(x, t) = f ψ (x − t) and 2 bψ n (a; f ; x)C = O(1)|ψ2 (n)|.

¯

¯

(4.41)

¯

If f ψ ∈ Hω , then Δ(x, t) = f ψ (x − t) − f ψ (x) and 2 bψ n (a; f ; x)C = O(1)|ψ2 (n)|ω(1/n).

¯

(4.42)

¯

If f ψ ∈ C, then Δ(x, t) = f ψ (x − t) − tn−1 (x − t), where tn−1 (·) is an arbitrary polynomial from Tn−1 and ¯

ψ 2 bψ n (a; f ; x)C = O(1)|ψ2 (n)| f (·) − tn−1 (·)C .

(4.43)

Proof. Since ψ2 (n) cos nt 1 J2 (ψ2 ; n; t)1 = + πt πt

∞

ψ2 (v) cos vtdt,

(4.44)

n

relations (4.11) and (4.40) yield 2 bψ n (a; f ; x) =

1 π

Δ(x; t)J3 (ψ2 ; t)1 dt |t|≥a/n

n = π

Δ(x, t/n)J3 (ψ2 ; t)dt,

(4.45)

|t|≥a

where 1 J3 (ψ2 ; t)1 = t

∞ n

ψ2 (v) cos vtdv,

(4.46)

Section 4

Auxiliary Statements J3 (ψ2 ; t)

1 = t

∞

ψ2 (v) cos vtdv.

217

(4.47)

1

Now if

¯ fψ

∈ C, then, by virtue of (4.45), we get 2 bψ n (a; f ; x)C

2 ≤ Δ(x, t)C π

∞ |J3 (ψ2 ; t)1 |dt.

(4.48)

a/n

Setting ϕ(v) = −ψ2 (v), β = 1, and x = n in (4.25), we have 1 |J3 (ψ2 ; t)1 | = | Φ1 (ψ2 ; n)|. t

(4.49)

Therefore, by analogy with obtaining of (4.29), we have |J3 (ψ2 ; t)| ≤ 2π|ψ2 (n)|/t2 . Consequently,

∞ |J3 (ψ2 ; t)1 |dt ≤

2π|ψ2 (n)|n . a

(4.50)

(4.51)

a/n

Inserting this estimate in (4.48), in view of (4.23), we obtain equalities (4.43) and (4.41). 4.7. We use the method from Subsection 4.4 to obtain estimate (4.42). On every interval (kπ, (k + 1)π), k = 0, 1, . . . , the integral cosine ∞ ci x =

cos t dt t

x

has a unique simple zero xk , which satisfies the following equality: xk = kπ + γk , 0 < γk < π/6.

(4.52)

This immediately follows from monotonicity of the function ci x on the intervals (kπ − π/2, kπ + π/2) and from equality which can easily be verified sign ci (kπ) = −sign ci (kπ + π/6), k = 0, 1, . . . . By analogy with the proof of Proposition 4.1, we establish the following statement:

Approximation by Fourier Sums in Spaces C and L1

218

Chapter 5

Proposition 4.2. On every interval (xk , xk+1 ), k = 0, 1, . . . , where xk are zeros of the integral cosine, the function Φ3 (x)

∞ =

J3 (ψ2 ; t)dt,

(4.53)

x

where J3 (ψ2 ; ·) is defined by formula (4.47), becomes zero with the change of its sign at a certain point x ¯k . Moreover, kπ < x ¯k < (k + 1)π + π/6.

(4.56)

In view of this, by using the second part of Lemma 1.3, we get n | π

∞

Δ(x; t/n)J3 (ψ2 ; t)dt|

∞ ≤ Kω(1/n)n

a

|J3 (ψ2 ; t)|dt

a

∞ = Kω(1/n)

|J3 (ψ2 ; t)1 |dt

(4.57)

a/n ¯

for all f ∈ C ψ Hω . Since the function J3 (ψ2 ; t) is odd, the same estimate also holds for the integral over the interval t < −a. Therefore, equality (4.45) and estimates (4.51) and (4.23) yield (4.42). 4.8. Further, we need the following statement: Lemma 4.5. If ϕ ∈ SM , then, for all b > 0 and any γ ∈ R1 , we have sin(nt + γ) dt| = O(1). (4.58) ϕ(x − t) | t |t|≥b

If ϕ ∈ Hω , then, for all b > 0 and any γ ∈ R1 , we have sin(nt + γ) (ϕ(x − t) − ϕ(x)) | dt| = O(1)ω(1/n). t

(4.59)

|t|≥b

If ϕ ∈ C, then, for all b > 0, any γ ∈ R1 , and for any tn−1 ∈ Tn−1 , we have sin(nt + γ) | (ϕ(x − t) − tn−1 (x − t)) dt| = O(1)ϕ(·) − tn−1 (·)C . (4.60) t |t|≥b

Section 4

Auxiliary Statements

219

In equalities (4.58)–(4.60), O(1) are quantities uniformly bounded in all parameters under consideration. Proof. Let m be an odd number which satisfies inequality (m − 2)π < b ≤ mπ.

(4.61)

Let y(t) be any function from L1 . Then sin(nt + γ) y(t) Yn (b) = dt t |t|≥b

=(

+

b≤|t|≤mπ

However,

y(t)

)y(t)

sin(nt + γ) dt. t

(4.62)

|t|≥mπ

sin(nt + γ) dt t

|t|≥mπ

= lim

k→∞

k i=0

−(m+2i)π

(

(m+2i+2)π

+

)y(t)

−(m+2i+2)π

sin(nt + γ) dt t

(m+2i)π

π y(t) sin(nt + γ)Am (t)dt,

=

(4.63)

−π

where Am (t) =

∞ i=(m+1)/2

t2

2t . − (2iπ)2

(4.64)

Thus, Yn (b) = b≤|t|≤mπ

sin(nt + γ) y(t) dt + t

π y(t)Am (t) sin(nt + γ)dt.

(4.65)

−π

Assume that y(t) = ϕ(x − t) − tn−1 (x − t) and ϕ ∈ C. Then (4.62) and (4.65) yield | |t|≥b

sin(nt + γ) y dt| ≤ ϕ(·) − tn−1 (·)( t

b≤|t|≤mπ

dt + t

π

−π

|Am (t)|dt).

Approximation by Fourier Sums in Spaces C and L1

220

Chapter 5

In view of the fact that the quantity Am (t) is bounded on the segment [−π, π], this yields relations (4.60) and (4.58). If y(t) = ϕ(x − t) − ϕ(x) and ϕ ∈ Hω , then y ∈ Hω and y(0) = 0. Denote by xk , k = 0, 1, . . . , zeros of the function ∞

sin(nt + γ) dt. t

x

By virtue of Lemma 1.3, we get sin(nt + γ) | y(t) dt| ≤ Kω(1/n). t

(4.66)

b≤|t|≤mπ

Since function Am (t) is monotone and bounded on the segment [o, π], we use the same method for estimating second term in (4.65). Thus, we have π

π y(t)Am (t) sin(nt+γ)dt| ≤ Kω(1/n)

| −π

|Am (t)|dt = O(1)ω(1/n). (4.67) 0

Combining relations (4.65)–(4.67), we arrive at equality (4.59). 4.9. Setting b = π/2 in Lemma 4.5, we conclude that the integrals over the intervals |t| ≥ a/n in equalities (4.12) and (4.40) can be replaced by integrals taken only over the intervals a/n ≤ |t| ≤ π/2. The error caused by this procedure does not exceed the remainders in these equalities. Therefore, combining the statements of Lemmas 4.2–4.5, we establish the validity of all assertions of Lemma 4.1. 4.10. We choose the number a in Lemma 4.1 with regard for the following statement: Lemma 4.6. For any function ±ψ2 ∈ M, there exists a number a > 0 such that, for any n ∈ N, on the interval (0, a/n) the following equality holds: sign J2 (ψ2 ; n; t)1 = sign ψ2 (n), t ∈ (0, a/n).

(4.68)

Proof. For definiteness, we assume that ψ2 (n) > 0 for every n ∈ N. To prove the lemma, it suffices to show that there exists a number a > 0 such that n J2 (ψ2 ; n; t/n)1 = π

∞ ψ2 (nv) sin vtdv > 0 ∀t ∈ (0, a), n ∈ N. 1

(4.69)

Section 4

Auxiliary Statements

221

For this purpose, we set ϕn (v) =

ψ2 (nv), v ≥ 1, ψ2 (n),

0 < v ≤ 1, n = 1, 2, . . . .

(4.70)

The function ϕn (v) does not increase. Therefore, for every t > 0, we have ∞ ϕn (v) sin vtdv > 0.

(4.71)

0

On the other hand, by virtue of (4.70), n π

∞ ϕn (v) sin vtdv = J2 (ψ2 ; n; t/n)1 + ψ2 (n)

2n sin2 t/2 . πt

0

In view of inequality (4.71), this yields the required statement because 2ψ2 (n) sin2 (t/2) < ψ2 (1)t/2. t In what follows, we assume that the number a in equalities (4.3)–(4.5) is chosen so that relation (4.68) is true. To simplify the right-hand side of equality (4.1) even more, we proceed as follows: Taking equalities (4.3)–(4.5) into account, we set xk = (kπ + γn )/n, tk = xk − π/2n, k = 0, ±1, . . . , n ∈ N.

(4.72)

Also denote by k0 the value of k for which tk0 is the point closest from the right to the point (a + π)/n among those for which sin(nt − γn ) = 1, and by k1 the largest of k for which tk < π/2. Further, we denote by k2 the number for which the point tk2 is the point closest from the left to the point −(a + π)/n among those for which sin(nt + γn ) = −1, and by k3 the least of k for which tk > −π/2. We set ln (t) = xk , t ∈ [tk , tk+1 ], k = k0 , . . . , k1 − 1, k = k3 , k3 + 1, . . . , k2 − 1. Let us prove the following auxiliary statement:

(4.73)

Approximation by Fourier Sums in Spaces C and L1

222

Chapter 5

Lemma 4.7. Let a be an arbitrary positive number, let γ ∈ R1 , and let n ∈ N. If ϕ ∈ SM , then, at every point x, we have ϕ(x − t)

sin(nt − γn ) dt t

a/n≤|t|≤π/2

ϕ(x − t)

=

sin(nt − γn ) dt + O(1). (4.74) ln (t)

i3,1

If ϕ ∈ Hω , then (ϕ(x − t) − ϕ(x))

sin(nt − γn ) dt t

a/n≤|t|≤π/2

(ϕ(x − t) − ϕ(x))

=

sin(nt − γn ) dt + O(1)ω(1/n). (4.75) ln (t)

i3,1

If ϕ ∈ C, then, for any tn−1 ∈ Tn−1 , at every point x, we have (ϕ(x − t) − tn−1 (x − t))

sin(nt − γn ) dt t

a/n≤|t|≤π/2

(ϕ(x − t) − tn−1 (x − t))

=

sin(nt − γn ) dt ln (t)

i3,1

+ O(1)ϕ(·) − tn−1 (·)C .

(4.76)

In equalities (4.74)–(4.76), i3,1 = (t3 , t2 ) ∪ (t0 , t1 ), and O(1) are quantities uniformly bounded in n and in the functions ϕ(·) from the corresponding classes under consideration. Proof. As before, let Δ(x, t) be either ϕ(x − t), or ϕ(x − t) − ϕ(x), or ϕ(x − t) − tn−1 (x − t). Consider the difference

Section 4

Auxiliary Statements

223

π/2 tk1 sin(nt − γn ) sin(nt − γn ) Δ(x, t) dt − Δ(x, t) dt Rn (ϕ, a)+ = t t a/n

tko

Δ(x, t)

=

sin(nt − γn ) dt t

(a/n,tk0 )∪(tk1 ,π/2)

tk1 +

Δ(x, t)

tk0 (1)

ln (t) − t sin(nt − γn )dt. tln (t)

(4.77)

(2)

Denote by Rn (ϕ; a) and Rn (ϕ; a) the first and second terms, respectively, on the right-hand side of (4.77). Then, by virtue of (4.77), tko − a/n ≤ 2π/n, and π/2 − x1 ≤ 2π/n,

(4.78)

and we get ⎧ O(1) ⎪ ⎪ ⎨ Rn(1) (ϕ; a) = O(1)ω(1/n) ⎪ ⎪ ⎩ O(1)ϕ(·) − tn−1 (·)C

if ϕ ∈ SM , if ϕ ∈ Hω ,

(4.79)

if ϕ ∈ C. (2)

Let us show that such a relation is also valid for the quantity Rn (ϕ; a). For this purpose, we note that tk1 ln (t) − t | |dt = O(1). (4.80) tln (t) tko

Therefore, Rn(2) (ϕ; a) =

O(1)

if ϕ ∈ SM ,

O(1)ϕ(·) − tn−1 (·)C

if ϕ ∈ C.

(4.81)

To prove that, for any ϕ ∈ Hω , Rn(2) (ϕ, a) = O(1)ω(1/n), we note that the function rn (t) =

ln (t) − t sin nt tln (t)

(4.82)

224

Approximation by Fourier Sums in Spaces C and L1

Chapter 5

preserves its sign on the intervals (tk , tk+1 ), k = k0 , k1 − 1, and changes it with a change in the number k. Furthermore, the quantities tk+1

dk =

rn (t)dt tk

strictly decrease as the number k increases. This implies that the function tk1 r¯n (x) =

rn (t)dt x

has a unique simple zero x ¯k on every interval (tk , tk+1 ), k = k0 , k1 − 1. Therefore, by using Lemma 1.3 and taking relation (4.80) into account, we obtain (4.82). Thus, ⎧ O(1) if ϕ ∈ SM , ⎪ ⎪ ⎨ (4.83) Rn (ϕ; a)+ = O(1)ω(1/n) if ϕ ∈ Hω , ⎪ ⎪ ⎩ O(1)ϕ(·) − tn−1 (·)C if ϕ ∈ C. It is clear that, by analogy, we can also obtain an estimate of the form (4.83) for the quantity −a/n

sin(nt − γn ) Δ(x, t) dt − t

Rn (ϕ; a)− = −π/2

tk2

tk3

Δ(x, t)

sin(nt − γn ) dt, ln (t)

which completes the proof of Lemma 4.7. By combining the statements of Lemmas 4.1 and 4.7, we obtain the following corollary: Corollary 4.1. Let ±ψ1 ∈ M0 , ±ψ2 ∈ M0 , and let a be an arbitrary posi¯ tive number. If f ∈ C ψ SM , then, for any n ∈ N, at every point x ∈ R1 , ¯ f ψ (x − t)J2 (ψ2 ; n; t)1 dt ρn (f ; x) = |t|≤a/n

¯ ψ(n) sin(nt − γn ) ¯ ¯ f ψ (x − t) + dt + O(1)ψ(n). (4.84) π ln (t) i3,1

Proofs of Theorems 3.1–3.3

Section 5

225

¯

If f ∈ C ψ Hω , then, for any n ∈ N, at every point x ∈ R1 ,

¯

(f ψ (x − t) − f (x))J2 (ψ2 ; n; t)1 dt

ρn (f ; x) = |t|≤a/n

¯ ψ(n) sin(nt − γn ) ¯ ¯ + (f ψ (x − t) − f ψ (x)) dt π ln (t) i3,1

¯ + O(1)ψ(n)ω(1/n).

(4.85)

¯

If f ∈ C ψ C, then, for any n ∈ N and any tn−1 ∈ Tn−1 , at every point x ∈ R1 , we have ¯ ρn (f ; x) = (f ψ (x − t) − tn−1 (x − t))J2 (ψ2 ; n; t)dt |t|≤a/n

+

¯ ψ(n) sin(nt − γn ) ¯ (f ψ (x − t) − tn−1 (x − t)) dt π ln (t) i3,1

ψ¯ ¯ + O(1)ψ(n)f (·) − tn−1 (·)C .

(4.86)

In equalities (4.84)–(4.86), i3,1 = (tk3 , tk2 ) ∪ (tk0 , tk1 ), ψ2 (n) ¯ ψ(n) = (ψ12 (n) + ψ22 (n))1/2 , γn = arctan , ψ1 (n) and quantities O(1) depend neither on n ∈ N nor on the functions f (·) from the corresponding classes under consideration.

5.

Proofs of Theorems 3.1–3.3

5.1. Let us pass directly to the proof of Theorems 3.1–3.3 . First of all, we ¯ note that the classes C ψ N are invariant under the shift of an argument, i.e., if ¯ ¯ f ∈ C ψ N, then the function f1 (x) = f (x + h) also belongs to C ψ N for any fixed h ∈ R1 . This yields ¯

¯

¯

En (C ψ N) = sup{|ρn (f ; x)| : f ∈ C ψ N} = sup{|ρn (f ; 0)| : f ∈ C ψ N}.

226

Approximation by Fourier Sums in Spaces C and L1

Chapter 5

¯

Thus, the value E(C ψ N) does not depend on the value of x. This result, the fact 0 , and (4.84) yield that both functions ϕ(t) and ϕ(−t) belong to the class SM ψ¯ ϕ(t)J2 (ψ2 ; n; t)1 dt| En (C∞ ) ≤ sup | 0 ϕ∈SM

|t|≤a/n

¯ ψ(n) sin(nt − γn ) ¯ + ϕ(t) sup | dt| + O(1)ψ(n) π ϕ∈S 0 ln (t) M

i3,1

|J2 (ψ2 , n; t)|dt

≤ |t|≤a/n

¯ ψ(n) sin(nt − γn ) ¯ | + |dt + O(1)ψ(n). π ln (t)

(5.1)

i3,1

5.2. On the set In = {t : t ∈ (−a/n, a/n) ∪ i3,1 }, we define a function ϕn (t) by setting ⎧ sign J2 (ψ; n; t)1 , |t| ≤ a/n, ⎪ ⎪ ⎨ (5.2) ϕn (t) = sin(nt − γn ) ⎪ ⎪ ⎩sign , t ∈ i3,1 , ln (t) 0 that coincides with ϕ (t) on the set and denote by ϕ∗n (t) a function from SM n In . It is clear that such a function ϕ∗ (t) always exists. In this case, by virtue of (4.84), the value ρn (J ψ (ϕ∗ ; ·); x) coincides exactly with the right-hand side of relation (5.1). This means that (5.1) cannot be strict inequalities. Hence, we have ψ¯ ) E(C∞

= |t|≤a/n

¯ ψ(n) sin(nt − γn ) |J2 (ψ2 ; n; t)1 |dt + | |dt π ln (t) i3,1

¯ + O(1)ψ(n)

(5.3)

and, in order to prove equality (3.3), it remains to show that |t|≤a/n

2 |J2 (ψ2 ; n; t)1 |dt = π

∞ n

|ψ2 (t)| ¯ dt + O(1)ψ(n) t

(5.4)

Section 5

Proofs of Theorems 3.1–3.3

227

sin(nt − γn ) 4 ¯ |dt = ln n + O(1)ψ(n). ln (t) π

(5.5)

and | i3,1

On the interval (0, a/n), equalities (4.68) are true. In view of this fact, we have

2 |J2 (ψ2 ; n; t)1 |dt ≤ | π

|t|≤a/n

a ∞

df

ψ2 (nv) sin vtdvdt| = In (ψ2 ; a). 0

(5.6)

1

Let us show that, in the integral In (ψ2 ; a), one can change the order of integration. The function |ψ2 (v)| monotonically tends to zero. Therefore, the function ∞ Φ1 (x) =

ψ2 (nv) sin vtdv, x > 0, x

is continuous for every fixed n and t. Moreover, it has one simple zero xk on every interval between neighboring zeros vk and vk+1 of the function sin vt (see Subsection 4.3.3). Let A be an arbitrary number, A ≥ 1. Denote by xk zero closest to the point x = A from the right. Then we have A ≤ xk < A + 2π/t. Hence, ∞ |

A+2π/t

ψ2 (nv) sin vtdv| ≤ | A

By virtue of these facts, we get

ψ2 (nv)dv|. A

(5.7)

228

Approximation by Fourier Sums in Spaces C and L1

a ∞

a ψ2 (nv) sin vtdvdt| ≤

| 0 A

Chapter 5

A+2π/t

|

ψ2 (nv)dv|dt

0

A

a

A+2π/t

=|

ψ2 (nv)dvdt| 0

A

A+2π/t

a

ϕ2 (nv)dv|a0 + 2π

=t

ψ2 (n(A + 2π/t)) dt t2

0

A

∞

ψ2 (t + An) dt). t

< 2π(ψ2 (An) + 2nπ/a

This implies that, for every ε > 0, one can always find a number A(ε) such that the inequality a ∞ ψ2 (nv) sin vtdvdt| < ε, | 0 A

holds for A ≥ A(ε). This inequality obviously guarantees the possibility of changing the order of integration in (5.6). Carrying out integration, we get 2 In (ψ2 ; a) = | π

∞

ψ2 (nv) dv − v

1

∞

ψ2 (nv) cos avdv|. v

(5.8)

1

Further, since the function |ψ2 (nv)|/v decreases, we obtain 2 | π

∞

ψ2 (nv) 2 cos avdv| ≤ | v π

1

1+2π/a

ψ2 (nv) dv| v

1

≤

2π 2 ψ2 (n) = O(1)ψ2 (n), π a

(5.9)

by analogy with the proof of estimate (5.7). By combining relations (5.8) and (5.9), we get (5.4).

Proofs of Theorems 3.1–3.3

Section 5

229

5.3. Let us prove equality (5.5). In view of relations (4.72) and (4.73), we have sin(nt − γn ) | |dt ln (t) i3,1

=

k 2 −1

1 |xk |

k=k3

tk+1

| sin(nt − γn )|dt +

k 1 −1 k=k0

tk

1 xk

tk+1

| sin(nt − γn )|dt. tk

However, for every k, tk+1

tk +π/2n

| sin(nt − γn )|dt = 2

| sin(nt − γn )|dt

tk

tk π/2n

| sin nt|dt = 2/n.

=2

(5.10)

0

Therefore, | i3,1

k k 2 −1 1 −1 sin(nt − γn ) 1 1 |dt = 2( + ) ln (t) |kπ − γn | (kπ − γn ) k=k3

= 2(

k=k0

|k3 |

k=|k2 |+1

k 1 −1 1 1 + ). kπ + γn kπ − γn

(5.11)

k=k0

By construction, |γn | ≤ 2π and k0 π − γn > a, |k2 |π + γn > a, k1 < n/2 + γn /π, |k3 | < n/2 + γn /π. (5.12) Taking these facts into account, we obtain 2(

|k3 |

k=|k2 |+1

=4

k 1 −1 1 1 + ) kπ + γn kπ − γn

k 1 −1 k=k0

k=k0

n 41 1 4 + O(1) = + O(1) = ln n + O(1). (5.13) kπ − γn π k π k=1

Approximation by Fourier Sums in Spaces C and L1

230

Chapter 5

Combining (5.11) and (5.13), we arrive at relation (5.5). Theorem 3.1 is proved.

¯

5.4. Proof of Theorem 3.2. As indicated above, the quantity En (C ψ Hω0 ) does not depend on the value of x. We also note that both ϕ(t) and ϕ1 (t) = ϕ(−t) belong to the class Hω0 . In this case, by virtue of equality (4.85), we have ¯

En (C ψ Hω0 ) =

sup

0 f ∈C ψ¯ Hω

|ρn (f ; 0)|

≤ sup | 0 ϕ∈Hω

(ϕ(t) − ϕ(0))J2 (ψ2 ; n; t)1 dt|

|t|≤a/n k k 2 −1 1 −1 ¯ ψ(n) 1 1 ek (ω)) + ( ek (ω) + π |xk | xk k=k3

k=k0

¯ + O(1)ψ(n)ω(1/n),

(5.14)

where tk+1

ek (ω) = sup | 0 ϕ∈Hω

ϕ(t) sin(nt − γn )dt|.

(5.15)

tk

The function J2 (ψ2 ; n; t)1 is odd. Hence, for any function ϕ ∈ Hω0 , we have |

(ϕ(t) − ϕ(0))J2 (ψ2 ; n; t)dt|

|t|≤a/n

a/n a/n = | (ϕ(t) − ϕ(−t))J2 (ψ2 ; n; t)1 dt| ≤ ω(2t)|J2 (ψ2 ; n; t)|1 dt 0

a/n =| ω(2t)J2 (ψ2 ; n; t)1 dt|. 0

0

(5.16)

Proofs of Theorems 3.1–3.3

Section 5

231

By taking (4.72) into account, for any ϕ ∈ Hω0 , we have tk+1

xk ϕ(t) sin(nt − γn )dt| = | (ϕ(t) − ϕ(2xk − t)) sin(nt − γn )dt|

| tk

tk

xk ≤

ω(2(xk − t))| sin(nt − γn )|dt tk π/2n

ω(2t) sin ntdt.

=

(5.17)

0

Thus, π/2n

ek (ω) ≤

ω(2t) sin ntdt.

(5.18)

0

Therefore, according to (5.14)–(5.18), we obtain a/n ω(2t)J2 (ψ2 ; n; t)dt| En (C Hω0 ) ≤ | ψ¯

0 π/2n k k 2 −1 1 −1 ¯ ψ(n) 1 1 + ω(2t) sin ntdt( ) + π |xk | xk 0

¯ + O(1)ψ(n)ω(1/n).

k=k3

k=k0

(5.19)

5.5. Now we construct a function f ∗ ∈ C ψ Hω0 for which the quantity ρn (f ∗ ; 0) coincides with the right-hand side of (5.19). To this end, we set ⎧ 1 ⎪ ⎨ ω(2(xk − t)), t ∈ [tk , xk ], ϕk (t) = 2 ⎪ ⎩− 1 ω(2(t − xk )), t ∈ [xk , tk+1 ], k = k0 , k1 − 1, k = k3 , k2 − 1. 2 1 π 2a ϕ+ (t) = (−1)k−k0 ϕk (t) − (ω( ) − ω( )), t ∈ [tk , tk+1 ], k = k0 , k1 − 1, 2 n n 1 π 2a ϕ− (t) = (−1)k−k2 +1 ϕk (t)+ (ω( )−ω( )), t ∈ [tk , tk+1 ], k = k3 , k2 − 1, 2 n n

232

Approximation by Fourier Sums in Spaces C and L1 ⎧ 1 ⎪ ⎪ ω(2|t|), ⎪ ⎪ 2 ⎪ ⎪ ⎪ ⎪ ⎪ 1 ⎪ ⎪ ω(2a/n), ⎪ ⎪ ⎪ ⎨2 ϕ(t) ˆ = ϕ+ (t), ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ 1 ⎪ ⎪ − ω(2a/n), ⎪ ⎪ ⎪ 2 ⎪ ⎪ ⎪ ⎩ ϕ− (t),

and

Chapter 5

|t| ≤ a/n, t ∈ [a/n, tk0 ], t ∈ [tk0 , tk1 ],

(5.20)

t ∈ [tk2 , −a/n], t ∈ [tk3 , tk2 ].

First, we assume that ω(t) is a convex modulus of continuity. Then, by virtue of the convexity of the function ω(t), the function ϕ(t) ˆ continuous on the interval [tk3 , tk1 ] satisfies the following relation: |ϕ(t) ˆ − ϕ(t ˆ )| ≤ ω(|t − t |), t, t ∈ [tk3 , tk1 ].

(5.21)

Moreover, −a/n

tk1 ϕ(t)dt ˆ = tk3

tk0

ϕ(t)dt ˆ + tk2

ϕ(t)dt ˆ = ω(

2a )(t0 + tk2 ). n

a/n

Since, by construction, a/n ≤ tk0 ≤ 2π/n, −a/n − 2π/n ≤ tk2 ≤ −a/n, thus, we have tk1 |

ϕ(t)dt| ˆ ≤

2π 2a ω( ). n n

(5.22)

tk3

It is clear that this estimate allows one to define function (5.20) on the remaining set of the period [−π, π] so that the resulting function (also denoted by ϕ(t) ˆ ) satisfies condition (5.21) on the interval [−π, π], takes equal values at the points −π and π, and its mean value on the period is equal to zero. In this case, we denote the 2π-periodic extension of the function ϕ(t) ˆ by ϕ∗ (t) and set f ∗ (·) = ¯ ∗ ψ J (ϕ ; ·). This is the required extremal function because, by its definition, f ∗ ∈ ¯ C ψ Hω0 . Moreover, direct calculations show that the value of quantity (4.84) for this function coincides with the right-hand side of (5.19). If ω(t) is a convex

Proofs of Theorems 3.1–3.3

Section 5

233

function, this reasoning implies that relation (5.19) is the equality. In view of equality (5.13), we can now prove Theorem 3.2 for convex moduli of continuity, taking into account that 1 rn = |

∞ ω(2t/n)

a

ψ2 (nv) sin vtdvdt| = O(1)ψ2 (n)ω(1/n) 1

for any a > 0. Indeed, by virtue of (5.7), we have 1 rn ≤ |

1+2π/t

ω(2t/n) a

ψ2 (nv)dv|dt 1

≤ |ψ2 (n)|ω(

2 max(a, 1) 2π|a − 1| ) = O(1)|ψ2 (n)|ω(1/n). n min(a, 1)

Assume that ω(t) is not necessarily a convex modulus of continuity. In this case, the constructed function f ∗ (·) also guarantees that the value of quantity (4.84) is equal to the right-hand side of (5.19). However, the functions in (5.20), together with the function ϕ(·), ˆ may not satisfy condition (5.21). Hence, the function ϕ∗ (·) does not necessarily belong to Hω0 . Moreover, one can show (see Subsection 1.2) that the function ϕ∗ (t) = 2ϕ∗ (t)/3 belongs to the set Hω0 . This implies that relation (5.19) is also the equality in the case of arbitrary moduli of continuity (with a certain undetermined value of θω , θω ∈ [2/3, 1] ). Theorem 3.2 is proved. ¯

5.6. Proof of Theorem 3.3. Let f (·) be an arbitrary function from C ψ C 0 . Then, by virtue of (4.86), we have ρn (f ; x)C

≤( |t|≤a/n

¯ ψ(n) sin(nt − γn ) |J2 (ψ2 ; n; t)1 |dt + | |dt π ln (t) i3,1 ¯

ψ ¯ (·) − tn−1 (·)C . + O(1)ψ(n))f

(5.23)

In view of equalities (5.4) and (5.5), this yields estimate (3.12) if, as tn−1 (·), we take the trigonometric polynomial t∗n−1 (·) that realizes the best approximation of ¯ a function f ψ (·) in the uniform metric.

234

Approximation by Fourier Sums in Spaces C and L1

Chapter 5

Now we prove the second part of Theorem 3.3. By virtue of relations (4.86) ¯ and (5.23), it suffices to show that, for any f ∈ C ψ C 0 and any n ∈ N, there ¯ ¯ exists a function F (x) for which En (F ψ ) = En (f ψ ) and, moreover, ¯ (F ψ (t) − t∗n−1 (t))J2 (ψ2 ; n; t)1 dt |t|≤a/n

¯

(F ψ (t) − t∗n−1 (t))

+ i3,1

=(

|J2 (ψ2 ; n; t)1 |dt +

|t|≤a/n

sin(nt − γn ) dt ln (t)

¯ ψ(n) sin(nt − γn ) | |dt π ln (t) i3,1

ψ¯ ¯ + O(1)ψ(n))E n (F ) ¯

¯

ψ ψ ¯ = (En (C∞ ) + O(1)ψ(n))E n (F ).

(5.24)

¯

For this purpose, we set En (f ψ ) = en and ⎧ ⎪ en sign J2 (ψ2 ; n; t)1 , ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨en sign J2 (ψ2 ; n; a/n)1 , ϕ0 (t) = ϕ0 (n; t) = e sign J (ψ ; n; −a/n) , n 2 2 1 ⎪ ⎪ ⎪ ⎪ ⎪ nt − γ n ⎪ ⎪ , ⎩en sign sin ln (t)

|t| ≤ a/n, a/n ≤ t ≤ xk0 , xk2 ≤ t ≤ −a/n,

(5.25)

t ∈ i3,1 .

Further, by ϕ1 (t) = ϕ1 (n; t; δ), we denote a function that coincides with ϕ0 (t) everywhere on [t3 , t1 ] except δ-neighborhoods (δ < π/4n) of these points 0 and xk , k ∈ k0 , k1 − 1, k ∈ k3 , k2 − 1. In these δ-neighborhoods, the function ϕ1 (t) is linear, and its graph connects the point (−δ, ϕ0 (−δ)) and (δ, ϕ0 (δ)) and also the points (xk − δ, ϕ0 (xk − δ)) and (xk + δ, ϕ0 (xk − δ)). On the remaining part of the period [−π, π], we complete the definition of ϕ1 (t) so that it is continuous and at least 2n points ck − π < c1 < c2 < . . . < c2n < π exist on the interval [−π, π) at which ϕ1 (t) attains its maximum absolute value equal to en . In this case, ϕ1 (k) = (−1)k en , which implies that ϕ1 (−π) = ϕ1 (π) and π ϕ1 (t)dt = 0. −π

Approximation by Fourier Sums on Classes Hω

Section 6

235

Finally, by ϕδ (t) = ϕδ (n; t) and Fδ (x), we denote the 2π-periodic extension of ¯ the function constructed and its ψ-integral, respectively. The function ϕδ (t) is continuous for any δ > 0 and, according to the Chebyshev criterion (see Section 7.1), En (ϕδ ) = en . Moreover, the polynomial of at most (n − 1)th degree that realizes the best uniform approximation of this function is identically equal to zero. Substituting the functions Fδ (·) into the right-hand side of (5.24), we obtain ¯ ¯ sin(nt − γn ) Fδψ J2 (ψ2 ; n; t)1 dt + Fδψ dt ln (t) |t|≤a/n

=

i3,1

ψ¯ ) (En (C∞

ψ¯ ¯ + O(1)ψ(n))E n (F ) +

(ϕδ (t) − ϕ0 (t))J2 (ψ2 ; n; t)1 dt

|t|≤a/n

+

(ϕδ (t) − ϕ0 (t))

sin(nt − γn ) dt. ln (t)

i3,1

Since δ is arbitrary, this implies the validity of the statement concerning relation (5.24). Theorem 3.3 is proved. Remark 5.1. The fact that the function ϕ(·) is orthogonal to the constant, in fact, was not used in the proof of estimate (5.1). Thus, this estimate holds for ¯ ψ¯ 0 ⊂ S , this implies the class C ψ SM as well as for the class C∞ . Since SM M ¯ ψ¯ ψ that equality (3.3) holds if we replace En (C∞ ) by En (C SM ) on its left-hand side. By analogy, we obtain that on the left-hand sides of equalities (3.10) and ¯ (3.10 ) one can use the quantity En (C ψ Hω ), and the Theorem 3.3 is valid for any ¯ f ∈ C ψ C.

6.

Approximation by Fourier Sums on Classes Hω

6.1. In this section, we obtain asymptotic equalities of the form (0.9) for the quantities En (Hω ) = sup |ρn (f ; x)| = sup |f (x) − Sn−1 (f ; x)|, f ∈Hω

f ∈Hω

where Hω is a set of continuous 2π-periodic functions f (·) which satisfy the condition |f (x) − f (x )| ≤ ω(|x − x |), x, x ∈ R1 , (6.1)

Approximation by Fourier Sums in Spaces C and L1

236

Chapter 5

where ω = ω(t) is an arbitrary modulus of continuity. Theorem 6.1. The quantity En (Hω ) does not depend on a point x. For any modulus of continuity ω = ω(t), the following equality holds as n → ∞: 2θω ln n En (Hω ) = π2

π/2 2t ω( ) sin tdt + O(1)ω(1/n), n

(6.2)

0

here θω ∈ [2/3, 1]; θω = 1 if ω(t) is a convex modulus of continuity and O(1) is a quantity uniformly bounded in n. Theorem 6.1 yields the following statement: Theorem 6.1. If ω = ω(t) is a convex modulus of continuity, then 2 En (Hω ) = 2 ln n π

π/2 2t ω( ) sin tdt + O(1)ω(1/n), n

(6.2 )

0

as n → ∞. In particular, if ω(t) = Ktα , 0 < α ≤ 1, then K21+α En (KH α ) = 2 α ln n π n

π/2 tα sin tdt + O(1)n−α ;

(6.3)

0

moreover, if α = 1, then En (KH 1 ) =

4K ln n + O(1)n−α . π2n

(6.3 )

Proof of Theorem 6.1. Representing the partial sum Sn−1 (f ; x) in the form (1.1.4) and taking into account that the Dirichlet kernel (1.1.5 ) satisfies the equality π 1 Dn (t)dt = 1, (6.4) π −π

we get 1 ρn (f ; x) = π

π (f (x) − f (x + t))Dn−1 (t)dt. −π

(6.5)

Section 6

Approximation by Fourier Sums on Classes Hω

237

Therefore, since the class Hω is invariant under the shift of an argument (see Subsection 5.1), we obtain En (Hω ) = sup |ρn (f ; x)| = sup |ρn (f ; 0)|, f ∈Hω

f ∈Hω

(6.6)

i.e., indeed, the quantity En (Hω ) does not depend on a point x. Let us represent the Dirichlet kernel in the following form: 1 sin(n + 1/2)t cos kt = + . 2 2 sin t/2 n

Dn (t) =

(6.7)

k=1

Then 1 1 sin nt 1 cos nt − cos nt = − cos nt. 2 2 2 tan(t/2) 2

Dn−1 (t) = Dn (t) − Therefore, in view of (6.5), 1 ρn (f ; 0) = π

π (f (0) − f (t)) −π

sin nt 1 dt + an (f ). 2 tan(t/2) 2

(6.8)

Using expansion (4.3.54) and performing obvious transformations, we obtain 1 ρn (f ; x) = π

∞ (f (0) − f (t)) −∞

sin nt 1 dt + an (f ), t 2

(6.9)

where the improper integral is understood as the limit of integrals over expanding symmetrical intervals. By virtue of (1.23 ), for any f ∈ Hω , we have 1 |an (f )| = O(1)ω(1/n). 2 Using Lemma 4.5 (relation (4.59)), we get sin nt | (f (0) − f (t)) dt| = O(1)ω(1/n). t

(6.10)

(6.11)

|t|≥π/2

Finally, note that, for any a > 0, sin nt (f (0) − f (t)) dt| = O(1)(ω(1/n)). | t |t|≤a/n

(6.12)

238

Approximation by Fourier Sums in Spaces C and L1

Chapter 5

By combining relations (6.9)–(6.12), we obtain that, for every f ∈ Hω and for any a > 0, 1 sin nt ρ(f ; 0) = (f (0) − f (t)) dt + O(1)ω(1/n). (6.13) π t a/n≤|t|≤π/2

Therefore, by applying Lemma 4.7 (see equality (4.75)) we have 1 sin nt (f (0) − f (t)) dt| + O(1)ω(1/n). En (Hω ) = sup | ln (t) f ∈Hω π

(6.14)

i3,1

Here the notation i3,1 and ln and below −xk and tk have the same sense as in Lemma 4.7 for γn = 0. Denoting by En (Hω ) the first term on the right-hand side of (6.14), we get En (Hω )

k2 −1 k 1 −1 1 1 1 ≤ ( ek (ω)) + O(1)ω(1/n), ek (ω) + π |xk | xk k=k3

(6.15)

k=k0

where, by virtue of (5.18), tk+1

ek (ω) = sup | ϕ∈Hω

π/2n

ϕ(t) sin ntdt| = tk

ω(2t) sin ntdt.

(6.16)

0

6.2. Let us construct the function f ∗ ∈ Hω , for which the value of ρn (f ∗ ; 0) coincides with the right-hand side of (6.15). Let (see Subsection 5.5 ) ⎧ 1 ⎪ ⎪ t ∈ [tk , xk ], ⎪ ⎨ 2 ω(2(xk − t)), ϕk (t) = ⎪ ⎪ 1 ⎪ ⎩− ω(2(t − xk )), t ∈ [xk , tk+1 ], k = k0 , k1 − 1, k = k3 , k2 − 1, 2 ϕ+ (t) = (−1)k−k0 ϕk (t), t ∈ [tk , tk+1 ], k = k0 , k1 − 1, ϕ− (t) = (−1)k−k2 +1 ϕk (t), t ∈ [tk , tk+1 ], k = k3 , k2 − 1 and

⎧ ϕ+ (t), t ∈ [xk0 , xk1 ], ⎪ ⎪ ⎨ ϕ(t) = ϕ− (t), t ∈ [xk3 , xk2 ], ⎪ ⎪ ⎩ 0, t ∈ [−π/2, π/2] \ ([xk0 , xk1 ] ∪ [xk3 , xk2 ]).

˜ω Approximation by Fourier Sums on Classes H

Section 7

239

Thus, the function ϕ(t) is defined on the segment [−π/2, π/2]. Denote by f ∗ (t) its 2π-periodic extension. Direct calculations show that the value of ρn (f ∗ ; 0) coincides with the right-hand side of (6.15) for any modulus of continuity ω(t). In the case where ω(t) is a convex function, ϕ ∈ Hω . Thus, in this case, relation (6.15) is, in fact, an equality. Therefore, by combining (6.15), (6.16) and (5.13), we obtain equality (6.2) (for θω = 1 ). If ω(t) is not necessarily a convex modulus of continuity, then (see Subsection 1.2) the function f∗ (t) = 2f ∗ (t)/3 belongs to Hω . Therefore, in the general case, equality (6.2) holds.

7.

˜ω Approximation by Fourier Sums on Classes H

7.1. Let us establish an analog of Theorem 6.1 for conjugate functions. When ˜ ω , we always assume that the following condition is considering the classes H satisfied: 1 w(t) dt < ∞. (7.1) t 0

By virtue of Theorem 3.5.2, this condition guarantees the existence and continuity of a conjugate function f˜(x) for any f ∈ Hω at any point x. Moreover, ˜ ] and, hence, the equality condition (7.1) guarantees the equality S[f˜] = S[f ˜ ˜ Sn (f ; x) = Sn (f ; x) to be valid. Theorem 7.1. The quantity df ˜ ω ) = sup |ρn (f˜; x)| = sup |f˜(x) − Sn−1 (f˜; x)| En (H f ∈Hω

f ∈Hω

df ρn (f ; x)| = sup |f˜(x) − Sñ−1 (f ; x)| = sup |˜ f ∈Hω

f ∈Hω

(7.2)

does not depend on a point x. Let ω = ω(t) be an arbitrary modulus of continuity satisfying condition (7.1). Then the following relations hold as n → ∞ : ˜ ω ) = θω ( 1 En (H π

1

2t dt 2 ω( ) + ln n n t π

0

=

θω π

1 ω( 0

π/2 2t ω( ) sin tdt) + O(1)ω(1/n) n 0

2t dt ) + En (Hω ) + O(1)ω(1/n), n t

(7.3)

Approximation by Fourier Sums in Spaces C and L1

240

Chapter 5

where θω ∈ [2/3, 1], and θω = 1, if ω(t) is a convex modulus of continuity and O(1) is a quantity uniformly bounded in n. Proof. We have Sn (f˜; x) =

n

(ak sin kx − bk cos kx) =

k=1

n

A˜k (f ; x),

(7.4)

k=1

where ak and bk , k = 1, 2, . . . , are the Fourier coefficients of the function f (·). Using the formulas for numbers ak and bk we obtain 1 Sn (f˜; x) = − π

π ˜ n (t)dt, f (x + t)D

(7.5)

−π

where ˜ n (t) = D

n

sin kt

k=1

is the trigonometric polynomial of the nth degree, which is called a conjugate ˜ n (t) : Dirichlet kernel. An analog of formula (6.7) holds for the kernel D ˜ n (t) = cos(t/2) − cos(n + 1/2)t , D 2 sin t/2

(7.6)

˜ n (t) − 1 sin nt = 1 − cos nt = D ˜ n−1 (t) + 1 sin nt. D 2 2 tan(t/2) 2

(7.7)

which yields

Therefore, by virtue of (7.5) and (7.6), 1 Sn−1 (f˜; x) = − π

π f (x + t) −π

1 − cos nt 1 dt + Añ (f ; x), 2 tan(t/2) 2

(7.8)

and, in view of (3.5.4), we have 1 ρn (f ; x) = − π

π f (x + t) −π

cos nt 1 dt − Añ (f ; x). 2 tan(t/2) 2

(7.9)

˜ω Approximation by Fourier Sums on Classes H

Section 7

241

By using expansion (4.3.54), we get 1 ρñ (f ; x) = − π

∞ f (x + t) −∞

cos nt 1 dt − Añ (f ; x). t 2

(7.10)

Since the class Hω is invariant under the shift of an argument, the quantity ˜ ω ), in fact, does not depend on a point x. Thus, in view of (7.10), in what En (H follows, it suffices to consider the quantity 1 ρñ (f ; 0) = − π

∞ f (t) −∞

cos nt 1 dt − bn . t 2

(7.11)

By virtue of (1.23 ) and (4.59), for every f ∈ Hω , we get 1 ρñ (f ; 0) = − π

π/2 f (t)

cos nt dt + O(1)ω(1/n) t

−π/2

1 =− π

1/n (f (t) − f (0)) −1/n

1 − π

cos nt dt t

(f (t) − f (0))

cos nt dt + O(1)ω(1/n). t

(7.12)

1/n≤|t|≤π/2

The last equality is obtained by virtue of the fact that the function (cos nt)/t is odd. Further, by using Lemma 4.7 (see equality (4.75)) for a = 1 and γ = −π/2, we obtain 1 ρñ (f ; 0) = − π

1/n f (t)

cos nt dt t

−1/n

1 − π

(f (t) − f (0))

cos nt dt + O(1)ω(1/n). (7.13) ln (t)

i3,1

In the case considered, k0 = 2, k2 = −2 and xk =

kπ k−1 π − , tk = π, k = 0, ±1, . . . , n ∈ N. n 2n n

(7.14)

Approximation by Fourier Sums in Spaces C and L1

242

Chapter 5

By virtue of (7.13), we find 1/n

1 ρn (f ; 0)| ≤ ( sup | sup |˜ π f ∈Hω f ∈Hω

f (t)

cos nt dt| t

−1/n

+

k 2 −1 k=k3

where

k 1 −1 ek (w) ek (w) ) + O(1)ω(1/n), (7.15) + |xk | xk k=k0

tk+1

ek (ω) = sup ϕ∈Hω

π/2n

ϕ(t) cos ntdt = tk

ω(2t) sin ntdt.

(7.16)

0

However, for every f ∈ Hω , we have 1/n |

cos nt f (t) dt| ≤ t

−1/n

1/n cos nt |f (t) − f (−t)| dt t 0

1/n cos nt ω(2t) dt. ≤ t

(7.17)

0

Thus, relations (7.13)–(7.17), with regard for (5.13), yield ˜ω) ≤ 1 ( En (H π

1/n cos nt ω(2t) dt t 0

2 + ( ln n + O(1)) π

π/2n

ω(2t) sin ntdt) + O(1)ω(1/n). (7.18) 0

On the other hand, for the function f ∗ (·) constructed in Subsection 5.5 (for a = 1 ), calculations show that the value ρ˜(f ∗ ; 0) exactly coincides with the righthand side of (7.18). Since, for the convex moduli of continuity ω(t), the function 2 f ∗ ∈ Hω , and in the general case f ∗ ∈ Hω , moreover, 3 1/n 1 cos nt 2t dt ω(2t) dt = ω( ) + O(1)ω(1/n), t n t 0

0

(7.19)

Analogs of Theorems 3.1–3.3 in Integral Metric

Section 8

243

Theorem 7.1 is proved.

8.

Analogs of Theorems 3.1–3.3 in Integral Metric 8.1. In this section, we obtain asymptotic equalities for the quantities ψ¯

df

π

En (L N)1 = sup ρn (f ; x)1 , ϕ1 = f ∈Lψ¯ N

|ϕ(t)|dt,

(8.1)

−π

where, as above, ρn (f ; x) = f (x) − Sn−1 (f ; x), Sn−1 (f ; x) is a partial Fourier sum of order (n − 1) of a function f (·), and N is either a unit ball S1 in the space L, S1 = {ϕ : ϕ1 ≤ 1}, or the class Hω1 of functions from L satisfying the condition ϕ(· + t) − ϕ(·)1 ≤ ω(t), (8.2) where ω(t) is a given modulus of continuity. We formulate these results in the following theorem: Theorem 8.1. If ±ψ1 ∈ M0 and ±ψ2 ∈ M0 , then the following asymptotic equalities hold as n → ∞ : ¯ En (Lψ S10 )1

2 = π

∞

|ψ2 (t)| 4 ¯ ¯ ln n + O(1)ψ(n), dt + 2 ψ(n) t π

(8.3)

n

ψ¯

En (L

Hω01 )1

1 = θω1 ( | π

1

2t ω( ) n

0

2 ¯ ln n + 2 ψ(n) π

∞ ψ2 (nv) sin vtdvdt| 1

π/2 2t ¯ ω( ) sin tdt) + O(1)ψ(n)ω(1/n), (8.4) n 0

where O(1) are quantities uniformly bounded in n, and θω1 ∈ [1/2, 1]. Moreover, θω1 = 1 if ω(t) is a convex modulus of continuity. Here we also prove the following analog of Theorem 3.3:

Approximation by Fourier Sums in Spaces C and L1

244

¯ Lψ ,

Chapter 5

Theorem 8.2. If ±ψ1 ∈ M0 and ±ψ2 ∈ M0 , then, for every function f ∈ the following inequality holds for any n ∈ N :

2 ρn (f ; x)1 ≤ ( π

∞

|ψ2 (t)| 4 ¯ ψ¯ ¯ ln n + O(1)ψ(n))E dt + 2 ψ(n) n (f )1 , (8.5) t π

n

where O(1) is a quantity uniformly bounded in n and ¯

En (f ψ )1 =

inf

tn−1 ∈Tn−1

¯

f ψ (·) − tn−1 (·)1 .

8.2. If ±ψ1 , ±ψ2 ∈ MC , then, by virtue of (3.5), equalities (6.3) and (6.4) have the form 4 ¯ ¯ ¯ En (Lψ S10 )1 = 2 ψ(n) ln n + O(1)ψ(n), (8.3 ) π 2θω ¯ ln n En (L Hω01 )1 = 2 1 ψ(n) π ψ¯

π/2 2t ¯ ω( ) sin tdt + O(1)ψ(n)ω(1/n), n

(8.4 )

0

and, in particular, if π π ψ1 (t) = ψ(t) cos β , ψ2 (t) = ψ(t) sin β , 2 2

(8.6)

where ψ ∈ MC and β ∈ R1 , then 0 En (Lψ β S1 )1 =

0 En (Lψ β Hω )1

4 ψ(n) ln n + O(1)ψ(n), π2

2θ1 = 2 ψ(n) ln n π

π/2 sin tdt + O(1)ψ(n)ω(1/n).

(8.7)

(8.7 )

0

The proof of Theorems 8.1 and 8.2 is analogous to that of Theorems 3.1–3.3. As before, we take as a starting point the part of Corollary 4.3.1 that relates the ¯ ¯ functions from Lψ : If f ∈ Lψ , ±ψ1 ∈ M, and ±ψ2 ∈ M , then equality (4.3.60) holds almost everywhere. Following the scheme of the proof of Lemma 4.1, using the well-known estimates for norms of a convolution of the form ϕ(x − t)K(t)dt1 ≤ ϕ1 |K(t)|dt, (8.8) and using Lemma 2.3 instead of Lemma 1.3 in the corresponding places, we arrive at the following statement:

Analogs of Theorems 3.1–3.3 in Integral Metric

Section 8

245

Lemma 8.1. Let ±ψ1 ∈ M0 , ±ψ2 ∈ M0 , and let a be an arbitrary pos¯ itive number. If f ∈ Lψ S1 , then, for any n ∈ N, equality (4.3) holds almost ¯ everywhere. If f ∈ Lψ Hω1 , then, for any n ∈ N, equality (4.4) holds almost ¯ everywhere. If f ∈ Lψ , then, for any n ∈ N and for any tn−1 ∈ Tn−1 , equality ¯ (4.5) holds almost everywhere if instead f ψ (x − t) − tn−1 (x − t)C we take ¯ f ψ (x − t) − tn−1 (x − t)1 . By virtue of the analog of Lemma 4.7, we arrive at the analog of Corollary 4.1: Corollary 8.1. Let ±ψ1 ∈ M0 , ±ψ2 ∈ M0 , and let a be an arbitrary ¯ positive number. If f ∈ Lψ S1 , then, for any n ∈ N, equality (4.84) holds ¯ almost everywhere. If f ∈ Lψ Hω1 , then, for any n ∈ N, equality (4.85) holds ¯ almost everywhere. If f ∈ Lψ L, then, for any n ∈ N and for any tn−1 ∈ Tn−1 , equality (4.86) holds almost everywhere. Now we pass directly to the proof of equality (8.2). According to Corollary 8.1, we have

¯

En (Lψ S10 )1 = sup ϕ∈S10

+

ϕ(x − t)J2 (ψ2 ; n; t)1 dt

|t|≤a/n

¯ ψ(n) sin(nt − γn ) ¯ ϕ(x − t) (8.9) dt1 + O(1)ψ(n). π ln (t) i3,1

This equality, estimate (8.8), and equalities (5.4) and (5.5) yield ρn (f ; x) ≤ |t|≤a/n

2 = π

∞

¯ ψ(n) sin(nt − γn ) ¯ |J2 (ψ2 ; n; t)|dt + | |dt + O(1)ψ(n) π ln (t) i3,1

|ψ2 (t)| 4 ¯ ¯ ln n + O(1)ψ(n) dt + 2 ψ(n) t π

(8.10)

n ¯

for every f ∈ Lψ S10 . 8.3. Let us show that this relation cannot be a strict inequality. Earlier there were solved such problems for the classes for which were taken upper bounds. In that case, we indicate extremal functions, i.e., the functions for which the relations

Approximation by Fourier Sums in Spaces C and L1

246

Chapter 5

of the form (8.10) transform to equalities. Existing of such functions is guaranteed by the property of local compactness of the classes. The set L0 does not possess this property. Thus, the required extremal function does not necessarily exist. However, the existence of such a function is not necessary for our purpose. It suffices to find a sequence of functions in L0 that transform the relations of the form (6.10) into equalities with given accuracy. For this purpose, we use the following statement: Lemma 8.2. Let K be a function continuous on [−π, π], and K(−π) = K(π). Then π E(K) = sup y∈S10

where

K ∗ (·)

y(t − x)K(t)dt1 ≥

−π

1 max K ∗ (x) − K(x + h)1 , (8.11) 2 |h|≤π

is the 2π-periodic extension of the function K(·).

Proof. Assume that h ∈ [−π, π], h = 0, δn is an arbitrary infinitesimal sequence of positive numbers, δn < |h|, un (t) is a neighborhood of length δn centered at the point t, and yn (t) is a 2π-periodic function defined on [−π, π] by the equality ⎧ (2δn )−1 , t ∈ Un (0), ⎪ ⎪ ⎨ yn (t) = −(2δn )−1 , t ∈ Un (h), ⎪ ⎪ ⎩ 0, t ∈ [−π, π] \ (Un (0) ∪ Un (h)). Then yn ∈ S10 and, thus, π E(K) ≥

yn (t − x)K(t)dt1 .

−π

Since the function yn (·) is periodic, π −π

1 yn (t − x)K(t)dt = 2δn

x+δ n /2

1 K ∗ (t)dt − 2δn

x−δn /2

x+h+δ n /2

K ∗ (t)dt.

x+h−δn /2

By using the mean-value theorem, we get π −π

1 yn (t − x)K(t)dt = (K ∗ (x) − K ∗ (x + h)) + αn (x), 2

Analogs of Theorems 3.1–3.3 in Integral Metric

Section 8

247

where, in view of the uniform continuity of the function K ∗ (·), the quantity αn (x) uniformly tends to zero as n → ∞. Consequently, for any h ∈ [−π, π], 1 E(K) ≥ K ∗ (x) − K ∗ (x + h)1 + εn , εn → 0, n → ∞. 2 Lemma 8.2 is proved. Corollary 8.2. The statement of the lemma is true for any function K ∈ L(−π, π). Indeed, let ε > 0 be a fixed arbitrary number. In view of the density of the space C(a, b) in L(a, b), we find continuous function Kε (·) for which K(t)− Kε (t)1 < ε. We can assume, without loss of generality, that Kε (−π) = Kε (π). Therefore, π E(K) = sup y∈S10

(K(t) + Kε (t) − Kε (t))y(t − x)dt1

−π

π ≥ sup ( y∈S10

π Kε (t)y(t)y(t − x)dt1 −

−π

(K(t) − Kε (t))y(t − x)dt1 )

−π

π ≥ sup y∈S10

Kε (t)y(t − x)dt1 − ε.

−π

By virtue of (6.11), this relation yields 1 E(K) ≥ max 2 |h|≤π

π

|Kε∗ (x) − Kε∗ (x + h)|dx − ε

−π

=

1 max {K ∗ (x) − K ∗ (x + h)1 2 |h|≤π π −

(Kε∗ (x) − K ∗ (x) + Kε∗ (x + h) − K ∗ (x + h))dx − ε

−π

≥

1 max K ∗ (x) − K ∗ (x + h)1 − 3ε/2. 2 |h|≤π

Since ε is arbitrary, relation (8.12) yields the required statement.

(8.12)

248

Approximation by Fourier Sums in Spaces C and L1

Chapter 5

8.4. In order to use Lemma 8.2, in view of Corollary 8.2, let

K(t) =

⎧ J2 (ψ2 ; n; t), ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨¯ ψ(n) sin(nt − γn ) ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ 0,

πln (t)

|t| ≤ a/n, , t ∈ i3,1 ,

(8.13)

t ∈ [−π, π] \ (i3,1 ∪ [−a/n, a/n]).

Denote by K ∗ (t) the 2π-periodic extension of the function K(t). Thus, in view of (8.11) and (8.9), we get 1 ¯ ¯ En (Lψ S10 )1 ≥ K ∗ (x) − K ∗ (x + π/n)1 + O(1)ψ(n). 2

(8.14)

In this connection, by virtue of (5.4) and (5.5), we obtain the estimate ¯ En (Lψ S10 )1

2 ≥ π

∞

|ψ2 (t)| 4 ¯ ¯ ln n + O(1)ψ(n) dt + 2 ψ(n) t π

n

because (8.13) yields 1 K ∗ (x) − K ∗ (x + π/n)1 2 ¯ ψ(n) sin(nt − γn ) = K(·)1 = |J2 (ψ2 ; n; t)|dt + | |dt. (8.15) π ln (t) |t|≤a/n

i3,1

Combining estimates (8.10), (8.14), and (8.15), we obtain equality (8.3). 8.5. We now prove equality (8.4). We begin with equality (4.85). According to Corollary 8.1, it holds almost everywhere in the case under consideration. First, we note that

tk+1

J2 (ψ2 ; n; t)dt = 0, |t|≤a/n

tk

sin(nt − γn ) dt = 0. ln (t)

(8.16)

Analogs of Theorems 3.1–3.3 in Integral Metric

Section 8

249

Therefore, according to (4.85), ψ¯

En (L

Hω01 )1

≤ sup 0 ϕ∈Hω 1

ϕ(x − t)J2 (ψ2 ; n; t)dt1

|t|≤a/n

k k 2 −1 1 −1 ¯ ψ(n) 1 1 + ek (ω)1 ) ( ek (ω)1 + π |xk | xk k=k3

k=k0

¯ + O(1)ψ(n)ω(1/n), where

(8.17)

tk+1

ek (ω)1 = sup 0 ϕ∈Hω 1

ϕ(x − t) sin(nt − γn )dt1 .

(8.18)

tk

Since the function J2 (ψ2 ; n; t) is odd, in view of equalities (4.72) for any ϕ ∈ Hω01 we get

a/n ϕ(x − t)J2 (ψ2 ; n; t)dt1 ≤ | ω(2t)J2 (ψ2 ; n; t)dt|

|t|≤a/n

(8.19)

0

and

π/2n

ek (ω)1 ≤

ω(2t) sin ntdt.

(8.20)

0

Hence, according to (8.17)–(8.20), we have ψ¯

En (L

Hω01 )1

a/n ≤| ω(2t)J2 (ψ2 ; n; t)dt| 0 π/2n k k 2 −1 1 −1 ¯ ψ(n) 1 1 + ω(2t) sin ntdt( ) + π |xk | xk 0

¯ + O(1)ψ(n)ω(1/n).

k=k3

k=k0

(8.21)

Let us show that relation (8.21) turns into the equality in the case where ω(t) is a convex modulus of continuity. By virtue of (4.85) and (8.21), it suffices to show

Approximation by Fourier Sums in Spaces C and L1

250

Chapter 5

that there exists a function f ∗ (t) = fω∗ (t) in the class Hω01 for which π

a/n f ∗ (x − t)K(t)dt1 = | ω(2t)J2 (ψ2 ; n; t)dt|

−π

0 π/2n k k 2 −1 1 −1 ¯ 1 1 ψ(n) ω(2t) sin ntdt( ) + + π |xk | xk k=k3

0

Let

k=k0

¯ + O(1)ψ(n)ω(1/n).

(8.22)

⎧1 ⎪ t ∈ [0, π/2n), ⎪ ⎪ 4 ω(2t), ⎨ f1 (t) = − 1 ω(2|t|), t ∈ (−π/2n, 0], ⎪ ⎪ ⎪ ⎩ 4 0, π/2n ≤ |t| ≤ π.

(8.23)

Denote by f2 (t) the 2π-periodic extension of the function f1 (t). The required function f ∗ (t) has the form f ∗ (t) = f2 (t) − ω(π/n)/2. Let us verify that f ∗ ∈ Hω0 . If t ∈ [0, π/n], then the difference Δt (x) = π ∗ f (x + t) − f ∗ (x) is positive on the interval (−t − 2n , − 2t ), is negative on the π interval (− 2t , 2n ), and is equal to zero for all other x ∈ [−π, π]. Therefore, ∗

−t/2

∗

π/2n

f (x + t) − f (x)1 =

Δt (x)dx − −t−π/2n

−t/2

(f2 (x + t) − f2 (x))dx −

= −t−π/2n

Δt (x)dx −t/2

π/2n

(f2 (x + t) − f2 (x))dx = ω(t).

−t/2

If t ∈ [π/n, π], then ∗

∗

π/2n

f (x + t) − f (x)1 = 2

∗

π/2n

f2 (t)dt

f (t)dt = 4

−π/2n

0 π/2n

= 4f1 (t)|0

π = ω( ) ≤ ω(t). n

Section 8

Analogs of Theorems 3.1–3.3 in Integral Metric

251

Consequently, f ∗ ∈ Hω0 . Finally, we have to show that equality (8.22) holds for it. Let π Φ(x) = f ∗ (x − t)K(t)dt. −π

The function Φ(x) is identically equal to zero on the intervals (−π, tk3 − π/2n), (xk2 , −(a/n + π/2n)), (a/n + π/2n, tk0 − π/2n), and (xk1 , π). The function Φ(x) preserves its sign, successively changing it on the intervals (tk3 − π/2n, xk3 ), (xk3 , xk3 +1 ), . . . , (xk2 , xk2 +1 ), and on the intervals (tk0 − π/2n, xk0 ), (xk0 , xk0 +1 ), . . . , (xk1 −1 , xk1 ). Equality sign Φ(x) = sign K(x) holds on the intervals (−(a/n + π/2n), 0) and (0, a/n + π/2n). By using this information and performing calculations, we obtain relation (8.23). Thus, if ω(t) is a convex modulus of continuity, then relation (8.21) turns into the equality. However, if ω(t) is an arbitrary (not necessarily convex) modulus of continuity, then the construction of the function f ∗ (t), which requires differentiation, is, generally speaking, incorrect. However, in this case, it is also possible to show that relation (8.21) turns into the equality if one multiplies its right-hand side by some value θω1 , θω1 ∈ [1/2, 1]. For this purpose, using Lemma 3.1.1, one can choose convex modulus of continuity ω∗ (t) satisfying condition (3.1.15) and construct the function f ∗ (t) = f∗∗ (t) for it. In this case, f∗∗ ∈ Hω0 and the value of the left-hand side of (8.22) for this function is equal to the value of the right-hand side of (8.22) multiplied by some value θω1 ∈ [1/2, 1]. By this and, in view of (5.13), we arrive at (8.4). Theorem 8.1 is proved. Proof of Theorem 8.2. According to Corollary 8.1, equality (4.86) holds ¯ almost everywhere for every f ∈ Lψ L0 , for any n ∈ N, and for any polynomial tn−1 (·) from Tn−1 . Thus, ρn (f ; x)1 ≤ ( |t|≤a/n

¯ ψ(n) sin(nt − γn ) |J2 (ψ2 ; n; t)|dt + | |dt π ln (t) i3,1 ψ¯ ¯ (·) − tn−1 (·)1 . + O(1)ψ(n))f

Hence, to obtain estimate (8.5), it remains to use equalities (5.4) and (5.5) and ¯ choose as tn−1 (·) the polynomial of best approximation of the function f ψ (·) in the space L.

252

9.

Approximation by Fourier Sums in Spaces C and L1

Chapter 5

Analogs of Theorems 6.1 and 7.1 in Integral Metric

9.1. In this section, we obtain asymptotic equalities for the quantities ˜ ω )1 . En (Hω1 )1 and En (H 1 Theorem 9.1. For any modulus of continuity ω = ω(t), the following asymptotic equality holds as n → ∞ : 2θω En (Hω1 )1 = 2 ln n π

π/2 2t 1 ω( ) sin tdt + O(1)ω( ), n n

(9.1)

0

where θ ∈ [1/2, 1], θω = 1 if ω(t) is a convex modulus of continuity, and O(1) are quantities uniformly bounded in n. Theorem 9.1 is an analog of Theorem 6.1 in the integral metric. It yields an analog of Theorem 6.1 : Theorem 9.1. If ω = ω(t) is a convex modulus of continuity, then 2 En (Hω1 )1 = 2 ln n π

π/2 2t ω( ) sin tdt + O(1)ω(1/n) n

(9.2)

0

as n → ∞. In particular, if ω(t) = Ktα , 0 < α ≤ 1, then En (Hω1 )1 =

En (KH1α )1

K2α+1 = 2 α ln n π n

π/2 tα sin tdt + O(1)n−α ;

(9.3)

0

moreover, if α = 1, then En (Hω1 )1 = En (KH11 )1 =

4K ln n + O(1)n−1 . π2n

(9.3 )

Comparing Theorems 6.1 and 9.1, we obtain the following statement: ∞

Theorem 9.2. If ω = ω(t) is a convex modulus of continuity, then as n → En (Hω ) − En (Hω1 )1 = O(1)ω(1/n),

where O(1) are quantities uniformly bounded in n.

(9.4)

Section 10

Approximations of Functions of High Smoothness

253

9.2. In the case of conjugate classes, the following statements are true: Theorem 9.3. Let ω(t) be an arbitrary modulus of continuity such that 1

ω(t) dt < ∞. t

(9.5)

0

Then the following asymptotic equality holds as n → ∞ : ˜ ω )1 = θω ( 1 En (H 1 π

1

2t dt 2 ω( ) + 2 ln n n t π

0

π/2 2t 1 ω( ) sin tdt) + O(1)ω( ), (9.6) n n 0

where θω ∈ [1/2, 1], θω = 1 if ω(t) is a convex modulus of continuity, and O(1) are quantities uniformly bounded in n. Comparing Theorems 7.1 and 9.3, we obtain the following statement: Theorem 9.4. If ω(t) is a convex modulus of continuity which satisfies condition (9.5), then, as n → ∞, ˜ ω ) − En (H ˜ ω ) = O(1)ω(1/n), En (H 1

(9.7)

where O(1) are quantities uniformly bounded in n. Proof of Theorems 9.1 and 9.3. The proof is carried out by analogy with the proof of Theorems 6.1 and 7.1. Here we use estimates of the form (8.8), Lemma 2.3, representation (6.9) written at an arbitrary point x, i.e., 1 ρn (f ; x) = π

π (f (x) − f (x + t)) −π

sin nt 1 dt + An (f ; x), t 2

(9.8)

and equality (7.10), following the proof of equality (8.4) in Theorem 8.1.

10.

Approximations of Functions of High Smoothness by Fourier Sums in Uniform Metric

10.1. We obtain analogs of Theorems 3.1–3.3 in the case, when both func¯ tions ψ1 (t) and ψ2 (t), which determine the sets Lψ , are such that ±ψ1 ∈ F

254

Approximation by Fourier Sums in Spaces C and L1

Chapter 5

and ±ψ2 ∈ F. Where F is a subset of functions ψ ∈ M determined in Subsection 3.13. We shall formulate these statements in two versions. The first version embraces the pairs ψ¯ = (ψ1 , ψ2 ) for which one can find positive constants K1 and K2 satisfying the following condition: 0 < K1 ≤

η(ψ1 ; t) − t ≤ K2 < ∞ ∀t ≥ 1. η(ψ2 ; t) − t

(10.1)

The second version is applied to the pairs ψ¯ = (ψ1 , ψ2 ) for which |ψ2 (t)| ln+ (η(ψ2 ; t) − t) ≤ O(1)|ψ1 (t)|, ln+ x = max(ln x, 0). |ψ1 (t)| ln+ (η(ψ1 ; t) − t) ≤ O(1)|ψ2 (t)|

(10.2) (10.2 )

as t → ∞. Note that, by virtue of Theorem 3.13.2 and the remark to it, condition (10.1) is equivalent to the following condition: 0 < K1
ψ1 (t)ϕ(t) = ψ2 (η(ψ2 ; t)). 2 Consequently, η(ψ2 ; t) < η(ψ1 ; t), which yields (10.3). On the other hand, by virtue of Theorem 3.13.2, there exists the constant K > 0 such that for all t ≥ 1 we have ψ1 (t) ψ2 (t) ψ1 (t) ϕ (t) R1,2 (t) ≤ K · = K(1 + ). (10.4) ψ2 (t) ψ1 (t) ψ1 (t) ϕ(t) For any ψ1 ∈ M+ ∞ and ϕ ∈ M0 , by virtue of Theorem 3.12.1 we get ψ1 (t) tϕ (t) · ≤ K1 ∀t ≥ 1. tψ1 (t) ϕ(t)

(10.5)

Combining relations (10.3)–(10.5), we obtain (10.1). Thus, for the pair of functions ψ1 and ψ2 from the set F, condition (10.1) is satisfied if and only if the ratio ψ1 (t)/ψ2 (t) is any function from M0 . In particular, these can be the functions ϕ(t) = t−r for any r > 0, the functions ϕ(t) = ln−ε (t + e) for any ε > 0, etc. The following statement is important in this direction: Proposition 10.3. Suppose that the functions ψ1 , ψ2 ∈ M satisfy the following inequality for all t ≥ t0 ≥ 1 : ψ1 (t) ψ2 (t) ≥ 1 + ε, ψ1 (t) ψ2 (t)

(10.6)

where ε is an arbitrary positive number. Then (ψ1 (t0 ))1+ε ψ1 (t) ≥ K(ψ1 (t))−ε , K = ψ2 (t) ψ2 (t0 ) for all t ≥ t0 .

(10.7)

256

Approximation by Fourier Sums in Spaces C and L1

Chapter 5

Proof. Rewriting (10.6) in the form −

ψ2 (t) ψ (t) ≥ −(1 + ε) 1 ψ2 (t) ψ1 (t)

and integrating both its parts over the segment [t0 , t], we get ln

ψ2 (t0 ) ψ1 (t0 ) ≥ (1 + ε) ln , ψ2 (t) ψ1 (t)

which yields (10.6). 10.2. The basic results of this subsection are in the following statements: Theorem 10.1. If ±ψ1 , ±ψ2 ∈ F, then the quantities ¯

¯

En (C ψ N) = sup{|ρn (f ; x)| : f ∈ C ψ N}, N = M 0 ∪ Hω0 , do not depend on the value x. Moreover, if, in addition, condition (10.1) (or (10.1 )) is satisfied, then, as n → ∞, the following asymptotic equalities hold: ¯

ψ En (C∞ )= ¯

En (C ψ Hω0 ) =

¯ 4ψ(n) ¯ ln+ (η(n) − n) + O(1)ψ(n), π2

¯ 2ψ(n)e n (ω) ¯ ln+ (η(n) − n) + O(1)ψ(n)ω(1/n), 2 π

(10.8) (10.9)

¯ where ψ(n) = (ψ12 (n) + ψ22 (n))1/2 , η(n) is either η(ψ1 ; n) = ψ1−1 ( 12 ψ1 (n)) 1 or η(ψ2 ; n) = ψ2−1 ( ψ2 (n)), en (ω) is the quantity determined by the equality 2 π/2 2t ω( ) sin tdt, en (ω) = θω n

(10.10)

0

θω ∈ [2/3, 1], θω = 1 if ω(t) is a convex modulus of continuity, and ln+ (t) = max{ln t, 0} and O(1) are quantities uniformly bounded in n. Theorem 10.2. Let ±ψ1 , ±ψ2 ∈ F. If condition (10.2) is satisfied, then, as n → ∞, the following asymptotic equalities hold: ¯

ψ En (C∞ )=

4|ψ1 (n)| + ln (η(ψ1 ; n) − n) + O(1)|ψ1 (n)|, π2

(10.11)

Section 10

Approximations of Functions of High Smoothness

257

¯

En (C ψ Hω0 ) =

2|ψ1 (n)| en (ω) ln+ (η(ψ1 ; n) − n) + O(1)|ψ1 (n)|ω(1/n), (10.12) π2

where the quantities η(ψ1 ; n), η(ψ2 ; n), en (ω), and O(1) have the same sense as in Theorem 10.1. If condition (10.2 ) is satisfied, then ψ1 (·) should be replaced by ψ2 (·) on the right-hand sides of (10.11) and (10.12). By virtue of Proposition 10.1, if ±ψ1 ∈ MC and ±ψ2 ∈ MC , then condition (10.1) is satisfied and equalities (10.8) and (10.9) coincide with equalities (3.4 ) and (3.10 ), respectively. Now, let the functions ψ1 (·) and ψ2 (·) be determined by the following equalities: π π ψ1 (t) = ψ(t) cos β , ψ2 (t) = ψ(t) sin β , (10.13) 2 2 where ψ ∈ F and β is an arbitrary real number. It is clear that condition (10.1) is satisfied in this case. Consequently, the following statement is valid: Theorem 10.1. Let ψ ∈ F and β ∈ R1 . Then, as n → ∞, ψ )= En (Cβ,∞

4ψ(n) + ln (η(n) − n) + O(1)ψ(n), π2

(10.14)

2ψ(n) en (ω) ln+ (η(n) − n) + O(1)ψ(n)ω(1/n), (10.15) π2 1 where η(t) = η(ψ; t) = ψ −1 ( ψ(t)), and the quantities en (ω) and O(1) have 2 the same sense as in Theorem 10.1. En (Cβψ Hω0 ) =

The function ψ(t) = exp(−αtr ), as noted above, belongs to M+ ∞ ; thus, it belongs to F for any α > 0 and r > 0. For this function, the following equality holds: ln 2 + O(1)). (10.16) η(ψ; t) − t = t1−r ( rα Therefore, Theorem 10.1 yields the following statement: Theorem 10.1. Let ψ(t) = exp(−αtr ), α > 0, r > 0, and β ∈ R1 . Then, as n → ∞, ψ En (Cβ,∞ )=

4 exp(−αnr ) ln+ n1−r + O(1) exp(−αnr ), π2

(10.17)

258

Approximation by Fourier Sums in Spaces C and L1

En (Cβψ Hω0 ) =

Chapter 5

2 exp(−αnr )en (ω) ln+ n1−r π2 + O(1) exp(−αnr )ω(1/n), (10.18)

where the quantities en (ω) and O(1) have the same meaning as in Theorem 10.1 . The following statement is an analog of Theorem 3.3: Theorem 10.3. Let ±ψ1 , ±ψ2 ∈ F. If condition (10.1) is satisfied, then, for ¯ any f ∈ C ψ C 0 and n ∈ N, the following inequality holds: 4 + ψ¯ ¯ ρn (f ; x)C ≤ ln (η(n) − n) + O(1) ψ(n)E (10.19) n (f )C , 2 π 1 ¯ where ψ(n) = (ψ12 (n) + ψ22 (n))1/2 , η(n) is either η(ψ1 ; n) = ψ1−1 ( ψ1 (n)) 2 −1 1 or η(ψ2 ; n) = ψ2 ( ψ2 (n)), and O(1) is a quantity uniformly bounded in n. 2 ¯ If condition (10.2) is satisfied, then, for any f ∈ C ψ C 0 and n ∈ N, 4 + ¯ ρn (f ; x)C ≤ ln (η(ψ1 ; n) − n) + O(1) ψ1 (n)En (f ψ )C . (10.20) 2 π If condition (10.2 ) is satisfied, then ψ1 should be replaced by ψ2 on the righthand side of (10.20). ¯ For any function f ∈ C ψ C 0 and every n ∈ N, there exists a function ¯ ¯ ¯ F (x) = F (f ; n; x) in the space C ψ C 0 such that En (F ψ )C = En (f ψ )C and relations (10.19) and (10.20) become equalities. The following theorem is obtained from Theorem 10.3 by analogy with Theorem 3.3 : Theorem 10.3. Let ±ψ1 , ±ψ2 ∈ F. If condition (10.1) is satisfied, then, for ¯ any class f ∈ C ψ Cn0 (ε) and any n ∈ N, the following equality holds: 4 + ln (η(n) − n) + O(1))εn . π2 If condition (10.2) is satisfied, then ¯

En (C ψ Cn0 (ε)) = (

¯

En (C ψ Cn0 (ε)) = (

4 + ln (η(ψ1 ; n) − n) + O(1))εn . π2

(10.21)

(10.22)

If condition (10.2 ) is satisfied, then ψ1 should be replaced by ψ2 on the righthand side of (10.22).

Section 11

11.

Auxiliary Statements

259

Auxiliary Statements

11.1. By analogy with the proof of Theorems 3.1–3.3 , we use Corollary 4.3.1, which establishes the equality for the quantity ρn (f ; x) (see (4.1)). Our purpose is to simplify the right-hand side of (4.1) and to select its principal value. We proceed by analogy with Section 4, in view of the fact that functions ψ1 and ψ2 are chosen from the set F and, consequently, they could tend to zero ¯ faster than the functions from M0 . In view of (4.1), for every f ∈ Lψ we set ∞ ρn (f ; x)1 = −∞

ψ1 (n) Δ(x, t)J2 (ψ1 ; n; t)0 dt + 2π

∞ ρn (f ; x)2 = −∞

π Δ(x, t) cos ntdt,

(11.1)

Δ(x, t) sin ntdt,

(11.2)

−π

ψ2 (n) Δ(x, t)J2 (ψ1 ; n; t)1 dt + 2π

π −π

so that ρn (f ; x) = ρn (f ; x)1 + ρn (f ; x)2 .

(11.3)

The notation in these equalities is the same as in Subsection 4.1. Let us prove the following auxiliary statement: Lemma 11.1. Let ±ψ1 , ±ψ2 ∈ M and let α = α(n) and α = α (n) be two arbitrary sequences of real numbers for which nα(n) ≥ α0 > 0 and ¯ df

ψ 0 , then nα (n) ≥ α0 > 0. If f ∈ C∞ = C ψ SM

ρn (f ; x)1 = −

ψ1 (n) π

sin nt ψ¯ 1 dt + bψ n (α; f ; x), t

(11.4)

cos nt ψ¯ 2 dt + bψ n (α ; f ; x), t

(11.4 )

¯

f ψ (x − t) |t|≥α(n)

ψ2 (n) ρn (f ; x)2 = π

¯

f ψ (x − t) |t|≥α (n)

for any n ∈ N at every point x. Moreover, ¯

ψ1 ψ 1 bψ n (α; f ; x)C ≤ O(1)(|ψ1 (n)| + Qn (α)), ¯

ψ ψ2 2 bψ n (α ; f ; x)C ≤ O(1)(|ψ2 (n)| + Qn (α )). ¯

If f ∈ C ψ Hω0 , then

260

Approximation by Fourier Sums in Spaces C and L1

ρn (f ; x)1 =−

ψ1 (n) π

¯

¯

(f ψ (x − t) − f ψ (x))

Chapter 5

sin nt ψ¯ 1 dt + dψ n (α; f ; x), (11.5) t

|t|≥α(n)

ρn (f ; x)2 ψ2 (n) = π

¯

¯

(f ψ (x − t) − f ψ (x)) |t|≥α (n)

cos nt ψ¯ 2 dt + dψ n (α ; f ; x), (11.5 ) t

for any n ∈ N at every point x. Moreover, ¯

ψ1 ψ 1 dψ n (α; f ; x)C ≤ O(1)(|ψ1 (n)| + Qn (α))ω(1/n), ¯

ψ ψ2 2 dψ n (α ; f ; x)C ≤ O(1)(|ψ2 (n)| + Qn (α ))ω(1/n). ¯

If f ∈ C ψ C 0 , then, for any polynomial tn−1 ∈ Tn−1 , ρn (f ; x)1 = −

ψ1 (n) π

sin nt ¯ dt + gnψ1 (α; f ψ ; x), t

(11.6)

cos nt ¯ dt + gnψ2 (α ; f ψ ; x), t

(11.6 )

δn (x − t) |t|≥α(n)

ρn (f ; x)2 =

ψ2 (n) π

δn (x − t) |t|≥α (n)

¯

for any n ∈ N at every point x. Here, δn (v) = f ψ (v) − tn−1 (v) and ¯

1 gnψ1 (α; f ψ ; x)C ≤ O(1)(|ψ1 (n)| + Qψ n (α))δn C ,

¯

2 gnψ2 (α ; f ψ ; x)C ≤ O(1)(|ψ2 (n)| + Qψ n (α ))δn C .

The quantities Qψ n (·) are determined by the equality ∞ Qψ n (α)

=| 1/α(n)

ψ(t + n) dt| + | t

∞

t−1 (ψ(n) − ψ(n + t−1 ))dt|,

α(n)

and O(1) are quantities uniformly bounded in n.

(11.7)

Section 11

Auxiliary Statements

261

Proof. This lemma is an analog of a combination of Lemmas 4.3 and 4.4. However, in this case, instead of the sequence a/n, n = 1, 2, . . . , where a is a certain positive number, we have arbitrary positive sequences α(n) and α (n) for which nα(n) ≥ α0 > 0 and nα (n) > 0. Generally speaking, these sequences may not tend to zero (in what follows, we set α(n) = (η(ψ1 ; n) − n)−1 and α (n) = (η(ψ2 ; n) − n)−1 ). Moreover, we consider the case where ±ψ1 , ±ψ2 ∈ F. Thus, the proof of the lemma is similar to the proofs of Lemmas 4.3 and 4.4, but it has some specific features. For example, relation (4.24) may not be valid for ψ1 ∈ F. First, we consider the quantities ρn (f ; x)1 . We have ψ1 (n) sin nt ρn (f ; x)1 = − Δ(x, t) (11.8) dt + rn (f ; x)1 , π t |t|≥α(n)

where

1 Δ(x, t)J2 (ψ1 ; n; t)0 dt − π

rn (f ; x)1 = |t|≤α(n)

Δ(x, t)J3 (ψ1 ; t)0 dt |t|≥α(n)

ψ1 (n) + 2π

π Δ(x, t) cos ntdt −π

df

= R1 (f ; x) + R2 (f ; x) + R3 (f ; x), where, as above, 1 J3 (ψ1 ; t)0 = t

∞

(11.9)

ψ1 (v) sin vtdv.

(11.10)

n

11.2. By virtue of Lemma 4.2, we have ¯

ψ , R3 (f ; x)C = O(1)|ψ1 (n)| ∀f ∈ C∞

(11.11)

R3 (f ; x)C ≤ O(1)|ψ1 (n)|ω(1/n) ∀f ∈ C ψ Hω ,

(11.12)

and ¯

R3 (f ; x)C ≤ O(1)|ψ1 (n)|f ψ (·) − tn−1 (·)M ∀f ∈ C ψ C. We obtain necessary estimates for R1 (f ; x) and R2 (f ; x). We get

(11.13)

Approximation by Fourier Sums in Spaces C and L1

262

Chapter 5

α(n)

R1 (f ; x)C ≤ 2Δ(x; t)C

|J2 (ψ1 ; n; t)0 |dt.

(11.14)

0

By setting ϕ(v) = ψ1 (v) and β = 1 in (4.3.25), by virtue of (3.28), we find 1 1 |J2 (ψ1 ; n; t)0 | = |Φ1 (ψ1 ; n)| ≤ π π

n+2π/t

ψ1 (v)dv.

(11.15)

n

Therefore, in view of relation (3.1), we get α(n)

1 |J2 (ψ1 ; n; t)0 |dt ≤ π

0

α(n) n+2π/t

ψ1 (v)dvdt n

0

1 = (t π

n+2π/t

α(n) ψ1 (v)dv|0

α(n)

+ 2π

n

ψ1 (n + 2π/t) dt) t

0

∞ ≤ 2(|ψ1 (n)| + π

ψ1 (t + n) dt). t

(11.16)

2π/α(n)

Combining relations (11.4) and (11.6) and taking (3.1) into account, we get ∞ R1 (f ; x)C = O(1)(|ψ1 (n)| + |

ψ1 (t + n) ψ¯ dt|) ∀f ∈ C∞ t

(11.17)

1/α(n)

and R1 (f ; x)C ψ¯

= O(1)f (·) − tn−1 (·)C (|ψ1 (n)| + | 1/α(n)

ψ1 (t + n) dt|). (11.18) t

Section 11

Auxiliary Statements

263

11.3. In order to estimate the quantity α(n)

R1 (f ; x) =

Δ(x, t)J2 (ψ; n; t)0 dt −α(n) nα(n)

1 = n

∞ Δ(x, t/n)

−nα(n)

on the class

¯ CψH

ψ1 (nv) cos vtdvdt

(11.19)

1

ω,

consider the function ∞ ∞ ψ1 (nv) cos vtdvdt Φ2 (ψ1 ; x) = x

1

∞ ∞ sin nt n ψ1 (nv) sin vtdv)dt − = − (ψ1 (n) t t x

1

= ψ1 (n)si x − nΦ3 (x), t > 0, x > 0,

(11.20)

where Φ3 (x) is a function determined by relation (4.30). Denote, as in Proposition 4.1, zeros of integral sine by xk , k = 0, 1, . . . . In view of relation (4.38), we obtain signΦ2 (ψ1 ; xk ) = −signΦ3 (xk ) = (−1)k+1 . (11.21) The function Φ2 (ψ; x) is continuous. Thus, by virtue of (11.21), on every interval ¯k . Setting x ¯0 ≤ (xk , xk+1 ) it vanishes with change of its sign at a certain point x nα(n), we choose k under condition x ¯k −1 ≤ nα(n) ≤ x ¯k . Then, using Lemma 1.3, we get 1 | n

nα(n)

t Δ(x, )J2 (ψ1 ; n; t/n)0 dt| n

0

1 ≤ max |Δ(x, t/n)| 0≤t≤¯ x0 n + ω(Δ)

1 n

x ¯k −1

x¯0 |J2 (ψ1 ; n; t/n)0 |dt 0

|J2 (ψ1 ; n; t/n)0 |dt x ¯0

1 |Δ(x, t/n)| + max n x ¯k−1 ≤t≤nα(n)

nα(n)

|J2 (ψ1 ; n; t/n)0 |dt x ¯k −1

(11.22)

Approximation by Fourier Sums in Spaces C and L1

264

Chapter 5

¯

for every f ∈ C ψ Hω . Taking into account that, in the case considered, Δ < 4π and Δ(x; 0) = 0 and using estimates (11.15) and (11.16) for the sum of the first two terms in (11.22), which is denoted by σ(f ), we obtain α(n)

σ(f ) ≤ Kω(1/n)

|J2 (ψ1 ; n; t)0 |dt 0

∞ ≤ O(1)ω(1/n)(|ψ1 (n)| + |

ψ1 (t + n) dt|) t

(11.23)

1/α(n) ¯

for every f ∈ C ψ Hω . Denoting the last term in (11.22) by i2 (f ), we get nα(n)

1 i2 (f ) ≤ ω(α(n)) n

|J2 (ψ1 ; n; t/n)|dt x ¯k −1

1 = ω(α(n)) π

nα(n) ∞

|

x ¯k −1

ψ1 (nv) cos vtdv|dt 1

≤ 2ψ1 (n)ω(α(n))| ln

nα(n) | x ¯k −1

(11.24)

because (see 11.15) ∞

1+2π/t

ψ1 (nv) cos vtdv| ≤

| 1

|ψ1 (nv)|dv ≤

2πψ1 (n) . t

(11.25)

1

In view of relation (4.31) and since ω(λt) ≤ (λ + 1)ω(t) for any λ > 0 (see Subsection 3.1.4), we have nα(n) ω(α(n)) ln x ¯k −1 ⎧ Kω(1/n), nα(n) ≤ 4π, ⎪ ⎪ ⎨ ≤ h nα(n) + 1 ⎪ ⎪ ⎩ω(α(n)) ln(1 + ) ≤ 2π ω(1/n), nα(n) > 4π. nα(n) + h nα(n) − 2π

Section 11

Auxiliary Statements

265

Here, h = nα(n) − x ¯k −1 . One can see that the following relation is always true: nα(n) ≤ O(1)ω(1/n). ω(α(n)) ln (11.26) x ¯k −1 Therefore, if nα(n) ≥ x ¯0 , then according to (11.24) and (11.26), we have i2 (f ) ≤ O(1)ψ1 (n)ω(1/n).

(11.27)

¯

¯0 < x0 < π/2, in view of (11.23), for every f ∈ C ψ Hω . Since α0 ≤ nα(n) < x we get α(n)

∞

Δ(x; t)J2 (ψ1 ; n; t)0 dt| ≤ O(1)ω(1/n)(|ψ1 (n)| + |

| 0

ψ1 (t + n) dt|). t

1/α(n)

Therefore, combining relations (11.22), (11.23), and (11.27), we obtain 1 | n

nα(n)

α(n)

Δ(x, t/n)J2 (ψ1 ; n; t/n)0 dt| = | 0

Δ(x, t)J2 (ψ1 ; n; t)0 dt| 0

∞

ψ1 (t + n) dt|) t

≤ O(1)ω(1/n)(|ψ1 (n)| + |

(11.28)

1/α(n) ¯

for every f ∈ C ψ Hω . It is clear that the fact that the function J2 (ψ1 ; n; t)0 is even yields the validity of such an estimate for the integral over the interval ¯ (−α(n), 0). Consequently, for every f ∈ C ψ Hω ∞ R1 (f ; x)C ≤ O(1)ω(1/n)(|ψ1 (n)| + |

ψ1 (t + n) dt|). t

1/α(n)

11.4. We obtain a necessary estimate for the quantity 1 Δ(x, t)J3 (ψ1 ; t)0 dt R2 (f ; x) = π |t|≥α(n)

1 = nπ

Δ(x, t/n)J3 (ψ1 ; t/n)0 dt, |t|≥nα(n)

(11.29)

Approximation by Fourier Sums in Spaces C and L1

266

where, as usual, 1 J3 (ψ1 ; t)0 = t

∞

Chapter 5

ψ 1 (v) sin vtdv.

n

By using the intermediate estimate in (4.3.29), we get 1 Δ(x, t)J3 (ψ1 ; t)0 dt| | π |t|≥α(n)

∞

2 ≤ Δ(x, t)C π

|ψ1 (n + 2π/t) − ψ(n)| dt. (11.30) t

α(n) ¯

ψ , In this connection, we obtain, for every f ∈ C∞

∞ R2 (f ; x)C ≤ O(1)|

t−1 (ψ1 (n) − ψ1 (n + 1/t))dt|,

(11.31)

α(n) ¯

and for every ψ ∈ C ψ C R2 (f ; x)C ψ¯

∞

≤ O(1)f (·) − tn−1 (·)C |

t−1 (ψ1 (n) − ψ1 (n + 1/t))dt|. (11.32)

α(n) ¯

Let us obtain an estimate for the quantity R2 (f ; x) on the class C ψ Hω . Assume, as above, that x ¯k are zeros of the function Φ2 (ψ1 ; x), and k is chosen from the condition x ¯k −1 ≤ nα(n) < x ¯k . By using Lemma 1.3, we get 1 | nπ

∞ Δ(x, t/n)J3 (ψ1 ; t/n)0 |dt nα(n)

1 ≤ max |Δ(x, t/n)| nπ nα(n)≤t≤¯ xk

x¯k |J3 (ψ1 ; t/n)0 |dt nα(n)

ω(Δ) + n

∞ |J3 (ψ1 ; t/n)0 |dt. (11.33) x ¯k

Section 11

Auxiliary Statements

267

¯

for every f ∈ C ψ Hω . Since Δ(x, 0) = 0, the first term of the right-hand side in (11.33) (denote it by i3 (f ) ) does not exceed the quantity x ¯k 1 ω( ) n nπ

x¯k |J3 (ψ2 ; t/n)0 |dt.

(11.34)

nα(n)

Since, by virtue of (4.31), x ¯k = nα(n) + h, 0 ≤ h ≤ 2π and nα(n) ≥ α0 > 0, using estimate from (4.3.29), we get ψ1 (n) 2π i3 (f ) ≤ ω(α(n) + ) nπ n

x¯k

dt t

nα(n)

=

ψ1 (n) 2π x ¯k ω(α(n) + ) ln . nπ n nα(n)

(11.35)

However,

ln

⎧ 2π ⎪ ⎪ ⎪ln(1 + α ), h ≥ nα(n), ⎨ 0

x ¯k h = ln(1 + )≤ ⎪ nα(n) nα(n) ⎪ 2π ⎪ ⎩ , nα(n)

(11.36)

h < nα(n).

Thus, x ¯k 2π ) ln n nα(n) ⎧ Kω(1/n), ⎪ ⎪ ⎨

ω(α(n) +

≤

h ≥ nα(n),

2π(nα(n) + 2π + 1) ⎪ ⎪ ⎩ ω(1/n) ≤ Kω(1/n), h < nα(n). nα(n)

(11.37)

Therefore, we always have i3 (f ) ≤ O(1)

|ψ1 (n)| ¯ ω(1/n) ∀f ∈ C ψ Hω . n

(11.38)

Approximation by Fourier Sums in Spaces C and L1

268

Chapter 5

To estimate the second term in (11.33), note that, by virtue of (4.31), Δ < 4π. Using the estimate from (4.3.29), we obtain ∞

ω(Δ) n

|J3 (ψ1 ; t/n)0 |dt x ¯k

∞ ≤ O(1)ω(1/n)(|

t−1 (ψ1 (n) − ψ1 (n + 1/t))dt| + ψ1 (n)). (11.39)

α(n) ¯

Relations (11.33), (11.38), and (11.39) imply that, for every f ∈ C ψ Hω , 1 | π

∞ Δ(x, t)J3 (ψ1 ; t)0 dt| α(n)

∞ ≤ O(1)ω(1/n)(|

t−1 (ψ1 (n) − ψ1 (n + 1/t))dt| + ψ1 (n)). (11.40)

α(n)

It is clear that the same estimate holds for the integral over the interval t < −α(n). Therefore, R2 (f ; x)C ∞ ≤ O(1)ω(1/n)(|

t−1 (ψ1 (n) − ψ1 (n + 1/t))dt| + ψ1 (n)) (11.41)

α(n) ¯

for every f ∈ C ψ Hω . 11.5. Combining estimates (11.10)–(11.12), (11.17)–(11.19), (11.31), (11.32), and (11.41), by virtue of (11.9) we obtain the proof of Lemma 11.1 for ρn (f ; x)1 . The proof of the lemma for the quantity ρn (f ; x)2 is based on the following equality: ψ2 (n) cos nt Δ(x, t) ρn (f ; x)2 = dt + rn (f ; x)2 , π t |t|≥α (n)

Section 11

Auxiliary Statements

where

Δ(x, t)J2 (ψ2 ; n; t)1 dt −

rn (f ; x)2 = |t|≤α (n)

1 π

269

Δ(x, t)J3 (ψ2 ; t)1 dt |t|≥α (n)

ψ2 (n) + 2π

π Δ(x, t) sin ntdt. −π

In this case, we use the same reasoning as in Subsections 11.2 and 11.3. However, instead of information about zeros of the integral sine and the function Φ2 (ψ1 ; x), we take into account information about zeros of the integral cosine and the function (1) Φ2 (ψ2 ; x)

∞ ∞ =

ψ2 (nv) sin vtdvdt x

1

∞ = −ψ2 (n)cix + x

n t

∞

ψ2 (nv ) cos vtdvdt.

1

11.6. Note that if the value of α(n) (and/or the value α (n) ) is greater than, e.g., π/2, then, in view of Lemma 4.5, the integrals in the representations of the quantities ρn (f ; x)i , i = 1, 2, in Lemma 11.1 have the order of remainders of corresponding representations. Moreover, the values of these integrals over the set |t| ≥ π/2 have the same order regardless of the values of α(n) and α (n). Thus, integrals over the sets |t| ≥ α(n) and |t| ≥ α (n) in Lemma 11.1 could be replaced by integrals over the sets α(n) ≤ |t| ≤ π/2 and over the sets α (n) ≤ t ≤ π/2, respectively (considering, naturally, if a > π/2, that the set a ≤ |t| ≤ π/2 is empty). Arising errors have the order of remainders. Let α(n) = an = an (ψ1 ) = (η(ψ1 ; n) − n)−1 ,

(11.42)

α (n) = an = an (ψ2 ) = (η(ψ2 ; n) − n)−1 ,

(11.42 )

and ±ψ1 , ±ψ2 ∈ F. Thus, in view of Theorem 3.13.4, we get ∞ | 1/an

ψ1 (t + n) dt| = | t

∞

η(ψ1 ;n)

ψ1 (t) |dt ≤ O(1)|ψ1 (n)|, t−n

(11.43)

270

Approximation by Fourier Sums in Spaces C and L1 ∞ |

t

−1

(ψ(n) − ψ(n + t

−1

η(ψ 1 ;n)

))dt| = |

an

n

Chapter 5

ψ1 (n) − ψ(t) dt| t−n (11.43 )

≤ O(1)|ψ1 (n)|. Therefore, in the case considered,

and

1 Qψ n (an ) ≤ O(1)|ψ1 (n)|

(11.44)

2 Qψ n (an ) ≤ O(1)|ψ2 (n)|.

(11.44 )

Finally, recall that, by virtue of Theorem 3.13.4 F ⊂ M and, by virtue of (3.13.18), we have nan =

n n ≥ a0 > 0, nan = ≥ a0 > 0. η(ψ1 ; n) − n η(ψ2 ; n) − n

(11.45)

11.7. These facts, by virtue of Lemma 11.1, yield the following: ¯

ψ Corollary 11.1. Let ±ψ1 , ±ψ2 ∈ F. If f ∈ C∞ , then ψ1 (n) sin nt ¯ ρn (f ; x)1 = − f ψ (x − t) dt + O(1)|ψ1 (n)|, π t

(11.46)

an ≤|t|≤π/2

ρn (f ; x)2 =

ψ2 (n) π

an = (η(ψ1 ; n) − n)−1 , cos nt ¯ f ψ (x − t) dt + O(1)|ψ2 (n)|, t

(11.46 )

an ≤|t|≤π/2

an = (η(ψ2 ; n) − n)−1 , for any n ∈ N at every point x. ¯ If f ∈ C ψ Hω0 , then ψ1 (n) ρn (f ; x)1 = − π

¯

¯

(f ψ (x − t) − f ψ (x))

sin nt dt t

an ≤|t|≤π/2

+O(1)|ψ1 (n)|ω(1/n),

(11.47)

Section 12

Proofs of Theorems 10.1–10.3

ψ2 (n) ρn (f ; x)2 = π

¯

¯

(f ψ (x − t) − f ψ (x)) an ≤|t|≤π/2

271 cos nt dt t (11.47 )

+O(1)|ψ2 (n)|ω(1/n)

for any n ∈ N at every point x. ¯ If f ∈ C ψ C 0 , then, for any polynomial tn−1 ∈ Tn−1 , ψ1 (n) sin nt ¯ (f ψ (x − t) − tn−1 (x − t)) dt ρn (f ; x)1 = − π t an ≤|t|≤π/2 ¯

(11.48) +O(1)|ψ1 (n)|f ψ (·) − tn−1 (·)C , ψ2 (n) cos nt ¯ ρn (f ; x)1 = (f ψ (x − t) − tn−1 (x − t)) dt π t an ≤|t|≤π/2

¯

+O(1)|ψ2 (n)|f ψ (·) − tn−1 (·)C

(11.48 )

for any n ∈ N at every point x. In equalities (11.46)–(11.48 ), O(1) are quantities uniformly bounded in all parameters. 11.8. In view of equalities (11.1) and (11.2), according to the scheme of the proof of Lemma 11.1, using estimates of the form (8.8) and Lemma 2.3 instead of Lemma 1.3, we obtain an analog of Lemma 11.1 in the integral metric. By virtue of the arguments presented in Subsection 11.6, this yields the following statement: ¯

Corollary 11.2. Let ±ψ1 , ±ψ2 ∈ F. If f ∈ Lψ S10 , then equalities (11.46) and (11.46 ) hold almost everywhere for any n ∈ N. ¯ If f ∈ Lψ Hω1 , then equalities (11.47) and (11.47 ) hold almost everywhere for any n ∈ N. ¯ If f ∈ Lψ , then, for any polynomial tn−1 ∈ Tn−1 for any n ∈ N, equalities ¯ (11.48) and (11.48 ) hold almost everywhere if we change f ψ (·) − tn−1 (·)C by ¯ f ψ (·) − tn−1 (·)1 .

12.

Proofs of Theorems 10.1–10.3 ¯

12.1. As noted above, the classes C ψ N are invariant under the shift of an ¯ argument, thus, the quantities En (C ψ N), when N = M 0 and Hω0 , are independent of x. Therefore, it remains to prove corresponding asymptotic equalities.

272

Approximation by Fourier Sums in Spaces C and L1

Chapter 5

We consider the representations of the quantities ρn (f ; x)i , i = 1, 2, obtained in Corollary 11.1 and do what we did in Sections 4 and 5. First, as in Subsection 4.10, we set xk = (kπ + γn )/n, (12.1) tk = xk − π/2n, k = 0, ±1, . . . , n ∈ N,

(12.2)

where γn is a certain number from [−π/2, π/2]. Denote by k0 the value of k for which tk0 is closest to the point an + π/n from the right between all points tk ; by k1 the greatest from k for which tk < π/2; by k2 the number such that the point tk2 is closest to the point −(an + π/n) from the left between all points tk ; by k3 the lowest value under condition tk3 > −π/2. Further, set ln (t) = xk , t ∈ [tk , tk+1 ], k = k0 , . . . , k1 − 1, k = k3 , k3 + 1, . . . , k2 − 1.

(12.3)

Analyzing the proof of Lemma 4.7, we note that its assertion remains valid if we replace a/n by arbitrary numbers under the condition nαn ≥ α0 > 0. Thus, Corollary 11.1 yields the following statements: ¯

ψ , then Lemma 12.1. Let ±ψ1 , ±ψ2 ∈ F. If f ∈ C∞ ψ1 (n) sin nt ¯ f ψ (x − t) ρn (f ; x)1 = − dt + O(1)|ψ1 (n)| π ln (t)

(12.4)

i3,1

for any n ∈ N at every point x. ¯ If f ∈ C ψ Hω0 , then ρn (f ; x)1 ψ1 (n) =− π

¯

(f ψ (x − t) − f (x))

sin nt dt + O(1)|ψ1 (n)|ω(1/n) (12.5) ln (t)

i3,1

for any n ∈ N at every point x. ¯ If f ∈ C ψ C 0 , then, for any polynomial tn−1 ∈ Tn−1 for any n ∈ N, at every point x, ψ1 (n) sin nt ¯ (f ψ (x − t) − tn−1 (x − t)) ρn (f ; x)1 = − dt+ π ln (t) i3,1 ¯

+ O(1)|ψ1 (n)|f ψ (·) − tn−1 (·)C ,

(12.6)

Proofs of Theorems 10.1–10.3

Section 12

273

where ln (t) is a function constructed by (12.3), i3,1 = (t3 , t2 ) ∪ (t0 , t1 ), tk and xk , k = 0, ±1, . . . , are given according to (12.1), (12.2) under condition γn ≡ 0, O(1) are quantities uniformly bounded in all parameters. ¯

ψ Lemma 12.2. Let ±ψ1 , ±ψ2 ∈ F. If f ∈ C∞ , then ψ2 (n) cos nt ¯ f ψ (x − t) ρn (f ; x)2 = dt + O(1)|ψ2 (n)| π ln (t)

(12.4 )

i3,1

for any n ∈ N at every point x. ¯ If f ∈ C ψ Hω0 , then ρn (f ; x)2 ψ2 (n) = π

¯

(f ψ (x − t) − f (x))

cos nt dt + O(1)|ψ2 (n)|ω(1/n) (12.5 ) ln (t)

i3,1

for any n ∈ N at every point x. ¯ If f ∈ C ψ C 0 , then, for any polynomial tn−1 ∈ Tn−1 for any n ∈ N, at every point x, ψ2 (n) cos nt ¯ ρn (f ; x)2 = (f ψ (x − t) − tn−1 (x − t)) dt π ln (t) i3,1 ¯

+ O(1)|ψ2 (n)|f ψ (·) − tn−1 (·)C ,

(12.6 )

where ln (t) is a function constructed by (12.3), i3,1 = (t3 , t1 ) ∪ (t0 , t1 ), tk and xk , k = 0, ±1, . . . , are given according to (12.1), (12.2) under condition γn ≡ π/2, O(1) are quantities uniformly bounded in all parameters. 12.2. It is clear that reasoning as in Subsection 11.8, we obtain the following assertions of Lemmas 12.1 and 12.2 in the integral metric: ¯

Lemma 12.3. Let ±ψ1 , ±ψ2 ∈ F. If f ∈ Lψ S10 , then equalities (12.4) and ¯ (12.4 ) hold almost everywhere for any n ∈ N. If f ∈ Lψ Hω1 , then equalities ¯ (12.5) and (12.5 ) hold almost everywhere for any n ∈ N. If f ∈ Lψ , then, for any polynomial tn−1 ∈ Tn−1 for any n ∈ N, equalities (12.6) and (12.6 ) hold ¯ ¯ almost everywhere if we change f ψ (·) − tn−1 (·)C by f ψ (·) − tn−1 (·)1 .

Approximation by Fourier Sums in Spaces C and L1

274

Chapter 5

12.3. By virtue of equalities (12.4) and (12.4 ), we find asymptotic equalities for the quantities ¯

¯

df

df

(2) ψ ψ E(1) n (C∞ ) = sup |ρn (f ; x)1 | and En (C∞ ) = sup |ρn (f ; x)2 |. ¯ ψ f ∈C∞

¯

ψ f ∈C∞

Reasoning as in Subsections 5.1–5.3, we obtain ⎧ k1 −1 ⎪ ⎪ 1 |ψ1 (n)| π 3π ⎪ ⎪ (4 + O(1)), nan < − , ⎪ ⎨ π2 k 2 n k=k0 (1) ψ¯ En (C∞ ) = ⎪ ⎪ ⎪ ⎪ π 3π ⎪ ⎩O(1)ψ1 (n), . nan ≥ − 2 n In this case,

an an π 1 3 π 3 + < k0 < + , − < k1 < . π n π n 2 n 2

(12.7)

(12.8)

Thus, k 1 −1 k=k0

π/2

1 = k

dt 1 + O(1) = ln(η(ψ1 , n) − n). + O(1) = ln t an

(12.9)

an /π

Combining relations (12.7) and (12.9), we get ¯

ψ E(1) n (C∞ ) =

4 |ψ1 (n)| ln+ (η(ψ1 ; n) − n) + O(1)|ψ1 (n)|. π2

(12.10)

4 |ψ2 (n)| ln+ (η(ψ2 ; n) − n) + O(1)|ψ2 (n)|. π2

(12.10 )

By analogy, we find ¯

ψ E(2) n (C∞ ) =

By analogy with Subsections 5.4 and 5.5, we obtain ¯

ψ 0 E(1) n (C Hω ) =

=

sup

0 f ∈C ψ¯ Hω

|ρn (f ; x)1 |

2|ψ1 (n)| en (ω) ln+ (η(ψ1 ; n) − n) π2 + O(1)|ψ1 (n)|ω(1/n)

and

(12.11)

Proofs of Theorems 10.1–10.3

Section 12 ¯

ψ 0 E(2) n (C Hω ) =

=

sup

0 f ∈C ψ¯ Hω

275

|ρn (f ; x)2 |

2|ψ2 (n)| en (ω) ln+ (η(ψ2 ; n) − n) π2 (12.11 )

+ O(1)|ψ2 (n)|ω(1/n). By analogy with Subsection 5.6, we find |ρn (f ; x)1 | ≤

4 ¯ ¯ |ψ1 (n)|(ln+ (η(ψ1 ; n) − n) + O(1))En (f ψ )C ∀f ∈ C ψ C 0 , (12.12) π2

and |ρn (f ; x)2 | ≤

4 ¯ ¯ |ψ2 (n)|(ln+ (η(ψ2 ; n) − n) + O(1))En (f ψ )C ∀f ∈ C ψ C 0 . (12.12 ) 2 π ¯

By analogy with Subsection 5.6, we verify that for any function f ∈ C ψ C 0 ¯ and every n ∈ N in the space C ψ C 0 there exists a function F (x) such that ¯ ¯ En (F ψ )C = En (f ψ )C and relations (12.12) and (12.12 ) becomes equalities for it. 12.4. Assume that functions ±ψ1 and ±ψ2 belong to F and the equality |ψ2 (n)| ln+ (η(ψ2 ; n) − n) ≤ O(1)|ψ1 (n)|

(12.13)

holds as n → ∞. Then, by virtue of (11.3) and equalities (12.10) — (12.12 ), we obtain ¯

¯

ψ ψ ) = E(1) En (C∞ n (C∞ ) + O(1)|ψ1 (n)|,

(12.14)

¯

ψ ψ 0 Hω0 ) = E(1) En (C∞ n (C Hω ) + O(1)|ψ1 (n)|ω(1/n),

and

¯

|ρn (f ; x)| ≤ |ρn (f ; x)1 | + O(1)|ψ1 (n)|En (f ψ )C ¯

(12.15)

(12.16)

for any f ∈ C ψ C 0 . Relations (12.14)–(12.16) and (12.10)–(12.12 ) yield all assertions of Theorem 10.2 and the assertions of Theorem 10.3 formulated under condition (10.2).

276

Approximation by Fourier Sums in Spaces C and L1

Chapter 5

12.5. To prove Theorem 10.1 and the remaining part of Theorem 10.3, we return to Corollary 11.1. First, note that if condition (10.1) is satisfied, then

an an

dt a = | ln n | = O(1). t an

(12.17)

This enables us to replace the integrals in equalities (11.46)–(11.48 ) by integrals over the equal intervals an ≤ |t| ≤ π/2 or an ≤ |t| ≤ π/2. By virtue of (10.17), under such replacements, errors do not exceed the corresponding remainders. Thus, Corollary 11.1 yields the following statement: Lemma 12.4. Suppose that ±ψ1 , ±ψ2 ∈ F and condition (10.1) is satisfied. ψ¯ , then If f ∈ C∞ ¯ ψ(n) sin(nt − γn ) ¯ ¯ f ψ (x − t) ρn (f ; x) = dt + O(1)ψ(n), (12.17) π t an ≤|t|≤π/2

for any n ∈ N at every point x. ¯ If f ∈ C ψ Hω0 , then ¯ ψ(n) ρn (f ; x) = π

¯

¯

(f ψ (x − t) − f ψ (x))

sin(nt + γn ) dt t

an ≤|t|≤π/2

¯ +O(1)ψ(n)ω(1/n)

(12.18)

for any n ∈ N at every point x. ¯ If f ∈ C ψ C 0 , then, for any polynomial tn−1 ∈ Tn−1 for any n ∈ N, at every point x, ¯ ψ(n) sin(nt − γn ) ¯ (f ψ (x − t) − tn−1 (x − t)) ρn (f ; x) = dt π t an ≤|t|≤π/2 ¯

ψ ¯ (·) − tn−1 (·)C . +O(1)ψ(n)f

(12.19)

In equalities (12.17)–(12.19), ¯ ψ(n) = (ψ12 (n) + ψ22 (n))1/2 ,

a(n) = (η(ψ1 ; n) − n)−1 ,

ψ1 (n) (12.20) ψ2 (n) and O(1) is a quantity uniformly bounded in n. In equalities (12.17)–(12.19), we can take α (n) = (η(ψ2 ; n))−1 instead of α(n). γn = arctan

Proofs of Theorems 10.1–10.3

Section 12

277

Further, reasoning as in Subsection 12.1, we obtain an analog of combination of Lemmas 12.1 and 12.2. Lemma 12.5. Suppose that ±ψ1 , ±ψ2 ∈ F and condition (10.1) is satisfied. ψ¯ , then If f ∈ C∞ ¯ ψ(n) sin(nt − γn ) ¯ ¯ f ψ (x − t) ρn (f ; x) = dt + O(1)ψ(n) (12.21) π ln (t) i3,1

for any n ∈ N at every point x. ¯ If f ∈ C ψ Hω0 , then ρn (f ; x) =

¯ ψ(n) sin(nt − γn ) ¯ ¯ (f ψ (x − t) − f (x)) dt + O(1)ψ(n)ω(1/n) (12.22) π ln (t) i3,1

for any n ∈ N at every point x. ¯ If f ∈ C ψ C 0 , then, for any polynomial tn−1 ∈ Tn−1 for any n ∈ N, at every point x, ¯ ψ(n) sin(nt − γn ) ¯ ρn (f ; x) = (f ψ (x − t) − tn−1 (x − t)) dt π ln (t) i3,1 ¯

ψ ¯ (·) − tn−1 (·)C . +O(1)ψ(n)f

(12.23)

In equalities (12.21)–(12.23), ln (t) is a function constructed by (12.3), i3,1 = (t3 , t2 ) ∪ (t0 , t1 ), tk and xk , k = 0, ±1, . . . are given according to equalities (12.1), (12.2), O(1) are quantities uniformly bounded in all parameters. 12.6. Reasoning as in Subsection 11.8, we obtain analogs of Lemmas 12.4 and 12.5 in the integral metric. Lemma 12.6. Suppose that ±ψ1 , ±ψ2 ∈ F and condition (10.1) is satisfied. ¯ If f ∈ Lψ S10 , then equality (12.21) holds almost everywhere for any n ∈ N. ¯ If f ∈ Lψ Hω1 , then equality (12.22) holds almost everywhere for any n ∈ N. ¯ If f ∈ Lψ , then, for any polynomial tn−1 ∈ Tn−1 for any n ∈ N, equality ¯ ¯ (12.23) holds almost everywhere if we replace f ψ (·) − tn−1 (·)C by f ψ (·) − tn−1 (·)1 .

Approximation by Fourier Sums in Spaces C and L1

278

Chapter 5

12.7. It is clear that the proofs of Theorem 10.1 and of the remaining part of Theorem 10.3 are obtained from Lemma 12.5 exactly as the proof of Theorem 10.2 and a part of Theorem 10.3 in Subsections 12.3 and 12.4.

13.

Analogs of Theorems 10.1–10.3 in Integral Metric

13.1. Here we obtain the complement to the results of Section 8, in the case, where ±ψ1 , ±ψ2 ∈ F we find analogs of the basic statements of Section 10 in the integral metric. Theorem 13.1. If ±ψ1 , ±ψ2 ∈ F and, moreover, there exist constants K1 and K2 such that for all t ≥ 1 0 < K1 ≤

η(ψ1 ; t) − t ≤ K2 < ∞. η(ψ2 ; t) − t

(13.1)

Then the following asymptotic equalities hold as n → ∞ : ¯

En (Lψ S1 )1 =

¯ 4ψ(n) ¯ ln+ (η(n) − n) + O(1)ψ(n), π2

(13.2)

¯ 2ψ(n) ¯ en (ω)1 ln+ (η(n) − n) + O(1)ψ(n)ω(1/n), (13.3) π2 1 ¯ where ψ(n) = (ψ12 (n) + ψ22 (n))1/2 , η(n) is either η(ψ1 , n) = ψ1−1 ( ψ1 (n)) or 2 −1 1 η(ψ2 , n) = ψ2 ( ψ2 (n)), O(1) is the value uniformly bounded in n and 2 ¯

En (Lψ Hω1 )1 =

θω en (ω)1 = 1 π

π/2 2t ω( ) sin tdt, θω1 ∈ [1/2, 1], n 0

moreover, θω1 = 1 if ω(t) is a convex modulus of continuity. Theorem 13.2. Let ±ψ1 , ±ψ2 ∈ F. If |ψ2 (t)| ln+ (η(ψ2 ; t) − t) ≤ O(1)|ψ1 (t)|,

(13.4)

as t → ∞, then the following asymptotic equalities hold as n → ∞: ¯

En (Lψ S1 ) =

4|ψ1 (n)| + ln (η(ψ1 ; n) − n) + O(1)|ψ1 (n)|, π2

(13.5)

Section 14

Remarks on the Solution of Kolmogorov–Nikol’skii Problem

279

¯

En (Lψ Hω1 )1 =

2|ψ1 (n)| en (ω)1 ln+ (η(ψ1 ; n) − n) + O(1)|ψ1 (n)|ω(1/n). (13.6) π2

The quantities η(ψ1 ; t), η(ψ2 ; t), en (ω)1 and O(1) have the same sense as in Theorem 13.1. If we interchange the functions ψ1 (·) and ψ2 (·) in condition (13.4), we have to do the same on the right-hand sides of relations (13.5) and (13.6). Theorem 13.3. Let ψ1 , ψ2 ∈ F. If condition (13.1) is satisfied, then the fol¯ lowing inequality holds for any f ∈ Lψ and n ∈ N : ρn (f ; x)1 ≤ (

4 + ψ¯ ¯ ln (η(n) − n) + O(1))ψ(n)E n (f )1 , π2

(13.7)

¯ where the quantities ψ(n), η(n), and O(1) have the same sense as in Theo¯ ¯ ψ rem 13.1; En (f )1 is the quantity of the best approximation of the function f ψ (·) by trigonometric polynomials of degree (n − 1) in the space L : ¯

En (f ψ )1 =

inf

tn−1 ∈Tn−1

¯

f ψ (·) − tn−1 (·)1 .

If condition (13.4) is satisfied, then ρn (f ; x)1 ≤ (

4 + ¯ ln (η(ψ1 ; n) − n) + O(1))|ψ1 (n)|En (f ψ )1 . 2 π

(13.8)

¯

for every f ∈ Lψ and any n ∈ N. If we interchange the functions ψ1 (·) and ψ2 (·) in condition (13.4), we have to do the same on the right-hand side of (13.8). Inequalities (13.7) and (13.8) are asymptotically exact: the constant 4/π 2 cannot be reduced. The proof of Theorems 13.1–13.3 is based on Lemmas 12.3 and 12.6 and is carried out by repeating the arguments of Section 8 with obvious changes.

14.

Remarks on the Solution of Kolmogorov–Nikol’skii Problem

14.1. Consider the results obtained in Sections 3–13 in view of a solution of the Kolmogorov–Nikol’skii problem, which is formulated in the beginning of this chapter.

280

Approximation by Fourier Sums in Spaces C and L1

Chapter 5

It is clear that if ±ψ1 ∈ M0 and ±ψ2 ∈ M0 , then equations (3.3) and (3.10) are always asymptotically exact. Thus, they solve the Kolmogorov–Nikol’skii ¯ ψ¯ and C ψ Hω in the space C. The same is true for problem for the classes C∞ Theorems 6.1 and 7.1. By analogy, equalities (8.3) and (8.4) and equalities (9.1) and (9.6) are the solutions of the Kolmogorov–Nikol’skii problem on corresponding classes in the space L1 . Therefore, if ±ψ1 ∈ M0 and ±ψ2 ∈ M0 , then results of Sections 3 and 8 give the complete solution of the Kolmogorov–Nikol’skii problem on corresponding classes in the spaces C and L1 . The case with statements in Sections 10 and 13 for the functions ψ1 and ψ2 from the set F is different. By Theorem 3.13.1, MC ⊂ F. Hence, if ±ψ1 , ±ψ2 ∈ MC , then, as was noted above, equalities (10.8) and (10.9) coincide with equalities (3.4 ) and (3.10 ). Consequently, they give the solutions of the corresponding Kolmogorov– Nikol’skii problem. The same is also true for equalities (13.2) and (13.3). If ψ ∈ F \ MC , then the quantity μ(ψ; t) = (η(ψ; t) − t)/t can increase with no limit. The following cases are possible: (a) lim (η(ψ; t) − t) = ∞,

(14.1)

(b) η(ψ; t) − t ≤ K < ∞.

(14.2)

t→∞

Moreover, it is clear that if ψ(·) and/or ψ2 (·) belong to F \ MC , then condition (10.1) is not automatically satisfied. The functions ψα,r (t) = exp(−αtr ) where α > 0, r > 0 are typical examples of functions ψ(·) from F \ MC . Here, as noted above, η(ψα,r ; t) − t = t1−r (

ln 2 + O(1)), αr

(14.3)

where O(1) is a quantity uniformly bounded in t. This implies that ψα,r (·) satisfies condition (14.1) and condition (14.2) for r ∈ (0, 1) and r ≥ 1, respectively. By taking these remarks into account, we conclude the following: If ±ψ1 , ±ψ2 ∈ F \ MC , then equalities (10.8) and (10.9) give a solution of the Kolmogorov–Nikol’skii problem if and only if ψ1 (·) and ψ2 (·) satisfy condition (14.1). Moreover, equalities (10.11) and (10.12) give a solution of the Kolmogorov–Nikol’skii problem if and only if ψ1 (·) satisfies condition (14.1).

Section 14

Remarks on the Solution of Kolmogorov–Nikol’skii Problem

281

In the case where condition (14.2) is satisfied instead of (14.1), both terms on the right-hand sides of equalities (10.8)–(10.12) are of the same order as n → ∞. Consequently, these relations should be improved for finding a solution of the Kolmogorov–Nikol’skii problem. Such an improvement is performed in the next sections; here, we only make several remarks. 14.2. If a function ψ ∈ M tends to zero faster than any power function, i.e., lim tr ψ(k) = 0 ∀r ∈ R1 ,

(14.4)

k→∞

then the series

∞

ψ(k) cos kx,

k=0

∞

ψ(k) sin kx

(14.5)

k=1

are infinitely differentiable. Moreover, after their differentiation, we again obtain uniformly convergent series. This implies that if condition (14.4) is satisfied for ¯ functions ±ψ1 , ±ψ2 ∈ M, then functions f (·) that belong to the sets C ψ and are convolutions of a function ϕ ∈ L with kernels of the form (14.5) are infinitely differentiable. Denote by F a subset of functions ψ ∈ F such that lim μ(t) = lim μ(ψ; t) = lim

t→∞

t→∞

t→∞

t = ∞. η(ψ; t) − t

(14.6)

Note that if ψ ∈ F, then, by virtue of (3.13.2), there exists a constant K such that η (ψ; t) ≤ K for any t ≥ 1. Consequently, η (ψ; t) =

ψ (t) 2ψ (η(ψ; t))

≤ K < ∞ ∀t ≥ 1.

In this case, by virtue of (3.12.24), for any ψ ∈ F at any point t ≥ 1 |ψ (t)| (η(t) − t) ≤ |ψ(t)| ≤ 2|ψ (t)|(η(t) − t), η(t) = η(ψ; t), K or

ψ (t) 1 (η(t) − t)−1 ≤ − ≤ K(η(t) − t)−1 . 2 ψ(t)

(14.7)

By integrating this relation over the interval [t0 , t], we find t ψ(t0 ) exp(−K t0

1 dτ ) ≤ |ψ(t)| ≤ |ψ(t0 )| exp(− η(τ ) − τ 2

t t0

dτ ). (14.8) η(τ ) − τ

282

Approximation by Fourier Sums in Spaces C and L1

Chapter 5

If ψ ∈ F , then, by virtue of (14.8), we have 1 t |ψ(t)| ≤ |ψ(1)| exp(− 2

t

r

1

1 = |ψ(1)| exp( 2

t 1

dτ +r η(τ ) − τ

t

dτ ) τ

1

1 τ (2r − ))dτ τ η(τ ) − τ

for all t ≥ 1 and r > 0. This, in view of (14.6), implies that tr ψ(t) → 0 as t → ∞. Thus, the following statement is true: ¯

Proposition 14.1. If ±ψ1 , ±ψ2 ∈ F , then the functions f ∈ Lψ are infinitely differentiable. 14.3. Denote by Fc the subset of functions ψ ∈ F such that lim (η(ψ; t) − t) = c, c ∈ [0, ∞).

t→∞

(14.9)

Let us prove the following statement: Proposition 14.2. Let ψ1 ∈ Fc1 and ψ2 ∈ Fc2 . Then any function f (·) that ¯ belongs to the set C ψ is the trace on the real axis for a function F (z), regular in the strip 1 (14.10) |Imz| < min(1/c1 , 1/c2 ). 2 Moreover, if c1 = c2 = 0, then F (z) is regular in the whole complex domain, i.e., F (z) is an entire function. Proof. The basic idea of the proof is to use the well-known fact from the theory of regular functions (see Subsection 3.8.3) according to which if, for the coefficients ck = ck (f ) of the Fourier series of a function f ∈ L, the conditions lim inf ln |ck |−1/k = a, lim inf ln |c−k |−1/k = b, k→∞

are satisfied, then the series

k→∞

∞ k=−∞

(14.11)

ck eikz converges uniformly and absolutely

inside the strip −a < Im z < b and is a regular function therein.

Section 14

Remarks on the Solution of Kolmogorov–Nikol’skii Problem

283

If ψ ∈ Fc , then, by virtue of (14.9), we have 1 lim t→∞ t

t 1

dτ = lim (η(ψ; t) − t)−1 = t→∞ η(ψ; τ ) − τ

1/c, c = 0, ∞, c = 0,

(14.12)

and, by virtue of (14.8), for any t ≥ 1, 1 2t

t 1

dτ ln |ψ(1)| ≤ + ln |ψ(t)|−1/t η(ψ; τ ) − τ t K ≤ t

t 1

dτ . η(ψ; τ ) − τ

(14.13)

Therefore, for any ψ ∈ Fc , we have K 1 ≤ lim inf ln |ψ(t)|−1/t ≤ t→∞ 2c c

(14.14)

lim inf ln |ψ(t)|−1/t = ∞

(14.14 )

for c = 0, and

t→0

for c = 0. First, we assume that c1 · c2 = 0 and consider the following trigonometric series: ∞ ψ1 (k) − iψ2 (k), k ≥ 1, (14.15) μk eikt , μk = ψ1 (k) + iψ2 (k), k ≤ 1. k=−∞ For every k ∈ N, we have 1 1 ¯ ln |μk |−1/k = − ln ψ(k) = − ln ψ12 (k) + ψ22 (k) k k 1 ≥ − ln(2 max(|ψ1 (k)|; |ψ2 (k)|)) k 1 = − ln 2 + min(ln |ψ1 (k)|−1/k , ln |ψ2 (k)|−1/k ). k Consequently, by using (14.13), we get 1 lim inf ln |μk |−1/k ≥ min(1/2c1 , 1/2c2 ) = min(1/c1 , 1/c2 ). k→∞ 2

(14.16)

Approximation by Fourier Sums in Spaces C and L1

284

Chapter 5

By analogy, we obtain 1 lim inf ln |μ−k |−1/k ≥ min(1/c1 , 1/c2 ). k→∞ 2 Thus, the series

∞

(14.16 )

μk eikz converges uniformly and absolutely in strip (14.10)

k=−∞

and is a regular function therein. In particular, series (14.15) converges uniformly ¯ to its sum Ψ(t). Therefore, according to Proposition 3.11.3, any function f ∈ Lψ is the convolution of the function Ψ(t) and a certain summable function ϕ(·). In this case, its Fourier series has the form S[f ] = A0 + dk eikx , dk = μk ck (ϕ), |k|≥1

where μk are the quantities defined in (14.5), and ck (ϕ) are the Fourier coefficients of the function ϕ(·). Since ck (ϕ) → 0 as |k| → ∞, the following analogs of estimates (14.16) and (14.16 ) are valid for the quantities dk : 1 lim inf ln |dk |−1/k ≥ min(1/c1 , 1/c2 ) k→∞ 2 and

1 lim inf ln |d−k |−1/k ≥ min(1/c1 , 1/c2 ). k→∞ 2 dk eikz converges uniformly and absoThis implies that the series A0 + |k|≥1

lutely in strip (14.10). Thus, we have proved the validity of the statement in the case c1 · c2 = 0. It is clear that the proof does not change in the case where either of the values c1 and c2 is equal to zero or both are equal to zero, but one should use equality (14.14 ) instead of (14.14).

15.

¯ Approximation of ψ-Integrals That Generate Entire Functions by Fourier Sums

¯ 15.1. In this section, we consider approximations of ψ-integrals by Fourier sums under the condition that the functions |ψ1 (·)| and |ψ2 (·)| belong to the set ¯ F0 , i.e., in the case where, according to Proposition 14.2, elements of the sets Lψ are restrictions of functions regular in the entire complex plane to the real axis. In this case, the functions ψ1 (·) and ψ2 (·) decrease so fast that, in the remainder

¯ Section 15 Approximation of ψ-Integrals That Generate Entire Functions 285 ρn (f ; x) = f (x) − Sn−1 (f ; x), the first term of its Fourier series dominates. This allows us to obtain asymptotic equalities for the quantities ¯

En (Lψ N)X = sup ρn (f ; x)X , N ⊂ L0 , f ∈Lψ¯ N

that give a solution of the corresponding Kolmogorov–Nikol’skii problem both for X = C and X = Lp , p ∈ [1, ∞]. By Lp , p ≥ 1, as usual, we denote the subsets of functions ϕ ∈ L with finite norm ϕp , where π |ϕ(t)|p dt)1/p if 1 ≤ p < ∞,

ϕp = ( −π

and

df

ϕ∞ = ϕM = ess sup |ϕ(t)|. If ±ψ1 , ±ψ2 ∈ F0 , then the series ∞

(ψ1 (k) cos kx + ψ2 (k) sin kx)

k=1

converges uniformly to the sum Ψ(x). Therefore, according to Proposition 4.1.1, ¯ for any f ∈ Lψ , the following equality holds almost everywhere: 1 ρn (f ; x) = π =

π

¯

f ψ (x − t)Ψn (t)dt, Ψn (t) −π

∞

(ψ1 (k) cos kt + ψ2 (k) sin kt).

k=n ¯

If f ∈ C ψ , then this equality is valid at any point x. The function Ψn (t) is orthogonal to any trigonometric polynomial tn−1 ∈ Tn−1 , hence 1 ρn (f ; x) = π

π

¯

(f ψ (x − t) − tn−1 (x − t))Ψn (t)dt. −π

By using the Young inequality for convolutions, we have 1 y∗z = π

2π y(x − t)z(t)dt, 0

(15.1)

Approximation by Fourier Sums in Spaces C and L1

286

Chapter 5

πy ∗ zs ≤ yp z||q , 1 ≤ p ≤ s ≤ ∞, q −1 = 1 − p−1 + s−1 , y ∈ Lp , z ∈ Lq . ¯

For any function f ∈ Lψ Lp , 1 ≤ p ≤ s < ∞, we obtain ¯

¯

ρ(f ; x)s (f ψ − tn−1 ) ∗ Ψn s ≤ π −1 f ψ − tn−1 p Ψn q . However, q −1 ∈ [0, 1] and, therefore, Ψn q ≤ ψn M (2π)

1/q

≤ 2π

∞

¯ ¯ ψ(k), ψ(k) = (ψ12 (k) + ψ22 (k))1/2 .

k=n

Consequently, ψ¯

ρn (f ; x)s ≤ 2f − tn−1 p

∞

¯ ψ(k).

k=n

If 1 ≤ s < p ≤ ∞, then, by virtue of the Hölder inequality, for any function ϕ ∈ Lp , we have ϕs ≤ (2π)(p−s)/ps ϕp . Therefore, ¯

¯

ϕn (f ; x)s ≤ 2π(f ψ − tn−1 ) ∗ Ψn p ≤ 2f ψ − tn−1 p Ψn 1 ¯

≤ 4πf ψ − tn−1 p Ψn M . ¯

Thus, if 1 ≤ p, s ≤ ∞, then the following inequality holds for any f ∈ Lψ Lp : ψ¯

ρn (f ; x)s ≤ 4πf − tn−1 p

∞

¯ ψ(k).

k=n

By choosing as tn−1 (·) the polynomial t∗n−1 (·) of the best approximation of the ¯ derivative f ψ (·) in Lp , we arrive at the following statement: ¯

Theorem 15.1. Let ±ψ1 , ±ψ2 ∈ F0 and 1 ≤ p, s ≤ ∞. If f ∈ Lψ Lp , then the following inequality is valid: ¯

ρn (f ; x)s ≤ 4πEn (f ψ )p

∞

¯ ψ(k),

k=n

where En (ϕ)p =

inf

tn−1 ∈Tn−1

ϕ(·) − tn−1 (·)p .

(15.2)

¯ Section 15 Approximation of ψ-Integrals That Generate Entire Functions 287 15.2. If B is some class of functions, then we set En (B)s = sup{ρn (ϕ; x)s : ϕ ∈ B}, 1 An (B)s = sup{ π

π ϕ(x − t) cos ntdts : ϕ ∈ B}. −π

Under the notation accepted, the following theorem is true: Theorem 15.2. Let ±ψ1 , ±ψ2 ∈ F0 . Then ¯

0 ¯ En (Lψ Sp0 )s = ψ(n)A n (Sp )s + O(1)

∞

¯ ψ(k), 1 ≤ p, s ≤ ∞,

(15.3)

k=n+1

¯ = (ψ12 (k) + ψ22 (k))1/2 , where Sp0 = {ϕ : ϕp ≤ 1, ϕ ⊥ 1}, ψ(k) ¯ (0) ¯ En (Lψ Hω0p )s = ψ(n)A n (Hωp )s

+ O(1)ω(1/n)

∞

¯ ψ(k), 1 ≤ p, s < ∞, (15.4)

k=n+1

where Hω0p = {ϕ : ϕ ⊥ 1, ϕ(· + t) − ϕ(·)p ≤ ω(t)}, and ω(t) is a given modulus of continuity. In particular, ¯ En (Lψ S10 )1

∞ ¯ 4ψ(n) ¯ ψ(k), = + O(1) π

(15.3 )

k=n+1

¯

En (Lψ Hω01 )1 ¯ 2θω1 = ψ(n) π

π/2 ∞ ¯ ψ(k), (15.4 ) ω(2t/n) sin tdt + O(1)ω(1/n) 0

k=n+1

where θω1 ∈ [1/2, 1] and, moreover, θω1 = 1 if ω(t) is a convex modulus of continuity, ¯ 0 )∞ En (C ψ SM

=

¯ 0 En (C ψ SM )

∞ ¯ 4ψ(n) ¯ ψ(k) = + O(1) π k=n+1

(15.5)

Approximation by Fourier Sums in Spaces C and L1

288

Chapter 5

and ¯

¯

En (C ψ Hω0 )∞ = En (C ψ Hω0 ) ¯ 2θω = ψ(n) π

π/2 ω(2t/n) sin tdt 0 ∞

+ O(1)ω(1/n)

¯ ψ(k),

(15.6)

k=n+1

where θω ∈ [2/3, 1], and, moreover, θω = 1 if ω(t) is a convex modulus of continuity. In equalities (15.3)–(15.6), O(1) are quantities uniformly bounded in all parameters. Proof. First of all, for any n ∈ N, we have 1 ρn (f ; x) = π

π

¯

f ψ (x−t)(ψ1 (n) cos nt+ψ2 (n) sin nt)dt+ρn+1 (f ; x). (15.7) −π

It is clear that, for any n ∈ N and p, s such that 1 ≤ p, s ≤ ∞, we have ¯

sup En (ϕψ )p = sup En (ϕ)p ≤ sup ϕp ≤ 1

f ∈Lψ¯ Sp0

ϕ∈Sp0

ϕ∈Sp0

(15.8)

and, as is well known (see Section 6.9), for every ϕ ∈ Hω0p , En (ϕ)p = O(1)ω(1/n).

(15.8 )

Further, note that 1 π

π

¯

f ψ (x − t)(ψ1 (n) cos nt + ψ2 (n) sin nt)dts −π

π ¯ ψ(n) ¯ = f ψ (x − t) cos ntdts , (15.9) π −π

1 sup f ∈Lψ¯ S 0 π p

π

¯

f ψ (x − t) cos ntdts = An (Sp0 )s , −π

(15.10)

¯ Section 15 Approximation of ψ-Integrals That Generate Entire Functions 289 and 1 π

sup

0 f ∈Lψ¯ Hω p

π

¯

(15.10 )

f ψ (x − t) cos ntdts = An (Hω0p )s . −π

¯

Considering the upper bounds of both sides of (15.7) in the classes Lψ Sp0 and ¯ Lψ Hω0p and taking relations (15.8)–(15.10 ) into account, we obtain equalities (15.3) and (15.4). 15.3. Let us show that An (S10 )1

An (Hω01 )1

= 4/π,

2θω1 = π

π/2 ω(2t/n) sin tdt.

(15.11)

0

Indeed, An (S10 )1

1 = sup ϕ1 ≤1 π

π π π 1 4 | ϕ(x − t) cos ntdt|dx ≤ ϕ1 | cos nt|dt = . π π

−π −π

−π

On the other hand, by virtue of Lemma 6.2, An (S10 )1

1 ≥ 2π

π | cos nt − cos(nt + −π

π π )|dt = . n 4

The first relation in (15.11) is proved. Let us show the second one. We have π An (Hω01 )1

= sup 0 ϕ∈Hω 1

−π

1 | π

2π

2n−1 1 ϕ(x − t) cos ntdt|dx ≤ ek (ω)1 , π k=0

0

where π |

ek (ω)1 = sup

0 ϕ∈Hω 1

(k+1)π/n

−π

π/2n

ϕ(x − t) cos ntdt|dx =

ω(2t) sin ntdt. 0

kπ/n

The last equality here is obtained with regard for Lemma 2.1. Thus, π/2 2 An (Hω01 )1 ≤ ω(2t/n) sin tdt. π 0

290

Approximation by Fourier Sums in Spaces C and L1

Chapter 5

On the other hand, for the function f ∗ (t) constructed in Subsection 8.5, we obtain (see reasoning in Subsection 8.5) π −π

1 | π

2π

2 f ∗ (x − t) cos ntdt|dx = π

0

π/2 ω(2t/n) sin tdt. 0

Further, reasoning as in Subsection 8.5, we verify the validity of the second equality of (15.11). It is clear that π 1 4 0 0 An (SM )∞ = An (SM )C = ϕ(x − t) cos ntdtC = (15.12) π π −π

and, in view of equality (1.23), An (Hω0 )

=

An (Hω0 )C

2θω = π

π/2 ω(2t/n) sin tdt.

(15.13)

0 ¯

0 Hence, considering the upper bounds of both sides of (15.7) in the classes C ψ SM ¯ and C ψ Hω0 and taking relations (15.8) and (15.8 ) into account, we obtain equalities (15.5) and (15.6). Theorem 15.2 is proved.

15.4. For any 1 ≤ p, s ≤ ∞, we have An (Sp0 )s = sup an (ϕ) cos nx + bn (ϕ) sin nxs ϕ∈Sp0

= sup (a2n (ϕ) + b2n (ϕ))1/2 cos(nx + γn )s ϕ∈Sp0

= cos xs sup (a2n (ϕ) + b2n (ϕ))1/2 . ϕ∈Sp0

(15.14)

By analogy, An (Hω0p )s = cos xs sup (a2n (ϕ) + b2n (ϕ))1/2 . 0 ϕ∈Hω p

(15.14 )

The value cos xs is known: π/2 Γ( s+1 ) cos xss = 4 | cos x|s dx = 2π 1/2 s 2 . Γ( 2 + 1) 0

(15.15)

¯ Section 15 Approximation of ψ-Integrals That Generate Entire Functions 291 Thus, to determine the quantities An (Sp0 )s and An (Hω0p ), it suffices to find the quantities αn (Sp0 )s = sup (a2n (ϕ)+b2n (ϕ))1/2 and αn (Hω0p )s = sup (a2n (ϕ)+ ϕ∈Sp0

0 ϕ∈Hω p

b2n (ϕ))1/2 . If ϕ2 ≤ 1, then, by virtue of the Parseval equality, (a2n (ϕ) + b2n (ϕ))1/2 ≤ −1/2 π . At the same time, the function ϕ∗ (t) = π −1/2 cos nt belongs to S20 . Therefore, sup (a2n (ϕ) + b2n (ϕ))1/2 = π −1/2 . ϕ∈S20

Consequently, by virtue of (15.4) and (15.5), we have An (S20 )s = 2

Γ( s+1 2 ) s Γ( 2 + 1)

(15.16)

for any s ≥ 1. In particular, An (S20 )2 = 1. 15.5. We do not know exact values of An (Sp0 )s and An (Hω0p )s except those indicated in relations (15.11), (15.12), and (15.13). In the general case, for the quantities An (Sp0 )s , we have the following estimates: cos s ≤ An (Sp0 )s ≤ 4/π, 1 ≤ p ≤ s ≤ ∞, cos p

(15.17)

1 ≤ An (Sp0 )s ≤ 8, 1 ≤ s < p ≤ ∞.

(15.17 )

Note that, for p = s = 2, p = s = 1, and p = s = ∞, the estimate (15.17) is exact. Hence, it cannot be improved in the general case. The estimate from above in (15.17) is obtained with the use of the Young inequality: An (Sp0 )s = sup ϕ(·) ∗ cos n(·)s ≤ π −1 ϕp · cos(·)q ≤ ϕ∈Sp0

4 . π

The upper bound in (15.17 ) can be obtained in exactly the same simple way. Let us obtain the corresponding lower bound. We have

An (Sp0 )s

1 ≥ sup π

2π ϕ(x − t) cos ntdts , 0

292

Approximation by Fourier Sums in Spaces C and L1

Chapter 5

where the upper bound is extended onto function ϕ(·) from Sp0 which are representable in the form ϕ(t) = αn cos nt + bn sin nt, where an and bn are certain numbers and ϕp = 1, i.e., An (Sp0 )s

1 ≥ sup{ π

2π (an cos n(x − t) + bn sin n(x − t)) cos ntdts : 0

an cos nt + bn sin ntp = 1} = sup{an cos nt + bn sin nts : an cos nt + bn sin ntp = 1} cos ts = cos ts sup{ a2n + b2n : ( a2n + b2n = ( cos t−1 . p )} cos tp This directly yields the validity of the lower bounds in (15.17) and (15.17 ). 15.6. For the quantities An (Hω0p )s , the following inequality holds: cos ts 3π 2

π/2 4 ω(2t/n) sin tdt ≤ An (Hω0p )s ≤ sup E0 (ϕ)p . π ϕ∈Hω0

(15.18)

p

0

Indeed, let ϕ ∈ Hω0p and t∗n−1 (·) be the polynomial of the best approximation of a function ϕ(·) in the space Lp . Then, by using of Young inequality, we get 1 An (Hω0p )s = sup π 0 ϕ∈Hω p

2π (ϕ(x − t) − t∗n−1 (x − t)) cos ntdts 0

≤ sup π −1 ϕ(·) − t∗n−1 (·)p cos tq ≤ 0 ϕ∈Hω p

4 sup En (ϕ)p . π ϕ∈Hω0 p

To obtain the lower bound, we use equality

(15.14 ),

which yields

An (Hω0p ) ≥ cos ts sup |an (ϕ)|,

(15.19)

ϕ∈H

where H is the subset of even functions from Hω0p . Now let Hω = {ϕ : ϕ ∈ C, ϕ(x + t) − ϕ(t)C ≤ ω(t)}. Then, for any modulus of continuity ω(t), there exists a function ϕ∗ ∈ Hω0 , such that (see Subsection 1.3) π/2 2 |an (ϕ∗ )| = ω(2t/n) sin tdt. (15.20) 3π 0

¯ Section 15 Approximation of ψ-Integrals That Generate Entire Functions 293 For the function ϕ∗ (·), we have ϕ∗ (x + t) − ϕ∗ (x)p ≤ ω(t)(2π)1/p ≤ 2πω(t). Hence, the function f ∗ (·) = ϕ∗ (·)/2π belongs to Hω0p . Therefore, according to (15.19) and (15.20), An (Hω0p ) ≥ cos ts |an (f ∗ )| = cos ts = 3π 2

cos ts |an (ϕ∗ )| 2π

π/2 ω(2t/n) sin tdt. 0

It is clear that one can always choose constants K1 and K2 independent of n ∈ N and such that see (15.8 ) π/2 ω(2t/n) sin tdt ≥ K1 ω(1/n), 0

sup En (ϕ)p ≤ K2 ω(1/n).

0 ϕ∈Hω p

By virtue of (15.18), we always have C1 ω(1/n) ≤ An (Hω0p )s ≤ C2 ω(1/n),

(15.21)

where C1 and C2 are the values independent of n ∈ N and C1 > 0. In view of relations (15.17), (15.17 ) and (15.21), we conclude that, for all p, s ≥ 1 and n ∈ N, 0 ¯ ¯ ¯ C1 ψ(n) ≤ ψ(n)A n (Sp )s ≤ C2 ψ(n), 0 ¯ ¯ ¯ ≤ ψ(n)A C1 ψ(n)ω(1/n) n (Hωp ) ≤ C2 ψ(n)ω(1/n).

(15.22) (15.22 )

15.7. We now show that if ±ψ1 , ±ψ2 ∈ F0 , then −1 ¯ lim (ψ(n))

n→∞

∞

¯ ψ(k) = 0.

(15.23)

k=n+1

We have −1 ¯ (ψ(n))

∞

¯ ψ(k)

k=n+1

≤ |ψ1 (n)|−1

∞ k=n+1

|ψ1 (x)| + |ψ2 (n)|−1

∞ k=n+1

|ψ2 (k)|. (15.24)

294

Approximation by Fourier Sums in Spaces C and L1

Chapter 5

Since limk→∞ (η(ψ1 ; k) − k) = 0, we have η(ψ1 ; k) − k < 1 for sufficiently large values of k. Hence, for sufficiently large n, ∞ k=n+1

1 1 |ψi (x)| ≤ |ψi (n + 1)| + |ψi (n + 1)| + |ψi (n + 1)| + . . . 2 4 ≤ 2|ψi (n + 1)|,

i = 1, 2.

(15.25)

At the same time, by virtue of (14.8), we obtain the following relation for all t ≥ 1: ψi (t + 1) 1 ≤ exp(− ψi (t) 2 where

t+1 t

dτ 1 ) ≤ exp(− ϕ∗i (t)), η(ψi ; τ ) − τ 2

ϕ∗i (t) = inf (η(ψi ; x) − x)−1 . x≥t

Therefore, lim

t→∞

ψi (t + 1) = 0, i = 1, 2. ψi (t)

(15.26)

Combining relations (15.24)–(15.26), we obtain equality (15.23). Then, by comparing relations (15.22), (15.22 ), and (15.13), we conclude that equalities (15.3)– (15.6) always present solutions of the Kolmogorov–Nikol’skii problem.

16.

Approximation of Poisson Integrals by Fourier Sums

16.1. Here we consider the approximations of the functions from the classes ψ Cβ,∞ and Cβψ Hω under following conditions: ψ(t) = q t , q ∈ (0, 1), and β is an arbitrary real number. ¯ Since (see Proposition 3.11.5) Cβψ = C ψ for ψ1 (t) = q t cos βπ/2, ψ2 (t) = q t sin βπ/2, and ln 2 η(ψi ; t) − t ≤ 1 , i = 1, 2, (16.1) ln q Proposition 14.2 implies that, in the case under consideration, the set Cβψ consists of functions f (·) that are restrictions of functions F (z) analytic in the strip |Imz| ≤

ln 1q 2 ln 2

Section 16

Approximation of Poisson Integrals by Fourier Sums

295

q ψ q ψ q to the real axis. If ψ(t) = q t , then we set Lψ β = Lβ , Cβ = Cβ , fβ = fβ . In this case, the function Ψβ (t) is the Poisson kernel

Pβq (t) =

∞

q k cos k(t +

k=1

βπ ), 2

and elements f ∈ Lqβ are representable in the form 1 f (x) = A0 + π

π

ϕ(x + t)Pβq (t)dt

(16.2)

−π

and are the well-known Poisson integrals of the functions ϕ(·). Theorem 16.1. Let q ∈ (0, 1) and β ∈ R1 . Then, as n → ∞, q ) = sup |ρn (f ; x)| = En (Cβ,∞ q f ∈Cβ,∞

8q n q (K(q) + O(1) ), 2 π (1 − q)n

(16.3)

where O(1) is a quantity uniformly bounded in n and β, and K(q) is a complete elliptic integral of the first kind: π/2 K(q) = 0

Proof. Let Pβq (t) =

dt , q ∈ [0, 1). 1 − q 2 sin2 t ∞

q k cos(kt + βπ/2).

(16.4)

(16.5)

k=1

This series converges absolutely and uniformly for all q ∈ (0, 1) and β ∈ R. Therefore, (see Subsections 3.7.5, 3.8.2, 3.8.3) for f ∈ Cβq the equality a0 (f ) 1 f (x) = + 2 π

π

fβq (x + t)Pβq (t)dt,

(16.6)

−π

holds at every point x. Hence, 1 ρn (f ; x) = f (x) − Sn−1 (f ; x) = π

π −π

q fβq (x + t)Pβ,n (t)dt,

(16.7)

Approximation by Fourier Sums in Spaces C and L1

296 where

q (t) = Pβ,n

∞

q k cos(kt + βπ/2).

Chapter 5

(16.8)

k=n

Performing obvious transformations, we get q (t) = q n (g(t) cos(nt + βπ/2) − h(t) sin(nt + βπ/2)), Pβ,n

where g(t) =

∞

q k cos kt =

1 − q cos t 1 − 2q cos t + q 2

(16.10)

q k sin kt =

q sin t . 1 − 2q cos t + q 2

(16.11)

k=0

and h(t) =

∞

(16.9)

k=1

Setting

π π Hnβ (t) = g(t) cos(nt + β ) − h(t) sin(nt + β ), 2 2 by virtue of (16.7) and (16.9) we obtain qn ρn (f ; x) = π

π

fβq (x + t)Hnβ (t)dt.

(16.12)

(16.13)

−π

Thus, taking into account that classes Cβq N (where N = SM or N = Hω ) are invariant under the shift of an argument, we conclude that En (Cβq N)

qn = sup | f ∈C q N π β

π

fβq (t)Hnβ (t)dt|

(16.14)

ϕ(t)Hnβ (t)dt|,

(16.15)

−π

and, consequently, q ) En (Cβ,∞

En (Cβq , Hω0 )

qn = sup | ϕ∈M 0 π

qn = sup | π 0 ϕ∈Hω

π −π

π (ϕ(t) − ϕ(0))Hnβ (t)dt|,

(16.15 )

−π

M 0 = {ϕ : ϕM ≤ 1, ϕ⊥1}, Hω0 = {ϕ : ϕ ∈ Hω ; ϕ⊥1}.

(16.16)

Section 16

Approximation of Poisson Integrals by Fourier Sums

297

16.2. Applying the duality principle (7.2.7) (for Tn−1 (·) ≡ c ) to the righthand side of (16.15), we get q ) En (Cβ,∞

qn = inf π c

π |Hnβ (t) − c|dt.

(16.17)

−π

16.3. We use the following statement to find the integral: Lemma 16.1 (Fejér–Stechkin). Let g(t) and h(t) be the 2π-periodic functions of bounded variation on a period: π

π

−π

−π

V g < ∞, V h < ∞

(16.18)

and let α be an arbitrary real number. We set ϕ(t) = g(t) cos(nt + α) + h(t) sin(nt + α),

(16.19)

π π g 2 (t) + h2 (t), K = V g + V h.

(16.20)

r(t) = Then

π I1 = −π

π I2 = inf c

−π

1 I3 = sup h 2

π −π

−π

2 |ϕ(t)|dt = π

2 |ϕ(t) − c|dt = π

π

−π

r(t)dt + O(1)Kn−1 ,

(16.21)

r(t)dt + O(1)Kn−1 ,

(16.22)

−π

π −π

2 |ϕ(t + h) − ϕ(t)|dt = π

π

r(t)dt + O(1)Kn−1 .

(16.23)

−π

In equalities (16.21)–(16.23), O(1) are quantities uniformly bounded in all parameters. Proof. Since 1 2

π

π |ϕ(t + h) − ϕ(t)|dt ≤

−π

|ϕ(t) − c|dt, −π

(16.24)

Approximation by Fourier Sums in Spaces C and L1

298

Chapter 5

for all h and c from R1 , we always have I1 ≥ I2 ≥ I3 . Thus, it suffices to prove formula (16.21) and inequality 2 I3 ≥ π

π

r(t)dt + O(1)Kn−1 .

(16.25)

−π

Let us verify formula (16.21). Setting ϕk (t) = g(kπ/n) cos(nt + α) + h(kπ/n) sin(nt + α),

(16.26)

we get kπ/n

n

I1 =

|ϕ(t)|dt

k=−n+1(k−1)π/n kπ/n

n

=

|ϕk (t)|dt + O(1)

k=−n+1(k−1)π/n

n

kπ/n

|ϕ(t) − ϕk (t)|dt.

k=−n+1(k−1)π/n

However, kπ/n

|ϕk (t)|dt =

2 kπ r( ) n n

(k−1)π/n

and

kπ/n

|ϕ(t) − ϕk (t)|dt ≤ (k−1)π/n

Therefore, 2 I1 = π Finally, since

n

r(

k=−n+1

kπ/n π kπ/n g+ V h). ( V n (k−1)π/n (k−1)π/n

kπ π ) + O(1)Kn−1 . n n

π

π

π

−π

−π

−π

V r ≤ V g + V h,

we have

n k=−n+1

kπ π r( ) = n n

π −π

r(t)dt + O(1)Kn−1

Section 16

Approximation of Poisson Integrals by Fourier Sums

299

Formula (16.21) is proved. Let us prove inequality (16.25). Since |ϕ(t + π/n) − ϕ(t)| = |2ϕ(t) + Δg cos(nt + α) + Δh sin(nt + α)|, where Δg = g(t + π/n) − g(t), Δh = h(t + π/n) − h(t), we have 1 2

π |ϕ(t + π/n) − ϕ(t)|dt −π

π

π |ϕ(t)|dt + O(1)

= −π

|Δg cos(nt + α) + Δh sin(nt + α)|dt. (16.27)

−π

Denoting by Qn (t) the integrand of the last integral, we find π Qn (t)dt =

n

kπ/n

Qn (t)dt

k=−n+1(k−1)π/n

−π

n

= O(1)

k=−n+1

kπ/n π kπ/n g+ V h) = O(1)Kn−1 . ( V n (k−1)π/n (k−1)π/n

Therefore, in view of (16.21), it follows from (16.27) that π Qn (t)dt = −π

n

kπ/n

Qn (t)dt

k=−n+1(k−1)π/n n

= O(1)

k=−n+1

kπ/n π kπ/n g+ V h) = O(1)Kn−1 . ( V n (k−1)π/n (k−1)π/n

Lemma 16.1 is proved. 16.4. Lemma 16.1 is true if ϕ(t) = Hnβ (t). Indeed, in this case, the function g(t) is even and monotonically decrease on the segment [0, π] from (1 − q)−1 to (1 + q)−1 . Hence, π

V g=

−π

4q 4q ≤ . (1 + q)(1 − q) 1−q

Approximation by Fourier Sums in Spaces C and L1

300

Chapter 5

Further, the function h(t) is odd, satisfies the inequality |h(t)| ≤ q(1 − q)−1 , and it have a unique extremum on a segment [0, π]. Therefore, π

V h≤

−π

At the same time r(t) =

4q . 1−q

g 2 (t) + h2 (t) = (1 − 2q cos t + q 2 )−1/2 .

(16.28)

Consequently, π

π r(t)dt = 2

−π

0

=4 1 − 2q cos t + q 2

4 = 1+q

π/2 0

=

π/2

dt

# 1−

0

du (1 + q)2 − 4q sin2 u

du 4q sin2 u (1 + q)2

√ 2 q 4 K( ) = 4K(q). 1+q 1+q

(16.29)

Thus, by virtue of (16.21), (16.22), and (16.29) π |Hnβ (t) − c|dt =

inf c

−π

π |Hnβ (t)|dt = −π

8 q K(q) + O(1) , π (1 − q)n

8 q K(q) + O(1) . π (1 − q)n

(16.30)

(16.30 )

By combining relations (16.17) and (16.30), we obtain equality (16.3). q 16.5. The function Pβ,n (t) and, hence, the function Hnβ (t) is orthogonal to any polynomial tn−1 ∈ Tn−1 : π Hnβ (t)tn−1 (t)dt = 0. −π

Therefore, by virtue of (16.13), we have qn ρn (f ; x)C ≤ En (fβq )C π

π |Hnβ (t)|dt −π

(16.31)

Section 16

Approximation of Poisson Integrals by Fourier Sums

301

for every f ∈ Cβq C. Combining relations (16.31) and (16.30 ), we arrive at the following statement: Theorem 16.2. Let q ∈ (0, 1) and β ∈ R1 . If f ∈ Cβq C, then ρn (f ; x)C ≤

8q n q En (fβq )C (K(q) + O(1) ), 2 π (1 − q)n

(16.32)

where K(q) is determined by formula (16.4). 16.6. One can judge about the value of K(q) by the well-known formula: π 1·3 2 4 (2k − 1)!! 2 2n 1 2 2 K(q) = (1 + ( ) q + ( q + . . .). (16.33) ) q + ...+ 2 2 2·4 2k k! 16.7. Considering the upper bounds of both sides of (16.32) over the class Cβq Hω0 , we obtain the estimate of the quantity (16.15 ): En (Cβq Hω0 ) ≤

8q n q (K(q) + O(1) ) sup En (ϕ)C . 2 π (1 − q)n ϕ∈Hω0

(16.34)

Thus, in view of (3.14), in the case where ω(t) is a convex modulus of continuity, we have 4 π q En (Cβq Hω0 ) ≤ q n ω( )(K(q) + O(1) ). (16.35) π n (1 − q)n 16.8. Setting q = e−α , α > 0, and performing simple transformations, we get 1 |Hnβ (t)| = eα cos(nt + βπ/2) − cos((n − 1)t + βπ/2)|(cosh α − cos t)−1 2 ≤ (eα + 1)/2(cosh α − cos t).

(16.36)

Substituting this estimate into (16.17) and integrating, we obtain q En (Cβ,∞ )

q n (eα + 1) ≤ π

π 0

2q n dt eα + 1 n = q = . cosh α − cos t shα 1−q

Thus, the following theorem is true:

(16.37)

302

Approximation by Fourier Sums in Spaces C and L1

Chapter 5

Theorem 16.1. For any q ∈ (0, 1), β ∈ R1 and n ∈ N q )≤ En (Cβ,∞

2q n . 1−q

(16.38)

It is clear that, using estimate (16.36) and inequality (16.31), we arrive at the following statement: Theorem 16.2. For any q ∈ (0, 1), β ∈ R1 and n ∈ N 2q n En (fβq )C 1−q

ρn (f ; x)C ≤

(16.39)

for every f ∈ Cβq C. In particular, if ω(t) is a convex modulus of continuity, then, for every function f (·) from the class Cβq Hω , ρn (f ; x)C ≤

qn π ω( ). 1−q n

(16.40)

16.9. Let us establish analogs of Theorems 16.1–16.2 in the integral metric. Theorem 16.3. Suppose that q ∈ (0, 1), β ∈ R1 and Lqβ,1 is a class of t functions Lψ β,1 when ψ(t) = q . Then, as n → ∞,

En (Lqβ,1 ) = sup ρn (f ; x)1 = f ∈Lqβ,1

8q n q (K(q) + O(1) ), 2 π (1 − q)n

(16.41)

where O(1) is a quantity uniformly bounded in n and β, and K(q) is determined by (16.4). Proof. Since f ∈ Lqβ , equality (16.13) holds almost everywhere. Hence, En (Lqβ,1 )1

qn ≤ fβq 1 π

π

qn |Hnβ (t)|dt ≤ π

−π

π |Hnβ (t)|dt. −π

On the other hand, by Lemma 6.2, En (Lqβ,1 )1

qn ≥ sup π ϕ∈L0

π ϕ(x − t)Hnβ (t)dt1

−π

≥

qn 2π

sup Hnβ (t + h) − Hnβ (t)1 .

|h| 1. Thus, y1 (t) strictly increases and, hence, it has an inverse function t = y(τ ) = y1−1 (τ ) defined on the entire real axis. Moreover, Zq,n (t) = (

y (τ ) =

2 (y(τ )) Zq,n . Zq2 (y(τ ))

(18.16)

Consequently, for all τ ∈ R1 and q ∈ (0, 1), we have 0 < y (τ ) < 1.

(18.17)

Denoting the integral in (18.14) by Jn (f ) and setting in it t = y(τ ), in view of (18.16), we get y1(2π)

Jn (f ) =

f (y(τ )) y1 (0)

2 (y(τ )) Zq,n cos(n − 1)τ dτ. Zq (y(τ ))

(18.18)

Section 18

Solution of Kolmogorov–Nikol’skii Problem

Let

2 (y(τ )) Zq,n − Zq (y(τ )). Zq (y(τ ))

rn(1) (τ ) =

313

(18.19)

Then, according to (18.18), y1(2π)

f (y(τ ))Zq (y(τ )) cos(n − 1)τ dτ + Rn(1) (f ),

Jn (f ) =

(18.20)

y1 (0)

where

y1(2π)

Rn(1) (f )

=

f (y(τ ))rn(1) (τ ) cos(n − 1)τ dτ.

(18.21)

ω(1/n) ∀f ∈ Hω0 , (1 − q)2 n

(18.22)

y1 (0)

Let us verify that, as n → ∞, |Rn(1) (f )| = O(1)

where O(1) is a quantity uniformly bounded in all parameters. For this purpose, we use Lemma 1.3. Note that, in view of (18.17), the function F (τ ) = f (y(τ )) belongs to the set Hω [y1 (0), y1 (2π)] and F (y1 (0)) = 0. Therefore, to estimate integral in (18.21) (1) one can use Lemma 1.3 for ϕ(t) = rn (t) cos(n − 1)t if it is possible to find corresponding zeros xk . Let us verify that such zeros exist in required amount. Performing calculations, we find (rn(1) (τ )) =

1 4 (y(τ ))q sin y(τ )y (τ )Qn (cos(y(τ )), Z 3 (y(τ ))Zq,n n−1 q

where Qn (x) = q 2

5n − 3 2n − 1 2n − 1 2 x −( q − q 3 )x + + q2 − q4. n−1 n−1 n−1

Analyzing the quadratic trinomial Qn (x), we conclude that for all q ∈ [0, 1) and for all x ∈ [−1, 1]Qn (x) > 0. Consequently, sign(rn(1) (τ )) = sign sin y(τ ), τ ∈ [y1 (0), y1 (2π)]. (1)

(18.23)

Hence, the function rn (τ ) strictly increases on the interval (y1 (0), y1 (π)) and strictly decreases on the interval (y1 (π), y1 (2π)).

314

Approximation by Fourier Sums in Spaces C and L1

Chapter 5

Let xk = kπ/(n − 1) and τk = xk + π/2(n − 1). Then, in view of (18.15), we have y1 (0) < x2 < τ2 < x3 < . . . < τn−1 < xn ≤ y1 (π) < xn+2 < τn+2 < . . . < τ2n−1 < x2n ≤ y1 (2π).

(18.24)

(1)

By virtue of the monotonicity of the function rn (τ ), the quantities τk+1

rn(1) (τ ) cos(n − 1)τ dτ

αk = τk

are nondecreasing in the absolute value for k = 2, 3, . . . , n−2 and nonincreasing for k = n + 2, . . . , 2n − 2. Moreover, sign αk = sign cos(n − 1)xk+1 = (−1)k . Therefore, the function τ rn+ (τ )

rn(1) (v) cos(n − 1)vdv

= τ2

has a unique simple zero τ¯k on every segment [τk , τk+1 ], k = 2, . . . , n − 2 and the function τ − rn (τ ) = rn(1) (v) cos(n − 1)vdv τ2n−1

has a unique simple zero τ¯k on every segment [τk , τk+1 ], k = n + 2, . . . , 2n − 2. The assumptions of Lemma 1.3 are indeed satisfied on the intervals [y1 (0), y1 (π)] and [y1 (π), y1 (2π)]; its application gives the following estimate: τ2 |Rn(1) (f )|

≤

max

y1 (ν)≤τ ≤τ2

|f (y(τ ))|

|rn(1) (τ )|dτ

y1 (0) τn+3

+

max

τn−1 ≤τ ≤τn+3

|f (y(τ ))| τn−1

|rn(1) (τ )|dτ

Section 18

Solution of Kolmogorov–Nikol’skii Problem

315

y1(2π)

+

max

τ2n−2 ≤t≤y1 (2π)

4π + ω( ) n−1

|f (y(τ ))|

|rn(1) (τ )|dτ τ2n−2

τ 2n−1

|rn(1) (τ )|dτ.

(18.25)

τ2

In view of (18.19), (18.11), and (18.8), we get |rn(1) (t)| = ≤ ≤

|1 − 2q cos y(τ ) − n ( n−1 −

2n−1 n−1 q cos y(τ )

n 2n−1 n−1 + n−1 q cos y(τ )| + q 2 ) 1 − 2q cos y(τ )

+ q2

1 − q cos y(τ )

(n − 1)(1 − 2q cos y(τ ) + q 2 ) 1 − 2q + q 2 1 (1 −

q)2 (n

− 1)

, n ≥ 2.

(18.26)

At the same time, according to (18.24), we have τ2 − y1 (0) ≤

3π 4π , τn+3 − τn−1 ≤ , (n − 1) (n − 1)

y1 (2π) − τ2n−2

5π ≤ . (n − 1)

(18.27)

Combining estimates (18.25)–(18.27), we obtain relation (18.22). The next step is a further simplification of the integral in (18.20). To this end, we define a function ln (τ ) = ln,q (τ ) by setting ⎧ ⎨Zq (y(τk )), τ ∈ [xk , xk+1 ], k = 2, 3, . . . , 2n − 1, ln (τ ) = (18.28) ⎩ 0, τ ∈ [y1 (0), x2 ) ∪ (x2n , y1 (2π)). Then, by virtue of (18.20), x2n Jn (f ) = f (y(τ ))ln (τ ) cos(n − 1)τ dτ + Rn(1) (f ) + Rn(2) (f ),

(18.29)

x2

where

y1(2π)

Rn(2) (f )

f (y(τ ))rn(2) (τ ) cos(n − 1)τ dτ

= y1 (0)

(18.30)

316

Approximation by Fourier Sums in Spaces C and L1

Chapter 5

and rn(2) (τ ) = Zq (y(τ )) − ln (τ ).

(18.31)

(2)

To estimate the quantity Rn (f ) we use Lemma 1.3 again, but before that (2) we have to find the required information about the function rn (τ ) cos(n − 1)τ. First, we prove the following statement: Lemma 18.1. The function Zq (y(τ )), 0 < q < 1, n ≥ 2, strictly decreases on the interval (y1 (0), y1 (π)), strictly increases on the interval (y1 (π), y1 (2π)), and it has a unique inflection point on each of these intervals. Proof. Since Zq (t) = −q sin t(1 + q 2 − 2q cos t)−3/2 ,

(18.32)

in view of (18.16), 2 (Zq (y(τ ))) = Zq (y(τ ))y (τ ) = −q sin y(τ )Zq (y(τ ))Zq,n (y(τ )).

Therefore,

sign(Zq (y(τ )) = −sign sin y(τ ),

(18.33) (18.34)

which yields the first part of the assertion of the lemma. Further, we have Zq (t) = −q 2 (1 + q 2 − 2q cos t)−5/2 (cos2 t + and y (τ ) =

1 + q2 cos t − 3) q

6 (y(τ )) Zq,n q(1 − q 2 ) sin y(τ ) 2 . n−1 Zq (y(τ ))

(18.35)

(18.36)

Thus, combining relations (18.32), (18.33), (18.35), and (18.36), we get (Zq (y(τ ))) = Zq (y(τ ))y (τ ) + Zq (y(τ ))y (τ ) 2

4 (y(τ ))[cos2 y(τ ) = −q 2 Zq (y(τ ))Zq,n

+

1 + q2 1 − q2 2 cos y(τ ) − 3 + Z (y(τ )) sin2 y(τ )]. q n − 1 q,n

It is easy to see that sign(Zq (y(τ ))) = −sign[cos2 y(τ ) + +

1 + q2 cos y(τ ) − 3 q 1 − q2 2 Z (y(τ )) sin2 y(τ )]. n − 1 q,n

(18.37)

Section 18

Solution of Kolmogorov–Nikol’skii Problem

317

Analyzing this equality, we conclude that the function (Zq (y(τ ))) , indeed, has a unique simple zero on each interval (y1 (0), y1 (π)) and (y1 (π), y1 (2π)), as required. Lemma 18.2. Let g(t) be a function with two continuous derivatives on the segment [a, a + 2h], h > 0, and let F (t) = |g(t) − g(a + h/2)| − |g(t + h) − g(a + 3h/2)|, t ∈ [a, a + h]. (18.38) If g(t) does not increase on [a, a + 2h] and g (t) ≥ 0 for all t ∈ [a, a + 2h], then F (t) ≥ 0 ∀t ∈ [a, a + h]. (18.39) If, in addition, g (t) ≤ 0 for all t ∈ [a, a + 2h], then F (t) ≤ 0 ∀t ∈ [a, a + h].

(18.39 )

In the case where g(t) does not decrease, the condition g (t) ≥ 0 yields relation (18.39 ), and the condition g (t) ≤ 0 yields relation (18.39). Proof. Assume that g(t) does not increase on [a, a + 2h]. Then, by virtue of (18.38), ⎧ (g(t) − g(t + h)) − (g(a + h/2) ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ − g(a + 3h/2)), t ∈ [a, a + h/2], ⎨ F (t) = ⎪ ⎪ (g(t + h) − g(t)) − (g(a + 3h/2) ⎪ ⎪ ⎪ ⎪ ⎩ − g(a + h/2)), t ∈ [a + h/2, a + h]. Therefore, if t ∈ [a, a + h/2], then t+h |g (τ )|dτ − F (t) = t

Let

a+3h/2

|g (τ )|dτ.

a+h/2

t+h |g (τ )|dτ, t ∈ [a, a + h]. Φ(t) = t

Then F (t) = Φ(t) − Φ(a + h/2), t ∈ [a, a + h/2].

(18.40)

318

Approximation by Fourier Sums in Spaces C and L1

Chapter 5

If g (t) ≥ 0, then the function |g (t)| does not increase. Thus, Φ (t) = |g (t + h)| − |g (t)| ≤ 0, i.e., the function Φ(t) does not increase. Hence, in view of (18.40), F (t) ≥ 0. If g (t) ≤ 0, then |g (t)| does not decrease and F (t) ≤ 0. This proves relations (18.39) and (18.39 ) for all t ∈ [a, a + h/2] in the case where g(t) does not increase. It is clear that one can establish these relations by similar reasoning in all other cases. (2) Let us pass directly to the estimation of |Rn (f )| and show that, for every f ∈ Hω0 , ω(1/n) |Rn(2) (f )| = O(1) . (18.41) (1 − q)2 n According to Lemma 18.1, denote by τ ∗ the inflection point of the function Zq (y(τ )) on the interval [y1 (0), y1 (π)]. Thus, the function Zq (y(τ )) is decreasing and convex upward on the interval (y1 (0), τ ∗ ), and it is decreasing and convex downward on the interval (τ ∗ , y1 (π)). Further, let k1 and k2 be the numbers such that the point xk1 is closest to τ ∗ from the left among the points xk and the point xk2 is closest from the right; let k ∈ [2, k1 − 1] and xk+1

rn(2) (τ ) cos(n − 1)τ dτ.

αk = xk

In this case, sign αk = (−1)k

(18.42)

and the numbers |αk | do not decrease. Indeed, assume, for definiteness, that k is an even number. Hence, according to (18.31), xk+1

|αk | − |αk+1 | = |

xk+2

rn(2) (τ ) cos(n xk

− 1)τ dτ | − |

rn(2) (τ ) cos(n − 1)τ dτ |

xk+1

xk+1

|Zq (τ ) − Zq (τk )|| cos(n − 1)τ |dτ

= xk

xk+2

− xk+1

|Zq (τ ) − Zq (τk+1 )|| cos(n − 1)τ |dτ

Section 18

Solution of Kolmogorov–Nikol’skii Problem

319

xk+1

(|Zq (τ ) − Zq (τk )|

= xk

− |Zq (τ +

π π ) − Zq (τk + )|)| cos(n − 1)τ |dτ. n−1 n−1

The function Zq (τ ) decreases on the interval (y1 (0), xk1 ) and Zq (τ ) ≤ 0. Therefore, by virtue of Lemma 18.2, for a = xk , h = kπ/(n − 1), and g(τ ) = Zq (τ ) we conclude that |αk | ≤ |αk+1 |, k = 2, k1 − 2.

(18.43)

By analogy, if k2 ≤ k < n − 1, then sign αk = (−1)k+1

(18.44)

|αk | ≥ |αk+1 |, k = k2 , n − 2.

(18.45)

and

Let

x rn(2) cos(n − 1)τ dτ

Φ1 (x) = x2

and

x rn(2) cos(n − 1)τ dτ.

Φ2 (x) = xn−1

¯k By virtue of (18.42)–(18.45), the function Φ1 (x) has a unique simple zero x on every segment [xk , xk+1 ], k = 2, k1 − 2, and the function Φ2 (x) has such zeros on every segment [xk , xk+1 ], k = k2 , n − 2. In view of this construction and setting G(τ ) = f (y(τ ))rn(2) (τ ) cos(n − 1)τ, according to (18.30), we have y 1 (π)

Rn(2) (f )

=

y1(2π)

y1 (0)

df

G(τ )dτ = Jn(1) (f ) + Jn(2) (f ).

G(τ )dτ + y1 (π)

(18.46)

320

Approximation by Fourier Sums in Spaces C and L1

Chapter 5

Moreover, x2 Jn(1) (f )

x¯k2

x ¯k1 −1

=

G(τ )dτ + y1 (0)

G(τ )dτ +

G(τ )dτ

x2

x ¯k1 −1

xn−1

y 1 (π)

+

G(τ )dτ + x ¯k2

df

G(t)dt =

5

ij (f )

(18.47)

j=1

xn−1

and, by virtue of (18.24), 2π ¯k1 −1 < xk1 < τ ∗ < xk2 < x ¯ k2 < xk1 −1 < x n−1 2+β < xk2 +1 < xn−1 = π < xn ≤ y1 (π) = π + π. 2(n − 1)

y1 (0) ≤ x2 =

(18.48)

Therefore, in view of (18.31), we have x2 |i1 (f )| = |

|f (y(τ ))Zq (y(τ ))dτ |

y1 (0)

≤ ω(

1 2π ω(1/n) 2π = O(1) ) n − 1 n − 1 1 + q 2 − 2q (1 − q)n

(18.49)

¯k1 −1 ] and for every f ∈ Hω0 . Lemma 1.3 is applicable on the segments [x2 , x [¯ xk2 , xn−1 ], and, therefore, 4π |i2 (f )| + |i4 (f )| ≤ ω( ) n−1

xn−1

|rn(2) (τ )|dτ. x2

For τ ∈ [x2 , x2n ], by virtue of (18.34) and (18.31), |rn(2) (τ )| ≤

max

2≤k≤2n−1

|Zq (y(xk+1 )) − Zq (y(xk ))| xk+1

=

|Zq (y(τ ))||y (τ )|dτ.

max

2≤k≤2n−1 xk

(18.50)

Section 18

Solution of Kolmogorov–Nikol’skii Problem

321

Hence, in view of (18.17) and the estimate |Zq (y(τ ))| ≤

1 , (1 − q)2

which follows from (18.32) and (18.9), we find 1 |rn(2) (τ )| = O(1) , τ ∈ [x2 , x2n ]. (1 − q)2 n

(18.51)

Therefore, |i2 (f )| + |i4 (f )| = O(1)

ω(1/n) . (1 − q)2 n

(18.52)

By virtue of (18.51) and (18.48), x¯k2 |i3 (f )| = |

x¯k2 G(τ )dτ | ≤

x ¯k1 −1

|rn(2) (τ )|dτ = O(1) x ¯k1 −1

1 (1 − q)2 n2

and y 1 (π)

|i5 (f )| = |

y 1 (π)

G(τ )dτ | ≤ ω(π)

xn−1

|rn(2) (τ )|dτ = O(1) π

1 . (18.53) (1 − q)2 n2

By combining relations (18.47)–(18.53), we get |Jn(1) (f )| = O(1)

ω(1/n) ∀f ∈ Hω0 . (1 − q)2 n

(18.54)

(2)

It is clear that such estimate holds for the quantity |Jn (f )| which, by virtue of (18.46), yields relation (18.41). Therefore, according to (18.29), (18.22), and (18.41), for any f ∈ Hω0 we have x2n ω(1/n) Jn (f ) = f (y(τ ))ln (τ ) cos(n − 1)τ dτ + O(1) . (18.55) (1 − q)2 n x2

Consequently, by virtue of (18.14) and (18.55), En (Cβq Hω )

qn = sup | π f ∈Hω0

x2n f (y(τ ))ln (τ ) cos(n − 1)τ dτ | x2

+ O(1)

q n ω(1/n) . (1 − q)2 n

(18.56)

Approximation by Fourier Sums in Spaces C and L1

322

Chapter 5

Denoting the integral in the last expression by iqn (f ) and taking into account (18.28), we have iqn (f ) =

2n−1

xk+1

=

2n−1

f (y(τ )) cos(n − 1)τ dτ

Zq (y(τk ))

k=2

xk tk+1

f (t) cos[(n − 1)y1 (t)]y1 (t)dt, tk = y(xk ).

Zq (y(τk ))

k=2

tk

Thus, sup

f ∈Hω0

|iqn (f )|

where

≤

2n−1

Zq (y(τk ))Sk (ω),

(18.57)

f (t) cos[(n − 1)y1 (t)]y1 (t)dt|.

(18.58)

k=2 tk+1

Sk (ω) = sup | f ∈Hω

tk

In order to find the quantities Sk (ω) we use Lemma 1.4. The function ψ(t) = cos[(n − 1)y1 (t)]y1 (t)

(18.59)

is continuous and changes its sign at the unique point ck = y(τk ) on every segment [tk , tk+1 ], k = 2, . . . , 2n − 1. Moreover, tk+1

xk+1

cos(n − 1)τ dτ = 0.

ψ(t)dt = xk

tk

Hence, one can apply Lemma 1.4 to find the quantities Sk (ω). Let ρk (t) be a function determined on the segment [tk , ck ] by the following equality: t

ρk (t)

ψ(v)dv, tk ≤ t ≤ ck ≤ ρk (t) ≤ tk+1 .

ψ(v)dv = tk

(18.60)

tk

Thus, according to Lemma 1.4, for arbitrary modulus of continuity ω(t) we have ck Sk (ω) ≤

|ψ(t)|ω(ρk (t) − t)dt. tk

(18.61)

Section 18

Solution of Kolmogorov–Nikol’skii Problem

323

Moreover, if ω(t) is a convex modulus of continuity, then (18.61) turns into equality and the upper bound is realized by the functions from the class Hω of the form sk ± fk (t), where sk are arbitrary constants and ⎧ ck ⎪ ⎪ ⎪ ⎪ − ω (ρk (v) − v)dv, t ∈ [tk , ck ], ⎪ ⎪ ⎨ fk (t) = t t ⎪ ⎪ ⎪ ⎪ ω (v − ρ−1 t ∈ [ck , tk+1 ], ⎪ k (v))dv, ⎪ ⎩ ck

here ρ−1 k (v) is the function inverse to ρk (v). By combining relations (18.57), (18.58), and (18.61), for arbitrary modulus of continuity ω = ω(t), we get sup

f ∈Hω0

|iqn (f )|

≤

2n−1

ck |ψ(t)|ω(ρk (t) − t)dt.

Zq (y(τk ))

k=2

(18.62)

tk

Now assume that ω(t) is a convex modulus of continuity. By using Lemma 1.4, we construct a function f ∗ ∈ Hω0 such that iqn (f ∗ )

=

2n−1 k=2

ck |ψ(t)|ω(ρk (t) − t)dt + O(1)

Zq (y(τk )) tk

ω(1/n) . (18.63) (1 − q)2 n

We construct the function f ∗ (·) from the functions fk (t) and, first, note that for dk = tk+1 − tk , tk = y(xk ) the following equality holds: xk+1

y (τ )dτ −

dk − dk+1 = xk

xk+2

y (τ )dτ

xk+1

xk+1

(y (τ ) − y (τ +

= xk

π ))dτ. n−1

(18.64)

Relation (18.36) implies that the function y (τ ) increases on the interval (y1 (0), y1 (π)) and decreases on the interval (y1 (π), y1 (2π)). Thus, in view of (18.64), when k increases, the numbers dk first increase and then decrease. Let k¯ be a value of index k at which the pattern changes, i.e., the quantities dk increase for k = 2, . . . , k¯ and decrease for k = k¯ + 1, . . . , 2n − 1.

Approximation by Fourier Sums in Spaces C and L1

324

Chapter 5

It is clear that n − 1 < k¯ < n + 5. We now set ⎧ ¯ ⎨(−1)k fk (t) + γk , t ∈ [tk , tk+1 ), k = 2, 3, . . . , k, f0 (t) = ⎩ (−1)k fk (t) + δk , t ∈ (tk , tk+1 ], k = k¯ + 1, k¯ + 2, . . . , 2n − 1, where γ2 = δ2n−1 = 0, and other numbers γk and δk are such that the function ∗ f0 (t) is continuous on the intervals (t2 , tk+1 ¯ ) and (tk+1 ¯ , t2n ). Then by f (t) denote a 2π-periodic function determined on the periodicity interval [0, 2π] by the following equality: ⎧ ⎪ 0, t ∈ [0, c2 ] ∪ [c2n−1 , 2π], ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪f0 (t), t ∈ [c2 , tk¯ ] ∪ [tk+2 ¯ , c2n−1 ], ⎨ ∗ f (t) = (18.65) ⎪ ⎪ ⎪ , t ], f (t − 0) > 0, m(t), t ∈ [t ¯ ¯ ¯ 0 k k+2 k+1 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩M (t), t ∈ [t , t ], f (t − 0) < 0, ¯ k+2 ¯ ¯ 0 k+1 k where m(t) = min{f0 (t), f0 (tk+1 − 0), f0 (tk+1 + 0)}, ¯ ¯ M (t) = max{f0 (t), f0 (tk+1 − 0), f0 (tk+1 + 0)}. ¯ ¯ The function f ∗ (·) is the required one. Indeed, a direct calculation shows that iqn (f0 )

=

2n−1 k=2

ck |ψ(t)|ω(ρk (t) − t)dt.

Zq (y(τk ))

(18.66)

tk

However, the function f0 (t) differs from the function f ∗ (t) only on the intervals (t2 , c2 ), (tk¯ , tk+2 ¯ ), and (c2n−1 , t2n ), which immediately yields iqn (f ∗ ) = iqn (f0 ) + O(1)

ω(1/n) . (1 − q)2 n

This in combination with (18.66) yields equality (18.63), thus, it remains to show that f ∗ ∈ Hω0 .

Section 18

Solution of Kolmogorov–Nikol’skii Problem

325

Since f ∗ (0) = 0, we only have to verify the validity of equality |f ∗ (t ) − f ∗ (t )| ≤ ω(|t − t |)

(18.67)

for every t , t ∈ [0, 2π]. According to Lemma 1.4, the functions fk (t) satisfy relation (18.67) if t and t belong to the segment [tk , tk+1 ]. Therefore, (18.67) holds for f ∗ (t) whenever t and t simultaneously belong to the segments [0, c2 ], [c2 , t3 ], [tk , tk+1 ], k = 3, 4, . . . , 2n−1, [t2n−1 , c2n−1 ], or [c2n−1 , 2π]. The function f ∗ (t) is monotone on each of these segments and the character of monotonicity changes on successive segments. Moreover, ρ k (tk )

ω (t − ρ−1 k (t))dt − fk (tk )

fk (tk+1 ) − fk (tk ) = fk (ρk (tk )) − fk (tk ) = ck

tk

ω (ρk (z) −

=

z)ρk (z)dz

ck

ck +

ω (ρk (z) − z)dz

tk

ck =−

ω (ρk (z) − z)d(ρk (z) − z)

tk

= ω(ρk (tk ) − tk ) = ω(dk ).

(18.68)

Therefore, by equalities (18.65) and (18.68), in view of monotonicity of the quantities dk , we conclude that f ∗ ([0, c2 ]) ⊂ f ∗ ([c2 , t3 ]) ⊂ f ∗ ([t3 , t4 ]) ⊂ . . . ⊂ f ∗ ([tk−1 ¯ , tk ¯ ])

(18.69)

and ∗ f ∗ ([tk+2 ¯ , tk+3 ¯ ]) ⊃ . . . ⊃ f ([t2n−2 , t2n−1 ])

⊃ f ∗ ([t2n−1 , c2n−1 ]) ⊃ f ∗ ([c2n−1 , 2π]),

(18.69 )

where by f ∗ [a, b] is denoted the range of the function f ∗ (t) on the segment [a, b]. Now relation (18.67) is obvious because, by virtue of (18.69) and (18.69 ), for arbitrary points t and t there exists the segment [tk , tk+1 ] which contains the points t¯ and t¯ such that |f ∗ (t ) − f ∗ (t )| ≤ |f ∗ (t¯ ) − f ∗ (t¯ )|

Approximation by Fourier Sums in Spaces C and L1

326

Chapter 5

at the same time |t − t | ≥ |t¯ − t¯ |. Hence, relation (18.67) for arbitrary t and t is a consequence of validity of this relation for the function fk (t) on the segment [tk , tk+1 ]. Therefore, f ∗ ∈ Hω0 . Hence, for the convex moduli of continuity, relation (18.62) is an equality up to the quantities O(1)ω(1/n)(1 − q)−2 n−1 , i.e., sup |iqn (f )|

f ∈Hω0

=

2n−1

ck |ψ(t)|ω(ρk (t) − t)dt + O(1)

Zq (y(τk ))

k=2

tk

ω(1/n) (1 − q)2 n

(18.70)

for an arbitrary convex modulus of continuity ω(t). Further, setting t = y(τ ) and using (18.59), we have ck

τk |ψ(t)|ω(ρk (t) − t)dt =

cos[(n − 1)τ ]ω(ρk (y(τ )) − y(τ ))dτ.

(18.71)

xk

tk

Assume that τ ∈ [xk , τk ] for some k = 2, 2n − 1. Then τ

2τk −τ

cos(n − 1)vdv = xk

cos(n − 1)vdv xk

or setting v = y1 (t) and recalling that y(xk ) = tk , we have y(τ ) cos[(n − 1)y1 (t)]y1 (t)dt = tk

y(2τ k −τ )

cos[(n − 1)y1 (t)]y1 (t)dt.

tk

By combining this equality and relation (18.60), we conclude that y(2τk − τ ) = ρk (y(τ )), tk ≤ y(τ ) ≤ ck ≤ ρk (y(τ )) ≤ tk+1 .

(18.72)

Therefore, according to (18.71) and (18.72), ck

τk |ψ(t)|ω(ρk (t) − t)dt = |

tk

cos[(n − 1)τ ]ω(y(2τk − τ ) − y(τ ))dτ |. (18.73) xk

Section 18

Solution of Kolmogorov–Nikol’skii Problem

327

By mean value theorem, we have y(2τk − τ ) − y(τ ) = y (ξ)2(τk − τ ), ξ ∈ [τ, 2τk − τ ] ⊂ [xk , xk+1 ] (18.74) in view of (18.16), y (ξ) = 1 − γn ,

(18.75)

where γn = 1 −

2 (y(ξ)) Zq,n 1 − q cos y(ξ) , = n 2 2 Zq (y(ξ)) (n − 1)( n−1 − 2n−1 n−1 q cos y(ξ) + q )

thus, in view of (18.8), 0 < γn
0) almost everywhere on (c, b), moreover, b ϕ(t)dt = 0. a

Further, let Hωp be the class of functions determined by equality (18.90) for arbitrary fixed modulus of continuity ω(t) and let y = y(t) be a nondecreasing function on R1 satisfying the following condition: |y(t) − y(t )| ≤ |t − t | ∀t, t ∈ R1 .

(18.96)

Then, for any p ∈ [1, ∞), b Eωp (ϕ, y) = sup f ∈Hωp

f (x + y(t))ϕ(t)dtp a

c ≤

b |ϕ(t)|ω(ρ(t) − t)dt =

a

|ϕ(t)|ω(ρ−1 (t) − t)dt,

(18.97)

c

where ρ(x) is the function determined on [a, c] by the equalities Φ(x) = Φ(ρ(x)), a ≤ x ≤ c ≤ ρ(x) ≤ b, x ϕ(t)dt,

Φ(x) = a

and ρ−1 (x) is the function inverse to ρ(x).

(18.98)

Approximation by Fourier Sums in Spaces C and L1

334

Chapter 5

Proof. For p = 1 this lemma, in fact, coincides with Lemma 2.2 proved earlier. For p ∈ (1, ∞) it is proved by analogy with the case where p = 1, by virtue of generalized Minkowski inequality: b df d b p 1/p ( | (x, y)dy| dx) ≤ ( |f (x, y)|p dx)1/p dy, 1 ≤ p ≤ ∞. a

c

c

(18.99)

a

Under the assumptions of the lemma the functions t = ρ(x) and x = ρ−1 (t) are strictly monotone and absolutely continuous on [a, c] and [c, b], respectively. Moreover, ρ(a) = b, ρ(c) = c, ρ−1 (c) = c, ρ−1 (b) = a.

(18.100)

These facts enable us to substitute the variables t = ρ(x) and x = ρ−1 (t) in the integrals and, by virtue of (18.98), guarantee that the following equalities hold almost everywhere: ϕ(t) = ϕ(ρ(t))ρ (t), t ∈ (a, c), ϕ(ρ−1 (t))(ρ−1 (t)) = ϕ(t), t ∈ (c, b).

(18.101)

Setting t = ρ(τ ), in view of (18.100) and (18.101), we get b

c f (x + y(t))ϕ(t)dt = −

c

f (x + y(ρ(τ )))ϕ(ρ(τ ))ρ (τ )dτ

a

c =−

f (x + y(ρ(τ ))ϕ(τ )dτ. a

Consequently, by virtue of (18.99) and (18.90), for every f ∈ Hωp we have b

c f (x + y(t))ϕ(t)dtp =

a

[f (x + y(t)) − f (x + y(ρ(t))]ϕ(t)dtp a

c ≤

f (· + y(t)) − f (· + y(ρ(t))p |ϕ(t)|dt a

c ω(y(ρ(t)) − y(t))|ϕ(t)|dt.

≤ a

Section 18

Solution of Kolmogorov–Nikol’skii Problem

335

Hence, taking into account (18.96), we get the first relation in (18.97). The equality on the right-hand side of (18.97) can be obtained by a change of variables. Since ρ(t) − t ≤ b − a, under assumptions of the lemma the following inequality holds: c Eωp (ϕ; y) ≤ ω(b − a) |ϕ(t)|dt. a

This and inequality (18.99) yield the following analog of Lemma 2.3 as a consequence of Lemma 18.3 in the space Lp for all p ∈ [1, ∞): Lemma 18.4. Let ϕ(t) be a summable function on [a, b] and let xk , k = 1, N , a ≤ x1 < x2 < . . . < xN ≤ b, be a set of points such that on every segment [xk , xk+1 ] the function satisfies all assertions of Lemma 18.3, i.e., there exists a point ck such the function ϕ(t) has opposite signs almost everywhere on the segments [xk , ck ] and [ck , bk ], moreover, xk+1

ϕ(t)dt = 0. xk

Further, let y(t) be a nondecreasing function on R1 that satisfies the conditions (18.96). If f ∈ Hωp , p ∈ [1, ∞), then b

f (x + y(t))ϕ(t)dtp a

x1 ≤ sup f (· + y(t)) − f (·)p a≤t≤x1

|ϕ(t)|dt a

xN

b |ϕ(t)|dt + sup f (· + y(t)) − f (·)p

+ ω(Δ) x1

where Δ = max(xk+1 − xk ). k

xN ≤t≤b

xN

|ϕ(t)|dt,

Approximation by Fourier Sums in Spaces C and L1

336

Chapter 5

Applying Lemma 18.4 to the right-hand side in (18.94), we get τ2 Rn(1) (f )p p ≤

sup y1 (0)≤τ ≤τ2

f (x + y(τ )) − f (x)p

|rn(1) (τ )|dτ

y1 (0) τn+3

+

sup

τn−1 ≤τ ≤τn+3

f (x + y(τ )) − f (x)p

|rn(1) (τ )|dτ

τn−1 y1(2π)

+

sup τ2n−2 ≤τ ≤y1 (2π)

4π + ω( ) n−1

f (x + y(τ )) − f (x)p

|rn(1) (τ )|dτ τ2n−2

τ 2n−1

|rn(1) (τ )|dτ.

(18.102)

τ2

By virtue of relations (18.27), we obtain sup y1 (0)≤τ ≤τ2

f (x + y(τ )) − f (x)p ≤ ω(y(τ2 )) = O(1)ω(1/n), sup

τn−1 ≤τ ≤τn+3

f (x + y(τ )) − f (x)p ≤ ω(π)

(18.103) (18.104)

and sup τ2n−2 ≤τ ≤y1 (2π)

=

f (x + y(τ )) − f (x)p

sup τ2n−2 ≤τ ≤y1 (2π)

f (x + y(τ )) − f (x + 2π)p = O(1)ω(1/n). (18.105)

Therefore, combining relations (18.26), (18.27), and (18.102)–(18.105), we obtain (18.95). Following the scheme of the proof of Theorem 18.1, we obtain analogs of relations (18.29) and (18.30): x2n δx (y(τ ))ln (τ ) cos(n − 1)τ dτ Jn (f ) = x2

+ Rn(1) (f )p + Rn(2) (f )p ,

f ∈ Hωp , (18.106)

Section 18

Solution of Kolmogorov–Nikol’skii Problem

where

337

y1(2π)

Rn(2) (f )p

δx (y(τ ))rn(2) cos(n − 1)τ dτ.

= y1 (0)

By analogy with the proof of estimate (18.41), using Lemma 18.4 instead of Lemma 1.3, we obtain ω(1/n) Rn(2) (f )p p = O(1) . (18.107) (1 − q)2 n Combining relations (18.92), (18.93), (18.95), (18.106), and (18.107), we arrive at formula (18.56): x2n qn q En (Lβ Hωp ) = f (x + y(τ ))ln (τ ) cos(n − 1)τ dτ p sup π f ∈Hωp x2

+ O(1) ≤

q n ω(1/n) (1 − q)2 n

2n−1 qn q n ω(1/n) Zq (y(τk ))Sk (ω)p + O(1) , π (1 − q)2 n

(18.108)

k=2

where

xk+1

Sk (ω)p = sup f ∈Hωp

f (x + y(τ )) cos(n − 1)τ dτ p . xk

Using Lemma 18.3 for ϕ(t) = cos(n−1)t, xk = kπ/(n−1), and ck = kπ/(n− π 1)+ 2(n−1) and noting that, in this case, ρ(x) = ρk (x) = 2k+1 n−1 π−x, x ∈ [xk , ck ], we get ck 2k + 1 π − 2t)dt Sk (ω)p = | cos(n − 1)t|ω( n−1 xk

π/(2(n−1))

sin(n − 1)tω(2t)dt

= 0

1 = n−1

π/2 2t ω(1/n) ω( ) sin tdt + O(1) . n n2 0

Substituting this value in (18.108) and taking into account equality (18.83), we obtain estimate (18.91).

338

Approximation by Fourier Sums in Spaces C and L1

19.

Lebesgue Inequalities for Poisson Integrals

Chapter 5

19.1. As usual, Lp , 1 ≤ p ≤ ∞, denotes the space of functions f ∈ L with finite norm f p , Up = {f : f ∈ Lp , f p ≤ 1} are the unit balls in the spaces Lp and Lqβ Up = Lqβ,p , C is the subset of continuous functions from the set L, and · C is the norm in the space C : f C = max |f (t)|. t

The space of trigonometric polynomials tn−1 (·) whose degree does not exceed n − 1 is denoted by T2n−1 and En (f )p =

inf

tn−1 ∈T2n−1

f (·) − tn−1 (·)p

is the best approximation of f (·) in the space Lp by trigonometric polynomials from T2n−1 . The main results of the current subsection are formulated in the following statement: Theorem 19.1. Let q ∈ (0, 1), β ∈ R1 , and p ≥ 1. Then, for an arbitrary function f ∈ Lqβ Lp , the following asymptotic inequality is true: ρn (f ; x)p ≤ (

8q n O(1)q n K(q) + )En (fβq )p , 2 2 π (1 − q) n

where

π/2 K(q) = 0

(19.1)

du , 1 − q 2 sin2 u

and O(1) is a quantity uniformly bounded in the parameters q, β, p, n, and in f ∈ Lqβ Lp . Inequalities of the form (19.1) for other classes of functions were established in Section 15. It was also noted there that such inequalities are exact on some important subsets of functions under consideration. We note several cases of this sort for estimate (19.1). If f ∈ Lqβ,p , then fβq p ≤ 1 and, consequently, En (fβq )p ≤ 1. By considering the upper bounds of both sides of (19.1) over the sets Lqβ,p , we obtain En (Lqβ,p )p ≤

8q n O(1)q n K(q) + , p ≥ 1. π2 (1 − q)2 n

(19.1 )

Section 19

Lebesgue Inequalities for Poisson Integrals

339

By comparing this relation with equalities (16.41) and (16.3), we conclude that, for p = 1 and p = ∞, relation (19.1 ) turns into the equality. Another example of the fact that estimate (19.1) cannot be improved is given by the following statements: Theorem 19.2. Let q ∈ (0, 1), β ∈ R1 , and let Lqβ C be the set of Poisson integrals of all continuous functions. If f ∈ Lqβ C, then ρn (f ; x)C = ρn (f ; x)∞ ≤ (

8q n O(1)q n K(q) + )En (fβq )C . π2 (1 − q)2 n

(19.2)

Moreover, for any function f ∈ Lqβ C and any natural n, there exists a function F (x) = F (f ; n; x) in the space Lqβ C such that En (Fβq )C = En (fβq )C and the following equality is true: ρn (F ; x)C = (

8q n O(1)q n K(q) + )En (Fβq )C . π2 (1 − q)2 n

(19.3)

In (19.2) and (19.3), O(1) are quantities uniformly bounded in parameters n, q, β, and in f ∈ Lqβ C. Let ε = εn , n ∈ N be an arbitrary sequence of nonnegative numbers monotonically decreasing to zero. By Cn (ε) we denote the set of continuous functions ϕ ∈ C for which, for a given n, the inequality En (ϕ)C ≤ εn is satisfied; by Lqβ Cn (ε) we denote the set of the Poisson integrals Jβq (ϕ; ·) of functions ϕ from Cn (ε). Then Theorem 19.2 yields the following statement: Theorem 19.2. Let q ∈ (0, 1), β ∈ R1 . Then, for any class Lqβ Cn (ε) and all n ∈ N, we have En (Lqβ Cn (ε)) = (

8q n O(1)q n K(q) + )εn , π2 (1 − q)2 n

(19.4)

where O(1) is a quantity uniformly bounded in n, q, and β. Indeed, if f ∈ Lqβ Cn (ε), then fβq (·) is continuous and En (fβq )C ≤ εn . Therefore, on the basis of (19.2), we have ρn (f ; x)C ≤ (

8q n O(1)q n K(q) + )εn ∀f ∈ Lqβ Cn (ε). 2 2 π (1 − q) n

(19.5)

Approximation by Fourier Sums in Spaces C and L1

340

Chapter 5

On the other hand, for the function F (x) from Theorem 19.2 constructed by a function ϕ ∈ Cn (ε) such that En (ϕ) = εn , this inequality turns into the equality, whence the statement of Theorem 19.2 follows. 19.2. Proof of Theorem 19.1. Let equality (18.3), which is valid for any q function f ∈ Lqβ , be a starting point. The function Pβ,n is orthogonal to any polynomial tn−1 from T2n−1 . Therefore, 1 ρn (f ; x) = π

2π

q δn (x + t)Pβ,n dt, δn (τ ) = fβq (τ ) − tn−1 (τ ),

(19.6)

0

where tn−1 (·) is an arbitrary polynomial from T2n−1 . By virtue of (18.13), we have qn ρn (f ; x) = π

2π

df

δn (x + t)Zq (t) cos(nt + θ(t) + βπ/2)dt =

qn ˆ Jn (f ). (19.7) π

0

Here, Jˆn (f ) is an analog of integral Jn (f ) determined in the proof of Theorem 18.1. We shall do the same things with Jˆn (f ) as we did with Jn (f ) in Subsection 18.2. Namely, by setting t = y(τ ) and using equality (18.19), we obtain an analog of relation (18.20): y1(2π)

Jˆn (f ) =

ˆ (1) (f ), δn (x + y(τ ))Zq (y(τ )) cos(n − 1)τ dτ + R n

(19.8)

y1 (0)

where y1(2π)

ˆ n(1) (f ) = R

δn (x + y(τ ))rn(1) (τ ) cos(n − 1)τ dτ.

(19.9)

y1 (0)

According to (18.26), |rn(1) (t)| ≤

1 (1 −

q)2 (n

− 1)

, n ≥ 2.

(19.10)

Therefore, using the Minkowski inequality (see (18.99)) for p ∈ [1, ∞), we get

Section 19

Lebesgue Inequalities for Poisson Integrals

341

2π y1(2π) (1) (1) ˆ |δn (x + y(τ ))|dτ )p dx)1/p Rn (f )p ≤ rn M ( ( 0

y1 (0)

y1(2π) 2π

≤

rn(1) M

|δn (x + y(τ ))|p dx)1/p dτ

( y1 (0)

0

= rn(1) M δn p (y1 (2π) − y1 (0)) ≤

2πδn p (1 + 1/(n − 1)) , n ≥ 2, (n − 1)(1 − q)2

because, by virtue of (18.15) and (18.12), we have y1 (2π) − y1 (0) = 2π(1 + 1/(n − 1)).

(19.11)

(1)

It is clear that the same estimate for Rn (f )p is also true for p = ∞. Thus, ˆ n(1) (f )p ≤ R

2πδn p (1 + 1/(n − 1)) (n − 1)(1 − q)2

∀p ≥ 1.

(19.12)

Further, in view of equality (18.28), we obtain an analog of relation (18.29): Jˆn (f ) =

x2n ˆ n(1) (f ) + R ˆ n(2) (f ), (19.13) δ(x + y(τ ))ln (τ ) cos(n − 1)τ dτ + R x2

where y1(2π)

ˆ n(2) (f ) = R

δ(x + y(τ ))rn(2) (τ ) cos(n − 1)τ dτ. y1 (0)

Taking into account relations (18.48) and (18.51), by virtue of the Minkowski inequality, we find n ˆ (2) (f )p = O(1) q δn p . R (19.14) n (1 − q)2 n Here and below, O(1) is a quantity uniformly bounded in all parameters under consideration. It is clear that the same estimate is also true for p = ∞. By

Approximation by Fourier Sums in Spaces C and L1

342

Chapter 5

combining relations (19.7), (19.8), and (19.12)–(19.14), we obtain the following relation: qn ρn (f ; x)p = π

x2n δn (x + y(τ ))ln (τ ) cos(n − 1)τ dτ p x2

+ O(1)

q n δn p (1 − q)2 n

(19.15)

for all p ≥ 1. Let iqn (f ) denote the integral in the last expression. Then, by virtue of (18.29), we get iqn (f )

=

2n−1

xk+1

δn (x + y(τ )) cos(n − 1)τ dτ.

Zq (y(τk ))

k=2

xk

Using the Minkowski inequality, we obtain xk+1

δn (x + y(τ )) cos(n − 1)τ dτ p

xk

xk+1

2π | cos(n − 1)τ |( |δn (x + y(τ ))|p dx)1/p dτ

≤ xk

0 (k+1)π/(n−1)

= δn p

| cos(n − 1)τ |dτ =

2δn p , 1 ≤ p < ∞. n−1

kπ/(n−1)

It is clear that the same estimate is also true for p = ∞. Therefore, for all p ≥ 1, we have 2n−1 2δn p q in (f )p ≤ Zq (y(τk )). (19.16) n−1 k=2

In view of relations (19.15), (19.16), and (18.83), we get ρn (f ; x)p ≤ (

8q n qn K(q) + O(1) )δn p . π2 (1 − q)2 n

(19.17)

Choosing as tn−1 (·) in (19.6), the polynomial t∗n−1 (·) that realizes the best approximation of the function fβq (·) in the space Lp , we get estimate (19.1).

Section 19

Lebesgue Inequalities for Poisson Integrals

343

19.3. Proof of Theorem 19.2. Since C ⊂ L∞ and f ∞ = f C for a function f ∈ C, estimate (19.2) is a consequence of (19.1). Therefore, it remains to prove the second part of the theorem. In the case under consideration, equality (19.15) takes the form qn ρn (f ; x)C = π

x2n q n δn C δn (x + y(τ ))ln (τ ) cos(n − 1)τ dτ C + O(1) . (1 − q)2 n

x2

Consequently, the required statement will be proved if we show that, for any function ϕ ∈ C, one can find a continuous function Φ(x) = Φ(ϕ; n; x) such that, for all n ≥ 2, the equalities En (Φ)C = En (ϕ)C and qn | π

x2n [Φ(y(τ )) − t∗n−1 (y(τ ))]ln (τ ) cos(n − 1)τ dτ | x2

=(

8q n qn K(q) + O(1) )En (ϕ)C , (19.18) π2 (1 − q)2 n

are true; here, t∗n−1 (·) is the polynomial of the best approximation of the function Φ(·) in the space C. For this purpose, we set tk = y(xk ), zk = y(τk ), k = 2, . . . , 2n. It is clear that 0 < t2 < z2 < t3 < . . . < z2n−1 < t2n ≤ 2π.

(19.19)

Beginning with a given function ϕ ∈ C, we define the function ϕ0 (t) on the segment [t2 , t2n ] by setting ϕ0 (t) = En (ϕ)C sign cos[(n − 1)y1 (t)]. For this function, we have x2n | ϕ0 (y(τ ))ln (τ ) cos(n − 1)τ dτ | x2

=

2n−1 k=2

tk+1

ϕ0 (t) cos[(n − 1)y1 (t)]y1 (t)dt

ln (τk ) tk

Approximation by Fourier Sums in Spaces C and L1

344

=

2n−1

Chapter 5

tk+1

| cos[(n − 1)y1 (t)]|y1 (t)dt

ln (τk )En (ϕ)C

k=2

tk

= En (ϕ)C

2n−1 k=2

xk+1

ln (τk ) xk

2n−1 2En (ϕ)C | cos(n − 1)τ |dτ = ln (τk ). (n − 1) k=2

Hence, taking into account equality (18.83), we get qn | π

x2n ϕ0 (y(τ ))ln (τ ) cos(n − 1)τ dτ | x2

=(

qn 8q n K(q) + O(1) )En (ϕ)C . (19.20) π2 (1 − q)2 n

Further, by ϕ1 (t) = ϕ1 (n, t) we denote the function coinciding with the function ϕ0 (t) everywhere on t ∈ [2π/(n−1), 2nπ/(n−1)] except the δ-neighborhoods (δ < π/2n) of the points zk , k = 2, . . . , 2n − 1, where it is linear and its graph joins the points (zk − δ, ϕ0 (zk − δ)) and (zk + δ, ϕ0 (zk + δ)). On the intervals [0, t2 ) and (t2n , 2π], we define the function ϕ1 (t) so that there exist at least 2n points ξk : 0 ≤ ξ1 < ξ2 < . . . < ξ2n < 2π on the interval [0, 2π) at which its modulus attains its maximum value equal to En (ϕ)C . It is clear that, by virtue of relation (19.19), this is always possible. Finally, by ϕδ (t) = ϕδ (n; t) we denote the 2π-periodic continuation of the function ϕ1 (t). The function ϕδ (t) is continuous for any δ > 0, and the polynomial t∗n−1 (·) of degree not higher than n − 1 that realizes its best uniform approximation according to the Chebyshev criterion is the polynomial identically equal to zero; in this case, En (ϕδ )C = En (ϕ)C . Therefore, x2n [ϕδ (y(τ )) − t∗n−1 (y(τ ))]ln (τ ) cos(n − 1)τ dτ x2

x2n = ϕδ (y(τ ))ln (τ ) cos(n − 1)τ dτ x2

x2n = ϕ0 (y(τ ))ln (τ ) cos(n − 1)τ dτ + Rn(3) (δ), x2

(19.21)

Section 20

Approximation on Classes of Analytic Functions

where Rn(3) (δ)

345

x2n = [ϕ0 (y(τ )) − ϕδ (y(τ ))]ln (τ ) cos(n − 1)τ dτ. x2

Hence, 2nδEn (ϕ)C . (19.22) (1 − q)2 Combining relations (19.20)–(19.22) and taking into account the arbitrariness of the number δ, we conclude that there exists a function Φ ∈ C such that En (Φ)C = En (ϕ)C and relation (19.18) is satisfied. |Rn(3) (δ)| ≤

20.

Approximation by Fourier Sums on Classes of Analytic Functions

20.1. By Dq , q ∈ [0, 1] we denote the set of sequences ψ(k), k ∈ N, for which ψ(k + 1) lim = q. (20.1) k→∞ ψ(k) ¯

The main results in this section are obtained for the classes Lψ N whose determining parameters ψ1 (k) and ψ2 (k) are such that the sequences ψ(k) = (ψ12 (k) + ψ22 (k))1/2 belong to the set Dq for certain q ∈ [0, 1). In this case, ¯ the sets C ψ and Cβψ¯ consist of 2π-periodic functions f (x) admitting a regular extension into the strip |Imz| ≤ ln 1/q (see Section 14). The kernels ∞ π Pβ¯q (t) = q k cos(kt − βk ), q ∈ (0, 1), βk ∈ R, (20.2) 2 k=1

are an important example of kernels whose coefficients ψ(k) satisfy condition (20.1); for βk ≡ β, kernels (20.2) are the well-known Poisson kernels and are denoted by Pβq (·). The classes Lψ N generated by kernels (20.2) are denoted by Lqβ¯N and the β¯ ¯ corresponding (ψ, β)-integrals are denoted by Jβ¯q (f ; x). As usual, let Lp , 1 ≤ p ≤ ∞, be the space of functions f ∈ L with finite norm f p ; here, for p ∈ [1, ∞), f p = f Lp

2π = ( |f (t)|p dt)1/p , 0

346

Approximation by Fourier Sums in Spaces C and L1

Chapter 5

so that L1 = L. For p = ∞, we have df

f ∞ = f M = ess sup |f (t)|. We denote a unit ball in Lp by Up and assume that ¯

¯

ψ 0 ψ 0 Lψ Up0 = Lψ ¯ , Up = {ϕ : ϕ ∈ Up , ϕ ⊥ 1}. p , Lβ¯ Up = Lβ,p

By ω(t) = ω(f ; t) and ωp (t) = ωp (f ; t) we denote the moduli of continuity of functions f ∈ C and f ∈ Lp , 1 ≤ p < ∞ in the spaces C and Lp , respectively, ω(t) = ω(f ; t) = sup f (· + h) − f (·)C , 0≤h≤t

ωp (t) = ωp (f ; t) = sup f (· + h) − f (·)p , 1 ≤ p < ∞. |h|≤t

Setting Hω = {ϕ ∈ C, ω(ϕ; t) ≤ ω(t)}, Hωp = {ϕ ∈ Lp , ωp (ϕ; t) ≤ ω(t)}, 1 ≤ p < ∞, where ω = ω(t) is the given modulus of continuity, we consider classes ¯ ¯ C ψ Hω , Cβψ¯ Hω , Lβ Hωp , and Lψ H . ψ¯ ωp Now let f ∈ L, let Sn (f ; x) = Sn (f ) be the partial Fourier sum of the function f of order n, and let, as usual, ρn (f ; x)=f (x) − Sn−1 (f ; x). Denote by T2n−1 the space of trigonometric polynomials tn−1 whose degree does not exceed n − 1. The quantity df

En (f )s = inf f − tn−1 s tn−1

is the best approximation of f in the metric of Ls by trigonometric polynomials of degree n − 1. ¯ Here, we investigate the quantities ρn (f ; x)s , f ∈ Lψ N, where N is a certain fixed subset of Lp , 0 ≤ p, s ≤ ∞, and the quantities ¯

En (Lψ N)s = sup f − Sn−1 (f )s f ∈Lψ¯ N

in order to obtain asymptotic inequalities for these quantities in the case ψ ∈ Dq , 0 < q < 1.

Section 20

Approximation on Classes of Analytic Functions

347

The main idea is reflected in Lemma 20.1 presented below, according to which, as n → ∞, the behavior of the remainders ρn (Ψβ¯) of the kernels Ψβ¯(t) which generate classes Lψ for ψ ∈ Dq , 0 < q < 1, is similar to the behavβ¯ q ior of the remainders ρn (Pβ¯ ) of the kernels Pβ¯q (t). In particular, this enables us to reduce the problem of determination of asymptotic equalities for the quanN)s to analogous problems for the quantities En (Lqβ¯N)s . In many tities En (Lψ β¯ important cases, asymptotic equalities for the quantities En (Lqβ¯N)s (and even their exact values) are known. In these cases, it is also possible to establish asymptotic equalities for the quantities En (Lψ N)s . In particular, by using this β¯ approach, on the basis of the results of Sections 16 and 18 one can obtain asympψ )C , En (Cβψ Hω )C , and En (Lψ totic equalities for the quantities En (Cβ,∞ β,1 )1 for all ψ ∈ Dq , 0 < q < 1. 20.2. The theorem below establishes the relationship between the norms of ¯ the remainders of the Fourier series of ψ-integrals Jβ¯ψ (ϕ), ψ ∈ Dq , 0 < q < 1, in the space Ls and the remainders of the Fourier series of integrals Jβ¯q (ϕ), ϕ ∈ Lp . Theorem 20.1. Let 1 ≤ p, s ≤ ∞ and let ψ ∈ Dq , 0 < q < 1, ψ(k) > 0. L , the following equality holds as n → ∞ : Then, for any function f ∈ Lψ β¯ p ρn (f )s = ψ(n)(q

−n

ρn (Jβ¯q (fβ¯ψ ))s

+ O(1)

εn En (fβ¯ψ )p (1 − q)2

),

(20.3)

where εn = sup | ψ(k+1) ψ(k) − q| and O(1) is a quantity uniformly bounded in k≥n

n, p, s, q, ψ(k), and βk . Prior to the proof of Theorem 20.1, we consider the following statement: Lemma 20.1. Let ψ(k), k ∈ N, be an arbitrary numerical sequence from Dq , 0 < q < 1. Then, for any sequence of real numbers γk , k = 1, 2, . . . , the following equality is true for any n ∈ N : ∞ k=n

ψ(k) cos(kt − γk ) = ψ(n)(q

−n

∞ k=n

q k cos(kt − γk ) + rn (t)).

(20.4)

Approximation by Fourier Sums in Spaces C and L1

348

Chapter 5

Furthermore, for the quantity rn (t) = rn (ψ, γ¯ , t), the following estimate holds beginning with a certain number n0 : εn |rn (t)| ≤ , (20.5) (1 − q − εn )(1 − q) εn = sup |δk |, δk = k≥n

ψ(k + 1) − q. ψ(k)

(20.5 )

Proof. We have ∞

ψ(n + i) cos((n + i)t − γn+i )

i=0

= ψ(n)(cos(nt − γn ) +

∞ $ i−1 ψ(n + l + 1) i=1 l=0

= ψ(n)(q

−n

∞

ψ(n + l)

cos((n + i)t − γn+i ))

q k cos(kt − γk ) + rn (t)),

k=n

where rn (t) = rn (ψ, γ¯ , t) =

∞ $ i−1 ψ(n + l + 1) − q i ) cos((n + i)t − γn+i ). (20.6) ( ψ(n + l) i=1 l=0

Let us prove inequality (20.5). In view of (20.6), we have |rn (t)| ≤

∞

|˜ qi − q i |,

(20.7)

i=1

where q˜i =

i−1 $ ψ(n + l + 1) l=0

ψ(n + l)

.

(20.8)

For q˜i ≥ q i , we have i−1 $ |˜ qi − q | = q˜i − q ≤ (q + εn ) − q i = (q + εn )i − q i . i

i

l=0

If q˜i < q i , then, by virtue of the convexity of the function tk , k = 1, 2, . . . , t > 0, we get i−1 $ |˜ qi − q | = q − q˜i ≤ (q − εn ) − q i = (q − εn )i − q i ≤ (q + εn )i − q i . i

i

l=0

Section 20

Approximation on Classes of Analytic Functions

349

Thus, the following inequality is always true: |˜ qi − q i | ≤ (q + εn )i − q i .

(20.9)

Taking into account that relations (20.1) and (20.5 ) imply that the sequence εn monotonically decreases to zero, we conclude that εn < 1 − q beginning with a certain number n0 . Therefore, taking (20.7) and (20.9) into account, for n ≥ n0 we obtain |rn (t)| ≤

∞

((q + εn )i − q i ) =

i=1

εn . (1 − q)(1 − q − εn )

The lemma is proved. Proof of Theorem 20.1. Let f ∈ Lψ L , 1 ≤ p ≤ ∞. Then the following β¯ p equality holds almost everywhere: 1 ρn (f ; x) = f (x) − Sn−1 (f ; x) = π where Ψβ,n ¯ (t) =

∞

π

ψ Ψβ,n ¯ (t)fβ¯ (x − t)dt,

(20.10)

−π

ψ(k) cos(kt −

k=n

βk π ). 2

Setting df

q Pβ,n ¯ (t) =

∞

q k cos(kt −

k=n

βk π ), 0 < q < 1, βk ∈ R, k ∈ N, 2

and using (20.10) and (20.4), we establish that 1 ρn (f ; x) = π

π

q ψ ψ(n)(q −n Pβ,n ¯ (t) + rn (t))fβ¯ (x − t)dt

−π

q −n = ψ(n)( π

π

q ψ Pβ,n ¯ (t)fβ¯ (x − t)dt + Rn (f ; x))

−π

= ψ(n)(q −n ρn (Jβ¯q (fβ¯ψ ; x)) + Rn (f ; x)),

(20.11)

Approximation by Fourier Sums in Spaces C and L1

350

Chapter 5

almost everywhere; here, 1 Rn (f ; x) = π df

π

rn (t)fβ¯ψ (x − t)dt,

−π

and the function rn (t) is defined by relation (20.6) for γk = βk , k ∈ N. Let us prove that Rn (f )s ≤ 4πfβ¯ψ − tn−1 p rn ∞ , 1 ≤ p, s ≤ ∞.

(20.12)

for any trigonometric polynomial tn−1 ∈ T2n−1 . By using the Young inequality for convolutions (see Subsection 15.1) for p ≤ s we obtain Rn (f )s = (fβ¯ψ − tn−1 ) ∗ rn s ≤ Since

1 r

= (1 −

1 p

1 ψ f − tn−1 p rn r . π β¯

(20.13)

+ 1s ) ∈ [0, 1] in the case under consideration, we have rn r ≤ rn ∞ (2π)1/r ≤ 2πrn ∞ .

(20.14)

Comparing estimates (20.13) and (20.14), we get Rn (f ; x)s ≤ 2fβ¯ψ − tn−1 p rn ∞ .

(20.15)

Now let 1 ≤ s ≤ p ≤ ∞. By virtue of the Hölder inequality, we have ∀f ∈ Lp f s ≤ (2π)

p−s ps

f p , 1 ≤ s ≤ p ≤ ∞.

Therefore, Rn (f )s = (fβ¯ψ − tn−1 ) ∗ rn s ≤ 2π(fβ¯ψ − tn−1 ) ∗ rn p ≤ 2fβ¯ψ − tn−1 p rn 1 ≤ 4πfβ¯ψ − tn−1 p rn ∞ .

(20.16)

Consequently, estimate (20.12) follows from (20.15) and (20.16). As tn−1 (·), we choose the polynomial t∗n−1 (t) of the best approximation of the function fβ¯ψ (·) in the space Lp . Then, by using inequality (20.5), we obtain the following estimate: Rn (f ; x)s = O(1)

εn En (fβ¯ψ )p (1 − q)2

, 1 ≤ p, s ≤ ∞.

(20.17)

Combining relations (20.11) and (20.17), we get equality (20.3). Setting s = p, βk ≡ β, β ∈ R in (20.3) and applying asymptotic inequality (18.91), we arrive at the following statement:

Section 20

Approximation on Classes of Analytic Functions

351

Corollary 20.1. Let 1 ≤ p ≤ ∞, ψ ∈ Dq , 0 < q < 1, ψk > 0, and β ∈ R. Then, for any function f ∈ Lψ β Lp , the following asymptotic inequality holds: ρn (f )p ≤ ψ(n)(

8 εn + 1/n K(q) + O(1) )En (fβψ )p , 2 2 π (1 − q)

where εn = sup | ψ(k+1) ψ(k) − q|, K(q) = k≥n

π/2

√

0

du , 1−q 2 sin2 u

and O(1) is a quantity

uniformly bounded in n, p, q, β, ψ(k), and in f ∈ Lψ β Lp . If ψ ∈ Dq , 0 < q < 1, then, by virtue of the d’Alembert criterion for the convergence of numerical positive series, series

π ψ(k) cos(kt − βk ) 2

converges absolutely and uniformly. Consequently, if f ∈ Cβψ¯ Lp , then equalities (20.10) and (20.11) are satisfied at every point x. Therefore, the above reasoning yields the following statement: Theorem 20.1. Let ψ ∈ Dq , 0 < q < 1, ψ(k) > 0. Then, for any function f ∈ Cβψ¯ X, where X is either Lp , 1 ≤ p ≤ ∞ or C the following equality holds as n → ∞ : ρn (f )C = ψ(n)(q

−n

ρn (Jβ¯q (fβ¯ψ ))C

+ O(1)

εn En (fβ¯ψ )X (1 − q)2

),

(20.19)

where εn = sup | ψ(k+1) ψ(k) − q| and O(1) is a quantity uniformly bounded in k≥n

n, p, q, ψ(k), and βk . Setting βk ≡ β, β ∈ R in (20.19) for f ∈ Cβψ C and using Theorem 19.2 , we arrive at the following statement: Corollary 20.2. Let ψ ∈ Dq , 0 < q < 1, ψ(k) > 0, and β ∈ R. If f ∈ Cβψ C, then ρn (f )C ≤ ψ(n)(

8 εn + 1/n K(q) + O(1) )En (fβψ )C . 2 2 π (1 − q)

Approximation by Fourier Sums in Spaces C and L1

352

Chapter 5

Moreover, for every function f ∈ Cβψ C and any natural number n, there exists a function F (x) = F (f ; n; x) in the space Cβψ C such that En (Fβψ )C = En (fβψ )C and 8 εn + 1/n ρn (F ; x)C = ψ(n)( 2 K(q) + O(1) )En (Fβψ )C . π (1 − q)2 In the last two relations, ψ(k + 1) − q|, εn = sup | ψ(k) k≥n

π/2 K(q) = 0

du , 1 − q 2 sin2 u

and O(1) are quantities uniformly bounded in n, q, β, ψ(k), and f ∈ Cβψ C. Considering the upper bounds for both sides of relation (20.3) over the classes ψ and for relation (20.19) over the classes Cβ,p ¯ and taking into account that

Lψ ¯ β,p

sup ρn (f ; ·)s = sup{ρn (Jβ¯ψ (ϕ; ·)s : ϕp ≤ 1, ϕ ⊥ 1},

f ∈Lψ ¯ β,p

we establish the following statement: Theorem 20.2. Let 1 ≤ p, s ≤ ∞, and ψ ∈ Dq , 0 < q < 1, ψ(k) > 0. Then the following relations hold as n → ∞ : −n En (Lψ En (Lqβ,p ¯ )s = ψ(n)(q ¯ )s + O(1) β,p

εn ), (1 − q)2

(20.20)

εn ), (1 − q)2

(20.21)

ψ q −n En (Cβ,p En (Cβ,p ¯ )C = ψ(n)(q ¯ )C + O(1)

where εn = sup | ψ(k+1) ψ(k) − q|, and O(1) are quantities uniformly bounded in k≥n

n, p, s, q, ψ(k), and βk . −n E (C q ) are uniformly As n → ∞, the quantities q −n En (Lqβ,p n ¯ )s and q ¯ C β,p bounded above and below by certain positive numbers depending, possibly, only on q, p, and s. Indeed,

Section 20

Approximation on Classes of Analytic Functions

En (Lqβ,p ¯ )s

1 = sup | ϕp ≤1 π ϕ⊥1

π ϕ(x − t)

ϕp ≤1 ϕ⊥1

(1) ≤ Cp,s

∞

∞

q k cos(kt −

k=n

(1) q k = Cp,s

k=n

and, consequently,

q k cos(kt −

k=n

−π

(1) ≤ sup Cp,s ϕp

∞

353

βk π )dts 2

βk π )C 2

qn , 1−q

(1) (1 − q)−1 . q −n En (Lqβ,p )s ≤ Cp,s

To obtain the required lower bound, we consider the function fn (x) = q n sin t−1 p sin(nx −

βn π ), 2

which obviously belongs to the class Lqβ,p ¯ : −n ρn (fn ; x)s = q −n En (Lqβ,p ¯ )s ≥ q

sin ts (2) = Cp,s . sin tp

Therefore, (2) (1) −1 (i) Cp,s ≤ q −n En (Lqβ,p ¯ )s ≤ Cp,s (1 − q) , Cp,s > 0, i = 1, 2.

(20.22)

By analogy, one can also prove that q (1) −1 (i) Cp(2) ≤ q −n En (Cβ,p ¯ )C ≤ Cp (1 − q) , Cp > 0, i = 1, 2.

(20.22 )

In view of relations (20.20) and (20.21) and the fact that the sequence εn tends to zero as n → ∞, relations (20.3) and (20.19) enable us to establish asymptotic ψ equalities for the quantities En (Lψ ¯ )s and En (Cβ,p ¯ )C in the cases where the corβ,p q q responding equalities for the quantities En (Lβ,p ¯ )s and En (Cβ,p ¯ )C , respectively, are known. 20.3. According to the definition of the sets Dq (see (20.1)), it follows from the fact ψ ∈ Dq1 and ϕ ∈ Dq2 , 0 ≤ q1 , q2 ≤ 1, that ψ · ϕ ∈ Dq3 for q3 = q1 q2 (in particular, if ψ ∈ Dq1 and ϕ ∈ D1 , then ψ · ϕ ∈ Dq , 0 ≤ q ≤ 1 ). Thus, the set Dq , 0 < q < 1, contains all sequences of the form ψ(k) = q k ϕ(k), where

354

Approximation by Fourier Sums in Spaces C and L1

Chapter 5

0 ≤ q < 1 and ϕ ∈ D1 . On the other hand, if ψ ∈ Dq , 0 ≤ q < 1, then, representing ψ(k) in the form ψ(k) = q k df ψ(k) , qk

where ϕ(k) =

ψ(k) = q k ϕ(k), qk

we establish that ϕ(k + 1) = 1. k→∞ ϕ(k) lim

Therefore, the following statement is true: Proposition 20.1. In order that a sequence ψ(k) belong to the set Dq , 0 ≤ q < 1, it is necessary and sufficient that the following representation be true: ψ(k) = q k ϕ(k), where ϕ(k) is a certain sequence from D1 . In particular, Dq , 0 < q < 1, contains the sequences ψ ∗ (k) = q k k r , r ∈ r (−∞, +∞), ψ ∗∗ (k) = q k eαk , α ∈ (−∞, +∞), r ∈ (0, 1), etc. As already noted, for βk = β, k ∈ N, β ∈ R, the kernels Pβ¯q (t) coincide q with the Poisson kernels. In this case, we denote the classes Lqβ,p ¯ by Lβ,p . Combining Theorem 20.2 and Theorems 16.1 and 16.3, we get the following statement: ψ Theorem 20.3. Suppose that the classes Cβ,∞ and Lψ β,1 are generated by the kernel ∞ βπ ψ(k) cos(kt − ), (20.23) Ψβ (t) = 2 k=1

β ∈ R, ψ(k) ≥ 0, ψ ∈ Dq , 0 < q < 1. Then the following asymptotic equalities hold as n → ∞ : ⎫ ψ (20.24) )⎬ En (Cβ,∞ q εn 8 )), + = ψ(n)( 2 K (q) + O(1)( ⎭ π n(1 − q) (1 − q)2 En (Lψ (20.24 ) β,1 )1 where εn = sup | ψ(k+1) ψ(k) − q|, and O(1) are quantities uniformly bounded in k≥n

n, β, and ψ.

Section 20

Approximation on Classes of Analytic Functions

355

Remark 20.1. The asymptotic equalities (20.24) and (20.24 ) remain true if the condition 0 < q < 1 in Theorem 20.3 is replaced by the condition 0 ≤ q < 1. Indeed, formally setting q = 0 and taking into account that K(0) = π/2, we can rewrite estimates (20.24) and (20.24 ) in the form ( (20.25) En (Cβ,∞ ) 4 = ψ(n) + O(1)ψ(n + 1). π En (Lψ (20.25 ) β,1 )1 The validity of the asymptotic equalities (20.25) and (20.25 ) for ψ ∈ D0 , ψ(k) > 0, follows, e.g., from equalities (15.3 ) and (15.5). The Poisson kernels of the biharmonic equation ∞

Bq,β (t) =

1 1 − q2 βπ (1 + + k)q k cos(kt − ), 0 < q < 1, β ∈ R, (20.26) 2 2 2 k=1

and the Neumann kernels Nq,β (t) =

∞ k q k=1

k

cos(kt −

βπ ), 0 < q < 1, β ∈ R 2

(20.27)

satisfy the conditions of Theorem 20.3. It is easy to verify that |εk | = |

ψ(k + 1) q − q| ≤ , k ∈ N. ψ(k) k

(20.28)

for the coefficients ψ(k) of the kernels Bq,β (t) and Nq,β (t). Consequently, Theorem 20.3 and relation (20.28) yield the following statements: ψ Corollary 20.3. Suppose that the classes Cβ,∞ and Lψ β,1 are generated by a kernel Bq,β (t) of the form (20.26) and n ∈ N. Then the following asymptotic equalities hold as n → ∞ : ⎫ ψ )⎬ En (Cβ,∞ 1 − q2 q 8 n = q (1 + ), n)( 2 K (q) + O(1) ψ ⎭ 2 π n(1 − q)2 En (L )1 β,1

where O(1) are quantities uniformly bounded in n, q, and β.

356

Approximation by Fourier Sums in Spaces C and L1

Chapter 5

ψ Corollary 20.4. Suppose that the classes Cβ,∞ and Lψ β,1 are generated by a kernel Nq,β (t) of the form (20.27) and n ∈ N. Then the following asymptotic equalities hold as n → ∞ : ⎫ ψ )⎬ q n 8 En (Cβ,∞ q ), = ( 2 K (q) + O(1) ψ n π n(1 − q)2 En (L )1 ⎭ β,1

where O(1) are quantities uniformly bounded in n, q, and β. 20.4. Analyzing the proof of Theorems 16.1 and 16.3, one can easily verify that the methods used there allow one to obtain asymptotic estimates for the quanq q q tities En (Cβ,∞ and Lqβ,1 ¯ ) and En (Lβ,1 ¯ )1 for the classes Cβ,∞ ¯ ¯ generated by the q kernels Pβ¯ (t) of the form (20.2) where βk = β + kπ, β ∈ R, k ∈ N. Moreover, the form of the estimates obtained is the same as in the case βk = β, k ∈ N, β ∈ R. Namely, the following statement is true: Theorem A. Let n ∈ N, 0 < q < 1, and βk = β + kπ, β ∈ R, k ∈ N. Then the following asymptotic equalities hold as n → ∞ : ( q (20.29) En (Cβ,∞ ¯ ) q n ( 8 K (q) + O(1) = q ), π2 n(1 − q) En (Lqβ,1 ¯ )1 (20.29 ) where π/2 du , K(q) = 1 − q 2 sin2 u 0

and O(1) are quantities uniformly bounded in n, q, and β. Comparing Theorems 20.2 and A, we obtain the following analog of Theorem 20.3: ψ Theorem 20.3. Suppose that the classes Cβ,∞ and Lψ ¯ ¯ are generated by β,1 the kernel ∞ βk π Ψβ¯(t) = ψ(k) cos(kt − ), ψ(k) > 0, 2 k=1

where ψ ∈ Dq , 0 < q < 1, and βk = β +kπ, β ∈ R, n ∈ N. Then the following equalities hold as n → ∞ :

Section 20

Approximation on Classes of Analytic Functions

q En (Cβ,∞ ¯ )

(

En (Lqβ,1 ¯ )1

= ψ(n)(

357

(20.30) q 8 εn K (q) + O(1)( ), + π2 n(1 − q) (1 − q)2 (20.30 )

where εn = max | k≥n

ψ(k + 1) − q|, ψ(k)

and O(1) are quantities uniformly bounded in n, β, and ψ. ¯

ψ and Theorem 20.3 admits the following generalization to the classes C∞ ¯ : Lψ 1 ¯

¯

ψ and Lψ Theorem 20.4. Suppose that the classes C∞ 1 are generated by the kernel ∞ Ψ(t) = (ψ1 (k) cos kt + ψ2 (k) sin kt), (20.31) k=1

where ψi ∈ Dqi , 0 < qi < 1, i = 1, 2.

(20.32)

Then the following asymptotic equalities hold as n → ∞ : 8 ψ¯ ) = ψ12 (n) + ψ22 (n)( 2 K(q) En (C∞ π q εn + O(1)( )), (20.33) + n(1 − q) (1 − q)2 ¯ En (Lψ 1 )1

=

ψ12 (n) + ψ22 (n)(

8 K(q) π2 + O(1)(

where q = max q1 , q2 ,

⎧ (i) ⎪ max{εn } if ⎪ ⎪ i=1,2 ⎨ εn =

(1)

εn ⎪ ⎪ ⎪ ⎩ (2) εn

ε(i) n = sup | k≥n

q εn )), (20.33 ) + n(1 − q) (1 − q)2

q1 = q2 ,

if

q1 > q2 ,

if

q1 < q2 ,

ψi (k + 1) − qi |, i = 1, 2, ψi (k)

and O(1) are quantities uniformly bounded in n, ψ1 and ψ2 .

Approximation by Fourier Sums in Spaces C and L1

358

Chapter 5

¯

ψ Proof. Let f ∈ C∞ . Then

1 ρn (f, x) = f (x) − Sn−1 (f, x) = π

π Ψn (t)ϕ(x − t)dt,

(20.34)

−π

where Ψn (t) =

∞

(ψ1 (k) cos kt + ψ2 (k) sin kt) = Gn (t) + Hn (t), n ∈ N,

k=n

Gn (t) =

∞

ψ1 (k) cos kt, Hn (t) =

k=n

∞

ψ2 (k) sin kt.

k=n

Using conditions (20.32) and applying Lemma 20.1 to each of the functions Gn (t) and Hn (t), we get Ψn (t) = Gn (t) + Hn (t) = ψ1 (n)(q1−n Pq1 ,0,n (t) + Rn (ψ1 ; t)) + ψ2 (n)(q2−n Pq2 ,1,n (t) + Rn (ψ2 ; t)) = ψ1 (n)q1−n

∞

q1k cos kt + ψ2 (n)q2−n

k=n

∞

q2k sin kt

k=n (1)

+ O(1)( where Pq1 ,0,n (t) =

∞

q1k cos kt, Pq2 ,1,n (t) =

k=n

(2)

εn ψ1 (n) εn ψ2 (n) + ), (1 − q1 )2 (1 − q2 )2 ∞

(20.35)

q2k sin kt,

k=n

ε(i) n = sup | k≥n

ψi (k + 1) − qi |, i = 1, 2. ψi (k)

First, let q1 = q2 = q. Then it follows from (20.35) that Ψn (t) = ψ(n)(q

−n

∞ k=n

= ψ(n)(q −n

∞ k=n

qk (

ψ1 (n) ψ2 (n) εn cos kt + sin kt) + O(1) ) ψ(n) ψ(n) (1 − q)2

q k cos(kt −

βn π εn ), ) + O(1) 2 (1 − q)2

(20.36)

Section 20

Approximation on Classes of Analytic Functions (i)

where εn = max εn , ψ(k) =

i=1,2

359

ψ12 (k) + ψ22 (k), and βn are the numbers from

the interval [0, 4) defined by the equalities cos

βn π ψ1 (n) βn π ψ2 (n) = , sin = . 2 ψ(n) 2 ψ(n)

It follows from (20.34) and (20.36) that ¯

ψ ) = sup ρn (f ; x)C En (C∞ ¯

ψ f ∈C∞

1 = sup ϕ∞ ≤1 π ϕ⊥1

π

ψ(n)(q −n

= ψ(n)( sup q ϕ∞ ≤1 ϕ⊥1

q k cos(kt −

k=n

−π

+ O(1)

∞

βn π ) 2

εn )ϕ(x − t)dtC (1 − q)2 −n

1 π

π Pq,βn ,t (t)ϕ(x − t)dtC + O(1) −π

= ψ(n)(q −n En (Cβqn ,∞ ) + O(1)

εn ). (1 − q)2

εn ) (1 − q)2 (20.37)

Taking into account the uniform boundedness of the quantity O(1) in equality (16.3) with respect to the parameter β, we can rewrite this equality in the form En (Cαq n ,∞ ) = q n (

8 q K (q) + O(1) ), 2 π n(1 − q)

(20.38)

where αn , n = 1, 2, . . . , are arbitrary real numbers. By using this equality for αn = βn , we derive equality (20.33) from (20.37). Now let, e.g., q1 < q2 = q. Then, by virtue of (20.32), we get ψ1 (k + 1)ψ2 (k) q1 < 1, = k→∞ ψ2 (k + 1)ψ1 (k) q2 lim

and, consequently, for any ε ∈ (0, 1 − q1 /q2 ), there exist a number n0 such that ϕ(k + 1) ψ1 (k) q1 + ε, ϕ(k) = ≤ ϕ(k) q2 ψ2 (k)

(20.39)

360

Approximation by Fourier Sums in Spaces C and L1

Chapter 5

for all k ≥ n0 . This implies that, for the sequences (1)

αk = |

ψ1 (k) ψ2 (k) (2) |, αk = 1 − | |, ψ(k) ψ(k)

the following equalities are true: q1 q1 (i) αk = O(1)( + ε)k , ε ∈ (0, 1 − ), i = 1, 2, q2 q2

(20.40)

where O(1) is a quantity uniformly bounded in k, q1 , q2 , and ε. It follows from relations (20.35) and (20.40) and obvious inequality 1 1 ≤ , β ∈ R, i = 1, 2, 1 − qi 1−q

qi−n Pqi ,β,n (t) ≤ that Ψn (t) = ψ(n)

∞ ψ1 (n) k−n ψ2 (n) k−n ( sin kt) q1 cos kt + q ψ(n) ψ(n) 2

k=n

(1)

+ O(1)(

(2)

εn ψ1 (n) εn ψ2 (n) + ) (1 − q1 )2 (1 − q2 )2

= ψ(n)(q2−n Pq2 ,1,n (t)signψ2 (n) (2)

(1)

(2)

αn εn + εn εn + ( + 1))) 2 (1 − q) 1−q 1−q εn αn + = ψ(n)(q2−n Pq2 ,1,n (t)signψ2 (n) + O(1)( )), (20.41) (1 − q)2 1 − q + O(1)(

(2)

(i)

where εn = εn and αn = max αn . i=1,2

Combining relations (20.34) and (20.41) and setting q2 = q, we get ¯

ψ En (C∞ ) = sup ρn (f ; x)C ¯

ψ f ∈C∞

= ψ(n)( sup q

−n

ϕ∞ ≤1 ϕ⊥1

1 π

π Pq,1,n (t)ϕ(x − t)dtC −π

εn αn + )) 2 (1 − q) 1−q εn αn q ) + O(1)( + )). = ψ(n)(q −n En (C1,∞ (1 − q)2 1 − q + O(1)(

Section 20

Approximation on Classes of Analytic Functions

361

Hence, taking into account equality (20.24) for β = 1 and the fact that, by virtue of (20.40), αn = o(1/n), we obtain ¯

ψ ) = ψ(n)( En (C∞

= ψ(n)(

8 q αn εn K(q) + O(1)( + + )) 2 2 π n(1 − q) (1 − q) 1−q q εn 8 K(q) + O(1)( )). + 2 π n(1 − q) (1 − q)2

This proves relation (20.33) for q1 < q2 . It is clear that, by using the same reasoning , we can also prove relation (20.33) for q1 > q2 . Following the proof of relation (20.33) and using equality (16.41) instead of (16.3), we establish equality (20.33 ). The theorem is proved. 20.5. As for conditions (20.32) in Theorem 20.4, we note the following fact: ψ1 (k) If q1 = q2 , then, as follows from (20.39), the ratio = ϕ(k) always has a ψ2 (k) limit as k → ∞, which is equal to either 0 (if q1 < q2 ) or ±∞ (if q1 > q2 ). This means that, having determined the sequence of real numbers βk from the interval [0, 4) by using equalities cos βk

π π = ψ1 (k)/ψ(k), sin βk = ψ2 (k)/ψ(k), 2 2 ψ(k) = ψ12 (k) + ψ22 (k),

(20.42)

we can guarantee the existence of the limit of the sequence βk as k → ∞ on this interval. If q1 = q2 = q, then one can indicate sequences ψ1 (k) and ψ2 (k) from Dq such that the sequence βk determined by these sequences on the interval [0, 4) according to (20.42) does not have a limit. For example, this is true for the following sequences: ψ1 (k) = q k , ψ2 (k) = q k ϕ(k), k ∈ N, 0 < q < 1, where

ϕ(k) =

⎧ 3 k ⎪ 2ν 2ν+1 , ⎪ ⎪ − 2ν+1 , 2 ≤ k < 2 ⎨ 2 2 ⎪ ⎪ ⎪ ⎩

k 22(ν+1)

,

22ν+1 ≤ k < 22(ν+1) , ν = 0, 1, 2, . . . .

362

Approximation by Fourier Sums in Spaces C and L1

Chapter 5

It easy to verify that ϕ(k + 1) = 1, k→∞ ϕ(k) lim

and, therefore, ψ1 , ψ2 ∈ Dq . At the same time, we have 12 ≤ ϕ(k) ≤ 1 for any k ∈ N, and ϕ(22ν ) = 1 and ϕ(22ν+1 ) = 1/2 for any ν ∈ N. Consequently, the function ϕ(k) does not have a limit as k → ∞. Hence, the corresponding sequence of numbers βk defined by the formula βk = π2 arctan ϕ(k) also belongs to the interval [ π2 arctan 12 , 12 ] and does not have a limit by virtue of the continuity of the function tan x on this interval. Remark 20.2. Following the proof of Theorem 20.4 and taking into account relations (20.25), (20.25 ), (20.29), and (20.29 ), one can easily verify that the asymptotic equalities (20.33) and (20.33 ) remain true if one replaces condition (20.32) in Theorem 20.4 by one of the following three conditions and sets q = max q1 , q2 : (i) limk→∞

ψ2 (k + 1) ψ1 (k + 1) = q1 , |q1 | < 1, | | ≤ q2 , 0 ≤ q2 < |q1 |; ψ1 (k) ψ2 (k)

(ii) limk→∞

ψ1 (k + 1) ψ2 (k + 1) = q2 , |q2 | < 1, | | ≤ q1 , 0 ≤ q1 < |q2 |; ψ2 (k) ψ1 (k)

(iii) limk→∞

ψi (k + 1) = qi , |qi | < 1, i = 1, 2, |q1 | = |q2 |. ψi (k)

An analog of Theorem 20.2 is also true for the classes determined by the moduli of continuity. Consider the upper bounds for both sides of relation (20.19) over the classes Lψ H and Cβψ¯ Hω . Taking into account that β¯ ωp sup f ∈Lψ ¯ Hωp β

ρn (f ; ·)s = sup{ρn (Jβ¯ψ (ϕ; ·)s : ϕ ∈ Hωp },

and, since, by virtue of Jackson inequalities, in the spaces Lp and C 1 sup En (ϕ)p ≤ Kωp ( ), 1 ≤ p < ∞, n ϕ∈Hωp

(20.43)

1 sup En (ϕ)C ≤ Kω( ), n ϕ∈Hω

(20.43 )

where K is an absolute constant, we arrive at the following statement:

Section 21

Convergence Rate of Group of Deviations

363

Theorem 20.2. Suppose that 1 ≤ s ≤ ∞, 1 ≤ p < ∞, and ψ ∈ Dq , 0 < q < 1, ψ(k) > 0. Then the following relations hold as n → ∞ : H ) = ψ(n)(q −n En (Lqβ¯Hωp )X + O(1) En (Lψ β¯ ωp X

εn ωp (n−1 ) ), (1 − q)2

(20.44)

where X is either Ls or C, and En (Cβψ¯ Hω )C = ψ(n)(q −n En (Cβq¯Hω )C + O(1)

εn ω(n−1 ) ). (1 − q)2

(20.44 )

In relations (20.44) and (20.44 ), the quantities εn and O(1) have the same sense as in Theorem 20.2. Equality (20.44) and Theorem 18.1 yield the following result: Theorem 20.5. Suppose that ψ ∈ Dq , 0 < q < 1, ψ(k) > 0, β ∈ R, and ω(t) is an arbitrary modulus of continuity. Then the following relation holds as n→∞: 4 ω(1/n)(εn + 1/n) En (Cβψ Hω ) = ψ(n)( 2 en (ω)K(q) + O(1) ), π (1 − q)2 where

π/2 2t ω( ) sin tdt, en (ω) = θω n 0

θω ∈ [1/2, 1], θω = 1 if ω(t) is a convex modulus of continuity, and O(1) is a quantity uniformly bounded in n, q, β, and ψ(k).

21.

Convergence Rate of Group of Deviations

21.1. Here, we continue the investigation of the quantities ρn (f ; x) = f (x) − Sn−1 (f ; x), n = 1, 2, . . . , ¯ over the sets of ψ-integrals. However, in this case, as a measure of the rate of convergence of the series S[f ] to f (·), we consider the quantities Rn(p) (f ; x) = (

1 γn

n+γ n −1

|ρn (f ; x)|p )1/p , p > 0,

(21.1)

k=n

where γn is a certain sequence of natural numbers. The main result of this section is formulated in the following statements:

364

Approximation by Fourier Sums in Spaces C and L1

Chapter 5

Theorem 21.1. Let ±ψ1 ∈ M0 , ±ψ2 ∈ M0 and let, for any p > 0, Rn(p) (f ; x) = (

2n−1 1 |ρk (f ; x)|p )1/p . n

(21.2)

k=n

¯

Then, for any function f (·) from the set C ψ C 0 at any point x, the following inequality holds: Rn(p) (f ; x)

∞ |ψ2 (t)| ψ¯ ¯ ≤ Cp ( dt + ψ(n))E n (f ), t

(21.3)

n

¯ = (ψ12 (n) + ψ22 (n))1/2 , En (ϕ) is the quantity of the best approxwhere ψ(n) imation of the function ϕ(·) in the uniform metric by trigonometric polynomials tn−1 (·) of degree ≤ n − 1, En (ϕ) = inf ϕ(x) − tn−1 (x)C , tn−1

¯

and Cp is a quantity uniformly bounded in n ∈ N and f ∈ C ψ C 0 . Theorem 21.2. Let ±ψ1 ∈ F and ±ψ2 ∈ F and let, for any p > 0, Rn(p) (f ; x) = (

[η(n)] 1 |ρk (f ; x)|p )1/p , η(n) = η(ψ1 ; n), γn

(21.4)

k=n

where γn = [η(ψ1 ; n)] − n + 1 and [α] is the integer part of α. If there exist constants K1 and K2 such that 0 < K1 ≤

η(ψ1 ; t) − t ≤ K2 < ∞ ∀t ≥ 1, η(ψ2 ; t) − t

(21.5)

¯

then, for any function f ∈ C ψ C 0 at any point x, ψ¯ ¯ Rn(p) (f ; x) ≤ Cp ψ(n)E n (f ).

(21.6)

If the condition |ψ2 (n)| ln+ (η(ψ2 ; n) − n) ≤ O(1)|ψ1 (n)|, n = 1, 2, . . . ,

(21.5 )

¯

is satisfied, then, for any function f ∈ C ψ C at any point x, ¯

Rn(p) (f ; x) ≤ Cp |ψ1 (n)|En (f ψ ).

(21.6 )

¯ The quantities ψ(n), En (ϕ), and Cp have the same sense as in Theorem 21.1.

Section 21

Convergence Rate of Group of Deviations

365

Proof of Theorem 21.1. We begin with Lemma 4.1, which yields the following statement: Lemma 21.1. Let ±ψ1 ∈ M0 , ±ψ2 ∈ M0 and let a be an arbitrary pos¯ itive number. If f ∈ C ψ C 0 , then, for any n ∈ N and for any trigonometric polynomial tn−1 (·) of degree n − 1, at every point x, the following equality holds: ρn (f ; x) =

Δn (x; t)J2 (ψ2 ; n; t)1 dt |t|≤a/n

+

¯ ψ(n) π

Δn (x; t)

sin(nt + θn ) dt t

a/n≤|t|≤π/2

¯ + O(1)ψ(n)Δ n (x; t)C ,

(21.7)

¯

where Δn (x; t) = f ψ (x − t) − tn−1 (x − t), 1 J2 (ψ2 ; n; t)1 = π

∞ ψ2 (v) sin vtdv, n

ψ1 (n) ψ2 (n) cos θn = ¯ , sin θn = ¯ ψ(n) ψ(n)

(21.8)

and O(1) is a quantity uniformly bounded in n. First, assume that p ≥ 2. Taking into account that, for arbitrary numbers a and b, |a + b|p ≤ 2p (|a|p + |b|p ) if p ≥ 1,

(21.9)

|a + b|p ≤ |a|p + |b|p if 0 ≤ p < 1,

(21.9 )

and

and using relations (21.2) and (21.7), we get

Approximation by Fourier Sums in Spaces C and L1

366

Rn(p) (f ; x)

2n−1 1 ≤ 4( | n

Chapter 5

Δn (x; t)J2 (ψ2 ; n; t)1 dt|p )1/p

k=n |t|≤a/k

2n−1 ¯ 1 ψ(k) + 4( | n π k=n

Δn (x; t)

sin(kt − θk ) p 1/p dt| ) t

a/k≤|t|≤π/2

2n−1 1 ¯p ψ (k)Δn (u)pC )1/p + O(1)( n k=n

df

= 4(σ1 + σ2 + O(1)σ3 ).

(21.10)

In what follows, as a polynomial tn−1 (·), we choose a polynomial t∗n−1 (·) that ¯ realizes the best approximation of the function f ψ : ¯

¯

f ψ (·) − t∗n−1 (·)C = En (f ψ ).

(21.11)

In view of equalities (5.4) and (21.11), we get σ1 = (

2n−1 1 | n

Δn (x; t)J2 (ψ2 ; k; t)dt|p )1/p

k=n |t|≤a/k

∞ 2n−1 2n−2 |ψ2 (t)| p 1/p 1 p ψ¯ 2 1 p ψ¯ ¯p ≤ 2( En (f )( En (f )ψ (k))1/p dt) ) + O(1)( n π t n k=n

2 ≤ 2En (f )( π ψ¯

k=n

k

∞

|ψ2 (t)| ¯ dt + O(1)ψ(n)). t

(21.12)

n

It is clear that σ3 ≤ O(1)(

2n−1 1 ¯p ¯ ψ¯ ¯ ψ (k)Enp (f ψ ))1/p ≤ O(1)ψ(n)E n (f ). n

(21.13)

k=n

It remains to verify that ¯

ψ ¯ σ2 ≤ Cp ψ(n)E n (f ).

We have

(21.14)

Section 21

Convergence Rate of Group of Deviations

2n−1 ¯ 1 ψ(k) | σ2 = ( n π k=n

≤

367

Δn (x; t) sin(kt + θk )dt|p )1/p t

a/k≤|t|≤π/2

2n−1 2 1 ¯ |ψ(k) ( π n k=n

Δn (x; t) sin(kt + θk )dt|p )1/p t

a/2n≤|t|≤π/2

2n−1 ¯ 2ψ(n) 1 + | ( π n

k=n a/2n≤|t|≤a/n

dt p 1/p ¯ | ) En (f ψ ). t

By virtue of (21.8), this yields 2n−1 4 1 σ2 ≤ [( |ψ1 (k)|p | π n k=n

Δn (x; t) sin ktdt|p )1/p t

a/2n≤|t|≤π/2

2n−1 1 +( |ψ2 (k)|p | n k=n

+

Δn (x; t) cos ktdt|p )1/p ] t

a/2n≤|t|≤π/2

4 ln 2 ¯ ¯ ψ(n)En (f ψ ). π

(21.15)

¯ Since |ψi (k)| ≤ ψ(n), i = 1, 2 for k ∈ [n, 2n − 1], relation (21.15) yields 1 ¯ σ2 ≤ Cp ψ(n)[( n 2n−1 1 +( | n

2n−1

|

Δn (x; t)t sin ktdt|p )1/p

k=n a/2n≤|t|≤π/2

k=n a/2n≤t≤π/2

Δn (x; t) ¯ cos ktdt|p )1/p + O(1)En (f ψ )]. (21.16) t

For all fixed x and n, let ⎧ a π Δn (x; t) ⎪ ⎪ , ≤ |t| ≤ , ⎨ t 2n 2 ϕn (t) = ⎪ ⎪ π a ⎩0, ] ∪ [ ≤ |t| ≤ π]. [|t| ≤ 2n 2 Let αk = αk (ϕn ) and βk = βk (ϕn ) denote the Fourier coefficients of this function.

Approximation by Fourier Sums in Spaces C and L1

368

Chapter 5

Then, by virtue of (21.16), we have 1 ¯ σ2 ≤ Cp ψ(n)[( n

2n−1

|αk | )

p 1/p

k=n

2n−1 1 ¯ +( |βk |p )1/p + O(1)En (f ψ )]. (21.17) n k=n

We now use the well-known Hausdorff–Young statement: If a function f belongs to Lr , 1 < r ≤ 2, and 1 ck = ck (f ) = 2π

π

f (t)e−int dt, n = 0, ±1, ±2, . . . ,

−π

are its Fourier coefficients, then (

∞

|ck |r )1/r ≤ (2π)−1 f r , 1/r + 1/r = 1.

(21.18)

k=−∞

Since α0 = c0 (ϕn ), αk = ck (ϕn ) + c−k (ϕn ), iβk = c−k (ϕn ) − ck (ϕn ), 2 and, in the case considered, p ≥ 2, by virtue of (21.18) we have ∞ ∞ ( |αk |p )1/p ≤ 2( |ck |p )1/p ≤ π −1 ϕn p = π( |ϕn (t)|p dt)1/p π

k=0

k=−∞

21/p ≤ ( π

−π

π/2

Δn (x; t)pC 21/p ¯ 1/p dt) = En (f ψ )( p π t

a/2n ¯

π/2

dt 1/p ) tp

a/2n

¯

≤ Cp En (f ψ )n(p −1)/p ≤ Cp En (f ψ )n1/p .

(21.19)

By analogy, we get (

∞

¯

|βk |p )1/p ≤ Cp En (f ψ )n1/p .

k=ν

Substituting estimates (21.19) and (21.20) in (21.17), we obtain (21.14).

(21.20)

Section 21

Convergence Rate of Group of Deviations

369

Combining relations (21.12)–(21.14), we obtain estimate (21.3) for all p ≥ 2. If 0 < p < 2, then (21.3) follows from what has been proved. Indeed, for any ak ≥ 0, p, and s such that 0 < p ≤ s, the Hölder inequality yields n+γ n+γ n p n 1 1 1/p ak ) ≤ ( ask )1/s , ( γn + 1 γn + 1 k=n

(21.21)

k=n

where γn is an arbitrary natural number. Theorem 21.1 is proved. Proof of Theorem 21.2. As a starting point in the proof of estimate (21.6), we use Lemma 12.4, which yields the following statement: Lemma 21.2. Suppose that ±ψ1 ∈ F, ±ψ2 ∈ F, and condition (21.5) is ¯ satisfied. If f ∈ C ψ C 0 , then, for any n ∈ N and for any trigonometric polynomial tn−1 (·) of degree n − 1, at every point x, the following equality holds: ¯ ψ(n) ρn (f ; x) = − π

Δn (x; t)

sin(nt + θn ) dt t

an ≤|t|≤π/2

¯ + O(1)ψ(n)Δ n (x; t)C ,

(21.22)

¯ where the quantities ψ(n), Δn (x; t), θn , and O(1) have the same sense as in Lemma 12.1 and an = (η(ψ1 ; n) − n)−1 or an = (η(ψ2 ; n) − n)−1 , n = 1, 2, . . . . If η(ψ1 ; n) < n + 1, then the sum in (21.4) contains the unique term corresponding to the value k = n. In this case, (21.6) follows from Theorem 10.3, according to which ρn (t; x)C ≤ (

4 + ψ¯ ¯ ln (η(ψ1 ; n) − n) + O(1))ψ(n)E n (f ). 2 π

(21.23)

Thus, further, it suffices to assume that η(ψ1 ; n) ≥ n + 1. Moreover, if all numbers ak are less than π/2, then relation (21.6) is proved because, by virtue of Lemma 4.5, for any b > 0, Δ(x; t) |t|>b

sin(nt + θn ) ¯ dt = O(1)En (f ψ ). t

(21.24)

Approximation by Fourier Sums in Spaces C and L1

370

Chapter 5

Taking this into account, following the proof of Theorem 21.1, and setting p ≥ 2, we obtain an analog of relation (21.10). In view of estimate (21.13), it can be written in the following way: Rn(p) (f ; x)

[η(n)] ¯ 1 ψ(k) ≤ 2( | γn π k=n

Δn (x; t)

sin(kt + θk ) p 1/p dt| ) t

ak ≤|t|≤π/2

ψ¯ ¯ + O(1)ψ(n)E n (f ).

(21.25)

To prove (21.6) it suffices to verify that σ2 = (

[η(n)] ¯ 1 ψ(k) | γn π k=n

Δn (x; t)

sin(kt + θk ) p 1/p dt| ) t

ak ≤|t|≤π/2

¯

ψ ¯ ≤ Cp ψ(n)E n (f ).

(21.26)

Let ¯n = max(ak : ak < π/2, k ∈ [n, n + γn ]). ak = a Then σ2

[η(n)] 1 ¯ ≤ Cp ( |ψ(k) γn k=n

Δ(x; t)

sin(kt + θk ) p 1/p dt| ) t

a ¯n ≤|t|≤π/2 [η(n)] 1 ψ¯ ¯ + Cp ψ(n)En (f )( ( γn

k=n a ≤|t|≤¯ an k

dt p 1/p ) ) . |t|

However, [η(n)] 1 ( γn

k=n a ≤|t|≤¯ an k

dt p 1/p a ¯n η(k) − k ≤ 2 max ln ) ) ≤ 2 max ln . |t| ak η(n) − n n≤k≤η(n) n≤k≤η(n)

If ψ ∈ F, then, by virtue of Theorem 3.13.2, we have η(τ ) − τ K2 ψ (η(t)) ψ(τ ) K2 ∀t ≥ 1 ∀τ ∈ [t, η(t)]. ≤ · ≤ η(t) − t ψ(t) K1 ψ (τ ) K1

(21.27)

Section 21

Convergence Rate of Group of Deviations

Consequently,

[η(n)] 1 ( ( γn

k=n a ≤|t|≤¯ an k

371

dt p 1/p ) ) ≤ Cp . |t|

Therefore, in view of (21.8), ¯ σ2 ≤ Cp ψ(n)((

[η(n)] 1 | γn

k=n a ¯n ≤|t|≤π/2

+(

[η(n)] 1 | γn

k=n a ¯n ≤|t|≤π/2

Δ(x; t) sin ktdt|p )1/p t

Δ(x; t) ψ¯ ¯ cos ktdt|p )1/p ) + Cp ψ(n)E n (f ). (21.28) t

Further, as in the proof of Theorem 21.1, let ⎧ t) ⎪ ⎨ Δ(x; , a ¯n ≤ |t| ≤ π/2, t ϕn (t) = ⎪ ⎩ an ≤ |t| ≤ π] 0, [|t| ≤ a ¯n ] ∪ [¯ and denote by αk and βk the Fourier coefficients of the function ϕn (t). Then (21.28) can be rewritten in the following way: σ2

[η(n)] [η(n)] 1 1 p 1/p ¯ ≤ Cp ψ(n)(( |αk | ) + ( |βk |p )1/p ) γn γn k=n

k=n

¯

ψ ¯ + Cp ψ(n)E n (f ).

(21.29)

By using estimate (21.18), we get ∞ dt 1/p ¯ ( |αk |p )1/p ≤ Cp En (f ψ )( ) ≤ Cp En (f ψ )(η(ψ1 ; k ) − k )1/p . tp π/2

k=0

a ¯n

However, if ψ1 ∈ F, then, by virtue of (21.27), 0 < K1 ≤

η(k) − k ≤ K2 < ∞ η(k ) − k

∀k, k ∈ [n, η(n)], η(n) = η(ψ1 ; n).

(21.30)

Approximation by Fourier Sums in Spaces C and L1

372 Hence,

Chapter 5

∞ ¯ |αk |p )1/p ≤ Cp En (f ψ )(η(n) − n)1/p . ( k=0

By analogy, we conclude that ∞ ¯ |βk |p )1/p ≤ Cp En (f ψ )(η(n) − n)1/p . ( k=0

Substituting these estimates in (21.29), we obtain (21.26) and, hence, (21.6). If 0 < p < 2, then (21.6) follows from what has been proved, in view of estimate (21.21). 21.2. Remark 21.1. Since Lemma 21.2 is also true in the case where an = (η(ψ2 ) − n), if condition (21.5) is satisfied, then estimate (21.6) holds for the quantity [η(n)] 1 (p) ¯ Rn (f ; x) = ( |ρk (f ; x)|p )1/p , (21.31) γn k=n

where η(n) = η(ψ2 ; n) and γn = [n(ψ2 ; n)] − n + 1, i.e., ψ¯ ¯ ¯ n(p) (f ; x) ≤ Cp ψ(n)E R n (f ).

(21.32)

Remark 21.2. Direct verification shows that if ±ψ1 ∈ F and ±ψ2 ∈ F, ¯ = (ψ 2 (t)+ψ 2 (t))1/2 belongs to the set M. Therefore, the then the function ψ(t) 1 2 ¯ t) and μ(ψ; ¯ t) are meaningful. Below, we show (see Proposition functions η(ψ; 24.1) that, in this case, ψ¯ ∈ F. Now let us prove following statement: Proposition 21.1. Let ±ψ1 ∈ M and ±ψ2 ∈ M. If, at a given point t, t ≥ ¯ t) satisfies the 1, inequality η(ψ1 ; t) < η(ψ2 ; t) holds, then the quantity η(ψ; following condition: ¯ t) ≤ η(ψ2 ; t). η(ψ1 ; t) ≤ η(ψ; (21.33) Indeed, since ψ22 (η(ψ1 ; t)) ≥ 14 ψ22 (t), we have ¯ ψ(η(ψ ; t)) = ψ12 (η(ψ1 ; t)) + ψ22 (η(ψ1 ; t)) 1 # 1¯ 1 2 = ψ1 (t) + ψ22 (η(ψ1 ; t)) > ψ(t). 4 2

Section 21

Convergence Rate of Group of Deviations

373

¯ t). By analogy, since ψ 2 (η(ψ2 ; t)) < 1 ψ 2 (t), we get Thus, η(ψ1 ; t) ≤ η(ψ; 1 4 1 1 ¯ ¯ ¯ ψ(η(ψ 2 ; t)) < 2 ψ(t) and, consequently, η(ψ; t) ≤ η(ψ2 ; t). Proposition 21.1 yields the following statement: Corollary 21.1. Let ±ψ1 ∈ F, ±ψ2 ∈ F, and let condition (21.5) be satisfied. Then there exist constants K1 and K2 such that 0 < K1 ≤

¯ t) − t ¯ t) − t η(ψ; η(ψ; ≤ K2 , 0 < K1 ≤ ≤ K2 ∀t ≥ 1. (21.34) η(ψ1 ; t) − t η(ψ2 ; t) − t

¯ n) − It follows from (21.34) that equality (21.22) also holds if an = (η(ψ; Therefore, we conclude that the following proposition is true:

n)−1 .

Proposition 21.2. Let ±ψ1 ∈ F and ±ψ2 ∈ F and let, for any p > 0, Rn(p) (f ; x)

[η(n)] 1 =( |ρk (f ; x)|p )1/p , γn

(21.35)

k=n

¯ v) and γn = [η(ψ; ¯ n)]−n+1. If condition (21.5) is satisfied, where η(v) = η(ψ; ¯ 0 ψ then, for any f ∈ C C , at every point x, ¯

(21.32 )

ψ ¯ Rn(p) (f ; x) ≤ Cp ψ(n)E n (f ), ¯

where Cp is quantity uniformly bounded in n ∈ N and f ∈ C ψ C 0 . 21.3. Let us continue the proof Theorem 21.2. Now let us verify that if condition (21.5 ) is satisfied, then (21.6 ) holds. By virtue of Corollary 11.1, we obtain the following statement: ¯

Lemma 21.3. Let ±ψ1 ∈ F and ±ψ2 ∈ F. If f ∈ C ψ C 0 , then, for any n ∈ N and for any trigonometric polynomial tn−1 (·) of degree n − 1, at every point x, the following equality holds: ψ1 (n) sin nt Δn (x; t) ρn (f ; x) = − dt π t an ≤|t|≤π/2

+

ψ2 (n) π

Δn (x; t)

an ≤|t|≤π/2

¯ + O(1)ψ(n)Δ n (x; t)C ,

cos nt dt t (21.36)

¯ where the quantities ψ(n), Δn (x; t), an , and O(1) have the same sense as in Lemmas 21.1 and 21.2, and an = (η(ψ2 ; n) − n)−1 .

Approximation by Fourier Sums in Spaces C and L1

374

Chapter 5

¯

Since, for every f ∈ C ψ C 0 , ψ2 (n) cos nt | Δn (x; t) dt| π t an ≤|t|≤π/2

|ψ2 (n)| ≤2 Δn (x; t)C π

π/2 an

dt t

≤ O(1)|ψ2 (n)|Δn (x; t)C ln+ (η(ψ2 ; n) − n)

(21.37)

¯

and Δn (x; t)C ≤ En (f ψ ), by virtue of (21.36) and (21.37) and in view of (21.5 ), we find |ψ1 (n)| sin nt ¯ ρn (f ; x) = − Δn (x; t) dt + O(1)|ψ1 (n)|En (f ψ ). π t an ≤|t|≤π/2

¯ This equality coincides with equality (21.22) if we set ψ(n) = |ψ1 (n)| and θn = 0 in the latter. Therefore, following the proof of estimate (21.6), we arrive at inequality (21.6 ).

22.

Corollaries of Theorems 21.1 and 21.2. Orders of Best Approximations

22.1. The expressions n (f ; x) = Vn−p

n 1 Sk (f ; x) p+1

(22.1)

k=n−p

are called de la Vallée Poussin sums of the function f (·) (see Section 1.1). They are trigonometric polynomials of degree n − 1. In this case, 2n−1 1 Sk (f ; x) = Vn2n−1 (f ; x) n k=n

and [η(n)] 1 Sk (f ; x) = Vn[η(n)] (f ; x), γn k=n

Section 22

Orders of Best Approximations

375

¯ n)), η(n) = η(ψ1 ; n) (or η(n) = η(ψ2 ; n), or η(n) = η(ψ; γn = [η(n)] − n + 1. Since Vn2n−1 (1; x) ≡ 1, inequality (21.6), for p = 1, yields |f (x) −

Vn2n−1 (f ; x)|

2n−1 1 =| ρn (f ; x)| ≤ Rn(1) (f ; x) n k=n ∞

≤ O(1)(

|ψ2 (t)| ψ¯ ¯ dt + ψ(n))E n (f ) t

(22.2)

n ¯

∀f ∈ C ψ C 0 , ±ψ1 ∈ M0 , ±ψ2 ∈ M0 . By analogy, in view of (21.6) and (21.6 ), we conclude that if ±ψ1 ∈ F, ±ψ2 ∈ ¯ F, and condition (21.5) is satisfied, then, for any f ∈ C ψ C 0 at any point x, ψ¯ ¯ ¯ |f (x) − Vn[η(n)] (f ; x)| ≤ O(1)ψ(n)E n (f ), η(n) = η(ψ; n).

(22.3)

If ±ψ1 ∈ F, ±ψ2 ∈ F, and condition (21.5 ) is satisfied, then, for any f ∈ ¯ C ψ C 0 at any point x, ¯

|f (x) − Vn[η1 (n)] (f ; x)| ≤ O(1)|ψ1 (n)|En (f ψ ), η1 (n) = η(ψ1 ; n).

(22.3 )

If ±ψ2 ∈ M0 , then, for any n ∈ N, ∞

|ψ2 (t)| dt = t

n

∞

|ψ2 (t)| dt + rn (ψ2 ), t

(22.4)

|ψ2 (t)| dt ≤ |ψ2 (n)| ln 2. t

(22.5)

2n

where

2n rn (ψ2 ) = n

Moreover, if ψ ∈ M0 , then, according to Theorem 3.16.1, for any q > 0 and m ∈ N, there exists a constant Cp depending only on q and such that ψ(m) ≤ Cq ψ(mq).

(22.6)

Therefore, by virtue of (22.2) and (22.4)–(22.6), if ±ψ1 ∈ M0 and ±ψ2 ∈ M0 , ¯ then, for any function f ∈ C ψ C at any point x, ∞ |f (x) −

Vn2n−1 (f ; x)|

≤ O(1)( 2n−1

|ψ2 (t)| ¯ ¯ dt + ψ(2n − 1))En (f ψ ). t

(22.7)

376

Approximation by Fourier Sums in Spaces C and L1 ¯

¯

Chapter 5

¯

ψ Hence, setting C∞ = {f : f ∈ C ψ C, f ψ C ≤ 1}, we arrive at the following statement: ¯

ψ Theorem 22.1. Let ±ψ1 ∈ M0 , ±ψ2 ∈ M0 , and f ∈ C∞ . Then, for any n ∈ N, there exists a trigonometric polynomial tn−1 (·) of the form (22.1) of degree ≤ n − 1 such that

∞ |ψ2 (t)| ¯ En (f ) ≤ f (x) − tn−1 (x)C ≤ O(1)( dt + ψ(n)), t

(22.8)

n ¯

ψ . where O(1) is a quantity uniformly bounded in n ∈ N and f ∈ C∞ ¯

ψ Let ±ψ1 ∈ F, ±ψ2 ∈ F, and f ∈ C∞ . If condition (21.5) is satisfied, then, by virtue of (22.3), we have

¯ f (x) − Vn[η(n)] (f ; x)C ≤ O(1)ψ(n);

(22.9)

if condition (21.5 ) is satisfied, then, by virtue of (22.3 ), we have (22.9 )

f (x) − Vn[η1 (n)] (f ; x)C ≤ O(1)|ψ1 (n)|.

¯ Since, ψ(η(ψ; t)) = 12 ψ(t) for any ψ ∈ M and t ≥ 1, one can replace ψ(n) ¯ ¯ n)]) and |ψ1 ([η(ψ1 ; n)])| on the right-hand sides in and |ψ1 (n)| by ψ([η( ψ; (22.9) and (22.9 ), respectively. Thus, in view of inequalities (22.9) and (22.9 ), we obtain the following statement: ¯

ψ . Then, for any Theorem 22.2. Let ±ψ1 ∈ F, ±ψ2 ∈ F, and f ∈ C∞ n ∈ N, there exists a trigonometric polynomial tn−1 (·) of the form (22.1) of degree ≤ n − 1 such that if condition (21.5) is satisfied, then

¯ En (f ) ≤ f (x) − tn−1 (x)C ≤ O(1)ψ(n),

(22.10)

and, if condition (21.5 ) is satisfied, then En (f ) ≤ f (x) − tn−1 (x)C ≤ O(1)|ψ1 (n)|.

(22.10 )

In equalities (22.10) and (22.10 ), O(1) are quantities uniformly bounded in n ∈ ψ¯ N and f ∈ C∞ .

Section 22

Orders of Best Approximations

377

Let Hω0 be the class of 2π-periodic continuous functions f (t) satisfying the condition π |f (t) − f (t )| ≤ ω(|t − t |), f (t)dt = 0, (22.11) −π

and ¯

¯

¯

C ψ Hω0 = {f : f ∈ C ψ , f ψ ∈ Hω0 }. It is well known (see Section 6.2) that 1 En (Hω0 ) ≤ O(1)ω( ). n

(22.12)

Therefore, by virtue of relations (22.3), (22.3 ), and (22.7), we obtain the follow¯ ing analog of Theorems 22.1 and 22.2 for the classes C ψ Hω0 : ¯

Theorem 22.3. Let ±ψ1 ∈ M0 , ±ψ2 ∈ M0 , and f ∈ C ψ Hω0 . Then, for any n ∈ N, there exists a trigonometric polynomial tn−1 (·) of the form (22.1) of degree ≤ n − 1 such that ∞ |ψ2 (t)| ¯ En (f ) ≤ f (x) − tn−1 (x)C ≤ O(1)( dt + ψ(n))ω(1/n). (22.13) t n ¯

Let ±ψ1 ∈ F, ±ψ2 ∈ F, and f ∈ C ψ Hω0 . Then, for any n ∈ N, there exists a trigonometric polynomial tn−1 (·) of the form (22.1) of degree ≤ n − 1 such that if condition (21.5) is satisfied, then ¯ En (f ) ≤ f (x) − tn−1 (x)C ≤ O(1)ψ(n)ω(1/n),

(22.14)

and, if condition (21.5 ) is satisfied, then En (f ) ≤ f (x) − tn−1 (x)C ≤ O(1)|ψ1 (n)|ω(1/n).

(22.14 )

In equalities (22.13)–(22.14 ), O(1) are quantities uniformly bounded in n and ¯ f ∈ C ψ Hω0 . Note that it will be shown in Chapter 7 that the orders of approximation on ¯ ψ¯ and C ψ Hω0 established in Theorems 22.1–22.3 are exact. the classes C∞

Approximation by Fourier Sums in Spaces C and L1

378

23.

Chapter 5

Analogs of Theorems 21.1 and 21.2 and Best Approximations in Integral Metric

23.1. Theorem 23.1. Let ±ψ1 ∈ M0 , ±ψ2 ∈ M0 , and Rn (f ; x) =

2n−1 1 ρk (f ; x). n

(23.1)

k=n

¯

Then, for any f ∈ Lψ , ∞ |ψ2 (t)| ψ¯ ¯ Rn (f ; x)1 ≤ O(1)( dt + ψ(n))E n (f )1 , t

(23.2)

n

where

π ¯ |ϕ(t)|dt, ψ(n) = (ψ12 (n) + ψ22 (n))1/2 ,

ϕ1 = −π

En (ϕ)1 is the quantity of the best approximation of the function ϕ(·) by a trigonometric polynomial tn−1 (·) of degree ≤ n − 1 in the space L1 , En (ϕ)1 = inf ϕ(·) − tn−1 (·)1 ,

(23.3)

tn−1

¯

and O(1) is a quantity uniformly bounded in n and f ∈ Lψ . Proof. Lemma 8.1 yields the following assertion: Lemma 23.1. Let ±ψ1 ∈ M0 , ±ψ2 ∈ M0 , and let a be an arbitrary posi¯ tive number. If f ∈ Lψ , then, for any n ∈ N, equality (21.7) with Δn (x; t)C replaced by Δn (x; t)1 holds almost everywhere. In view of this lemma, we obtain 2n−1 1 λk ρk (f ; x)1 Rn (f ; x)1 = n k=n

≤

2n−1 1 n

k=n |t|≤a/k

Δn (x; t)J2 (ψ2 ; n; t)1 dt1

Section 23

Analogs of Theorems 21.1 and 21.2

2n−1 ¯ 1 ψ(k) + n π k=n

+ O(1)

Δn (x; t)

379

sin(kt + θk ) dt1 t

a/k≤|t|≤π/2

2n−1 1 ¯ ψ(k)Δn (x; t)1 n k=n

= σ1 + σ2 + O(1)σ3 . df

(23.4)

By analogy with the proof of estimates (21.12) and (21.13), we get σ1

2 ≤ 2En (f )1 ( π ψ¯

∞

|ψ2 (t)| ¯ dt + O(1)ψ(n)) t

(23.5)

n

and Let us show that

ψ¯ ¯ σ3 ≤ O(1)ψ(n)E n (f )1 .

(23.6)

ψ¯ ¯ σ2 ≤ O(1)ψ(n)E n (f )1 .

(23.7)

We have σ2 ≤

2n−1 ¯ 1 ψ(n) n π k=n

Δn (x; t) ψ¯ ¯ sin(kt+θk )dt1 +O(1)ψ(n)E n (f )1 . t

a/2n≤|t|≤π/2

This, by virtue of equalities (21.8), yields σ2

1 ≤ π

2n Δn (x; t) (ψ1 (k) sin kt − ψ2 (k) cos kt)dt1 t

a/2n≤|t|≤π/2

k=n

ψ¯ ¯ + O(1)ψ(n)E n (f )1 .

(23.8)

Performing the Abel transformation, in view of the monotonicity of functions ψ1 and ψ2 , we conclude that |

∞

ψ1 (k) sin kt| ≤ O(1)ψ1 (n)|t|−1 ,

k=n

|

∞ k=n

(23.9) ψ2 (k) cos kt| ≤ O(1)ψ2 (n)|t|−1 .

380

Approximation by Fourier Sums in Spaces C and L1

Therefore, σ2

1 ¯ ≤ O(1)En (f )1 ψ(n)( n ψ¯

Chapter 5

dt + 1), t2

a/2n≤|t|≤π/2

which, immediately, yields (23.7). By combining relations (23.4)–(23.7), we obtain (23.2). Theorem 23.1. Let ±ψ1 ∈ F, ±ψ2 ∈ F and let λk , k ∈ N, be a sequence ¯ such that λk ≥ 0 and the numbers βk = λk ψ(k) do not increase. Further, let [η(n)] 1 λk ρk (f ; x), Rn (f ; x; λ) = γn

(23.10)

k=n

¯ t), γn = [η(n)] − n + 1, and [α] is the integer part of α. where η(t) = η(ψ; ¯ If condition (21.5) is satisfied, then, for any f ∈ Lψ , ψ¯ ¯ Rn (f ; x; λ)1 ≤ Cλ ψ(n)λ n En (f )1 ,

(23.11)

where Cλ is a quantity uniformly bounded in n ∈ N and in f ∈

¯ Lψ .

Proof. Lemma 12.6 yields the following statement: Lemma 23.2. Let ±ψ1 ∈ F, ±ψ2 ∈ F, and let condition (21.5) be satisfied. ¯ Then, for any f ∈ Lψ and n ∈ N, equality (21.22) with Δn (x; t)C replaced by Δn (x; t)1 holds almost everywhere. We assume, without loss of generality, that η(n) ≥ n + 1 because, otherwise, estimate (23.11) follows from Theorem 13.3, according to which, in the case considered, we have 4 + ψ¯ ¯ ln (η(n) − n) + O(1))ψ(n)E n (f )1 . π2 In view of Lemma 23.2, we get ρn (f ; x)1 ≤ (

[η(n)] 1 Rn (f ; x; λ)1 = λk ρk (f ; x)1 γn k=n

≤

[η(n)] 1 ¯ λk ψ(k) πγn k=n

Δn (x; t)

sin(kt + θk ) dt1 t

ak ≤|t|≤π/2

ψ¯ ¯ + O(1)ψ(n)λ n En (f )1 .

(23.12)

Section 23

Analogs of Theorems 21.1 and 21.2

381

Setting, as in the proof of Theorem 21.2, a ¯n = max(ak : ak < π/2, k ∈ [n, [η(n)]) and using estimate (21.27), we obtain Rn (f ; x; λ)1 ≤ O(1)( a ¯n ≤|t|≤π/2

[η(n)] Δn (x, t) 1 ¯ sin(kt + θk )dt1 λk ψ(k) t γn k=n

¯

ψ ¯ +O(1)ψ(n)λ n En (f )1 ).

(23.13)

¯ According to the monotonicity of the numbers βk = λk ψ(k) and by virtue of relations (23.9), this yields dt ¯ −1 ¯ Rn (f ; x; λ)1 ≤ O(1)En (f ψ )1 λn ψ(n)(γ + 1) (23.14) n t2 a ¯n ≤t≤π/2

To obtain (23.11), it suffices to prove that ¯n )−1 ≤ O(1). (γn a

(23.15)

It follows from (21.27) that if ψ ∈ F, then 0 < K1
1, then (24.8) holds almost everywhere for any p > 0. Later, Marcinkiewicz and Zygmund proved that (24.8) holds almost everywhere for any f ∈ L and p > 0. These results laid the foundation of the theory of strong summability, which now occupies an important place in the theory of Fourier series. Later, the statement of the problem has been extended, for example, in the direction of using, instead of the expressions 1 |f (x) − Sk (f ; x)|p , n n−1

(24.10)

k=0

series of the form

∞

λk |f (x) − Sk (f ; x)|p ,

(24.11)

k=1

where λk are arbitrary numbers which may depend on some parameter. In this section, the presentation of these interesting investigations is restricted ¯ to studying the convergence rate of series (24.11) over the sets of ψ-integrals in the uniform and the integral metric. 24.2. Assume that p is an arbitrary positive number, p > 0, and λk , k ∈ N, is an arbitrary number sequence. Then, for every f ∈ L, we set H

(p)

(f ; x; λ) =

∞ k=1

λk |ρk (f ; x)|p , ρk (f ; x) = f (x) − Sn−1 (f ; x),

(24.12)

386

Approximation by Fourier Sums in Spaces C and L1

and Hn(p) (f ; x; λ)

=

∞

Chapter 5

λk |ρk (f ; x)|p , n ∈ N.

(24.13)

k=n

In this notation, the following assertion is true: Theorem 24.1. Let ±ψ1 ∈ M0 , ±ψ2 ∈ M0 , and p > 0. Suppose that a sequence λk , k ∈ N, is such that λk ≥ 0 and the numbers βk = λk αkp , where ∞ αk =

|ψ2 (t)| ¯ dt + ψ(k), t

(24.14)

k ¯

do not increase for all k ≥ n. Then, for every f ∈ C ψ C 0 at any point x ¯

Hn(p) (f ; x; λ) ≤ Cp (nλn αnp Enp (f ψ ) +

∞

¯

λk αkp Ekp (f ψ )),

(24.15)

k=n

n = 1, 2, . . . , ¯

where Cp is a quantity uniformly bounded in n ∈ N and f ∈ C ψ C 0 . Proof. Let n0 = n, n1 = 2n0 , . . . , ni = 2ni−1 . Then, in view of inequality (21.3), we have Hn(p) (f ; x; λ) =

∞

λk |ρk (f ; x)|p

k=n

=

∞ 2n i −1

λk |ρk (f ; x)| ≤ p

i=0 k=ni

≤ Cp

∞

∞ i=0

¯i λ

2n i −1

|ρk (f ; x)|p

k=ni

¯ i ni αp E p (f ψ¯ ), λ ni ni

(24.16)

i=0

¯ i = max{λk : ni ≤ k ≤ 2ni − 1}. λ Note that one can find a constant A such that αk ≤ Aα2k

(24.17)

Section 24

Strong Summability of Fourier Series

387

for all k ∈ N. Indeed, by virtue of (22.6) (for q = 2 ), we get ∞ αk =

|ψ2 (t)| ¯ dt + ψ(k) = t

≤

2k

|ψ2 (t)| dt + t

2k

k

∞

∞

|ψ2 (t)| ¯ dt + ψ(k) t

k

∞

|ψ2 (t)| ¯ dt + ψ(k)(1 + ln 2) ≤ t

2k

|ψ2 (t)| ¯ dt + Aψ(2k) ≤ Aα2k . t

2k

¯ i = λk . Since the numbers βk = λk αp Further let ki be a number such that λ i k do not increase, in view of (24.17) we obtain ¯ i αp ≤ Cp λk αp ≤ Cp λk αp ≤ Cp λn αp . λ ni i ni i 2ni i ki

(24.18)

Substituting this estimate in (24.16), we get Hn(p) (f ; x; λ)

≤ Cp

∞

¯

ni λni αnp i Enpi (f ψ )

i=0 ¯

≤ Cp (n0 λn0 αnp 0 Enp0 (f ψ ) +

∞ ni ni−1 i=1

¯

≤ Cp (nλn αnp Enp (f ψ ) +

∞

2ni−1 −1

¯

λk αkp Ekp (f ψ ))

k=ni−1 ¯

λk αkp Ekp (f ψ )).

k=n

Theorem 24.1 is proved. Theorem 24.2. Suppose that ±ψ1 ∈ F, ±ψ2 ∈ F, p > 0, the sequence λk , k ∈ N, is such that λk ≥ 0, and the numbers βk = λk ψ¯p (k) do not ¯ increase for all k ≥ n. If condition (21.5) is satisfied, then, for any f ∈ C ψ C 0 and n ∈ N, at every point x, Hn(p) (f ; x; λ) ¯ n) − n)ψ¯p (n)E p (f ψ¯ ) + ≤ Cp (λn (η(ψ; n

∞

¯ λk ψ¯p (k)Ekp (f ψ )). (24.19)

k=n

Suppose that ±ψ1 ∈ F, ±ψ2 ∈ F, p > 0, the sequence λk , k ∈ N, is such that λk ≥ 0, and the numbers βk = λk |ψ1 (k)|p do not increase for all k ≥ n.

Approximation by Fourier Sums in Spaces C and L1

388

Chapter 5

¯

If condition (21.5 ) is satisfied, then, for any f ∈ C ψ C 0 and n ∈ N, at every point x, ¯

Hn(p) (f ; x; λ) ≤ Cp (λn (η(ψ1 ; n) − n)|ψ1 (n)|p Enp (f ψ ) +

∞

¯ λk |ψ¯1 (k)|p Ekp (f ψ )).

(24.19 )

k=n

In inequalities (24.19) and (24.19 ), Cp are quantities uniformly bounded in n ¯ and f ∈ C ψ C 0 . ¯ t). Then Proof. Let n0 = n, ni = [η(ni−1 )] + 1, i ∈ N, where η(t) = η(ψ; Hn(p) (f ; x; λ) =

∞

λk |ρk (f ; x)|p =

∞ i=0

λk |ρk (f ; x)|p

i=0 k=ni

k=n

≤

∞ [η(n i )]

[η(ni )]

¯i λ

|ρk (f ; x)|p ,

(24.20)

k=ni

¯ i = max{λk : ni ≤ k ≤ [η(ni )]}. λ If condition (21.5) is satisfied, then, by virtue of (21.6),

[η(ni )]

df ¯ |ρk (f ; x)|p ≤ Cp ψ¯p (ni )γni , γni = [η(ni )] − ni + 1 = Δi (ψ). (24.21)

k=ni

Hence, Hnp (f ; x; λ)

≤ Cp

∞

¯ i ψ¯p E p (f ψ¯ )Δi (ψ) ¯ λ ni ni

i=0

¯ n ψ¯p (n0 )E p (f ψ¯ )Δ0 (ψ) ¯ = Cp (λ n0 0 +

∞

¯ i ψ¯p (ni )E p (f ψ¯ )Δi−1 (ψ)Δ ¯ i (ψ)/Δ ¯ ¯ λ i−1 (ψ). ni

(24.22)

i=1

Further, let us show that ¯ ¯ Δi (ψ)/Δ i−1 (ψ) ≤ K < ∞, i = 1, 2, . . . . This estimate is obtained from the following statement:

(24.23)

Section 24

Strong Summability of Fourier Series

389

Proposition 24.1. If ±ψ1 ∈ F, ±ψ2 ∈ F, and condition (24.5) is satisfied, ¯ = (ψ 2 (t) + ψ 2 (t))1/2 also belongs to F. then the function ψ(t) 1 2 Proof. Theorem 3.13.3 and Remark 3.13.2 imply that if ψ ∈ M, then the condition 0 < K1 ≤

η(η(t)) − η(t) ≤ K2 < ∞, η(t) = η(ψ; t), t ≥ 1, η(t) − t

(24.24)

is necessary and sufficient for ψ(·) to belong to F. According to Remark 21.2, ¯ ψ¯ ∈ M. Thus, it remains to verify condition (24.24) for ψ(·). ¯ t). By virtue of Let η1 (t) = η(ψ1 ; t), η2 (t) = η(ψ2 ; t), and η(t) = η(ψ; Proposition 21.1, for any t ≥ 1 the value of η(t) lies between the values of η1 (t) and η2 (t). Let us show that the value of η(η(t)) also lies between η1 (η1 (t)) and η2 (η2 (t)). Assume, e.g., that η1 (η1 (t)) ≤ η2 (η2 (t)).

(24.25)

1 Since, for every ϕ ∈ M, one has ϕ(η(η(t))) = ϕ(t) and η(t) = η(ϕ, t), we 4 get ¯ 1 (η1 (t)) = (ψ 2 (η1 (η1 (t)) + ψ 2 (η1 (η1 (t)))1/2 ψ(η 1 2 1 ¯ ≥ (ψ12 (t) + ψ22 (t))1/2 = ψ(η(η(t))). 4 Therefore, η1 (η1 (t)) ≤ η(η(t)). Similarly, η(η(t)) ≤ η2 (η2 (t)) for any t ≥ 1. Consequently, η(η(t)) indeed lies between η1 (η1 (t)) and η2 (η2 (t)). Assume, as before, that condition (24.25) is satisfied at a given point t and, e.g., η2 (t) ≤ η(t) ≤ η1 (t). (24.26) Since ±ψ2 ∈ F and, consequently, it satisfies condition (24.24), we have η(η(t)) − η(t) η2 (η2 (t)) − t η2 (η2 (t)) − η2 (t) ≤ = + 1 ≤ K2 + 1. (24.27) η(t) − t η2 (t) − t η2 (t) − t On the other hand, by virtue of (24.25), (24.26), and (24.5), η1 (η1 (t)) − η1 (t) η1 (t) − t η(η(t)) − η(t) ≥ η(t) − t η1 (t) − t η(t) − t ≥ K1

η1 (t) − t ≥ α > 0. η2 (t) − t

(24.27 )

390

Approximation by Fourier Sums in Spaces C and L1

Chapter 5

Combining (24.27) and (24.27 ), we conclude that condition (24.24) is satisfied ¯ for the function ψ(t), and, consequently, it belongs to F. ¯ = [η(ni )] − ni + 1 = Let us continue the proof of the theorem. Since Δi (ψ) [η(ni )] − [η(ni−1 )], we get ¯ Δi (ψ) [η(ni )] − [η(ni−1 )] [η([η(z + 1)] + 1)] − [η(z + 1)] = = ¯ [η(ni−1 )] − [η(ni−2 )] [η(z + 1)] − z Δi−1 (ψ) ≤

η([η(z + 1)] + 1) − η(z + 1) + 1 , z = z(i) = [η(ni−2 )]. [η(z + 1)] − z

According to Proposition 24.1, ψ¯ ∈ F. Consequently (see (3.13.2)), the deriva¯ t) is uniformly bounded for any t ≥ 1. Therefore, tive η (t) = η (ψ; ¯ Δi (ψ) 1 . ≤K+ ¯ [η(z + 1)] − z Δi−1 (ψ) This immediately yields (24.23) because we always have η(t) > t and, in the case considered, z ∈ N. ¯ i = λk , ki ∈ [ni , [η(ni )]]. Since the Further, let ki be a number for which λ i p numbers λk ψ¯ (k) do not increase, we have ¯ i ψ¯p (ni ) = λk 2p ψ¯p (η(ni )) ≤ 2p λk ψ¯p (ki ) ≤ 2p λn ψ¯p (ni ). λ i i i

(24.28)

Substituting estimates (24.23) and (24.28) in (24.22), we get Hnp (f ; x; λ) ¯ ≤ Cp (λn ψ¯p (n)Enp (f ψ )(η(n) − n) +

∞

¯ ¯ λni ψ¯p (ni )Enpi (f ψ )Δi−1 (ψ))

i=1 ¯ ≤ Cp (λn ψ¯p (n)Enp (f ψ )(η(n) − n) +

∞ [η(n i )]

¯ λk ψ¯p (k)Enpi (f ψ ))

i=0 k=ni ¯ = Cp (λn ψ¯p (n)Enp (f ψ (η(n) − n) +

∞

¯ λk ψ¯p (k)Ekp (f ψ )).

(24.29)

k=n

Inequality (24.19) is proved. It is clear that, setting n0 = n, ni = [η(ni−1 )] + 1, i ∈ N, where η(t) = η(ψ1 ; t), and following the proof of estimate (24.19), we obtain inequality (24.19 ). Theorem 24.2 is proved. 24.3. We obtain the following analog of Theorem 24.2 in the integral metric:

Section 24

Strong Summability of Fourier Series

391

Theorem 24.3. Suppose that ±ψ1 ∈ F, ±ψ2 ∈ F, the sequence λk , k ∈ ¯ N, is such that λk ≥ 0, and the numbers βk = λk ψ(k) do not increase for all k ≥ n. Further, let ∞ Hn (f ; x; λ)1 = λk ρk (f ; x)1 , n = 1, 2, . . . . (24.30) k=n ¯

If condition (24.5) is satisfied, then, for every f ∈ Lψ , Hn (f ; x; λ)1 ¯

ψ ¯ n) − n)ψ(n)E ¯ ≤ Cλ (λn (η(ψ; n (f )1 +

∞

¯

ψ ¯ λk ψ(k)E k (f )1 ). (24.31)

k=n

Suppose that ±ψ1 ∈ F, ±ψ2 ∈ F, the sequence λk , k ∈ N, is such that λk ≥ 0, and the numbers βk = λk |ψ1 (k)| do not increase for all k ≥ 1. If ¯ condition (24.5 ) is satisfied, then, for every f ∈ Lψ , ¯

Hn (f ; x; λ)1 ≤ Cλ (λn (η(ψ1 ; n) − n)|ψ1 (n)|En (f ψ )1 +

∞

¯

(24.31 )

λk |ψ1 (k)|Ek (f ψ )1 ).

k=n

In inequalities (24.31) and ¯ N and f ∈ Lψ .

(24.31 ),

Cλ are quantities uniformly bounded in n ∈

Proof. As in the proof of Theorem 24.2, let n0 = n, ni = [η(ni−1 )] + 1, i ∈ ¯ t). Then, in view of (23.11), we get N, where η(t) = η(ψ; Hn (f ; x; λ)1 ≤

∞ i=0

[η(ni )]

λk ρk (f ; x)1 ≤ Cλ

∞

¯ i )En (f ψ¯ )1 Δi (ψ). ¯ λni ψ(n i

i=0

k=ni

Therefore, taking into account relation (24.23) and the monotonicity of the num¯ bers βk = λk ψ(k), we get ψ¯ ¯ Hn (f ; x; λ)1 ≤ Cλ (γn λn ψ(n)E n (f )1 +

∞

¯ i )En (f ψ¯ )1 ) Δi−1 λni ψ(n i

i=1 ¯

ψ ¯ ≤ Cλ (γn λn ψ(n)E n (f )1 +

∞ [η(n i−1 )]

¯

ψ ¯ λk ψ(k)E k (f )1 )

i=1 k=ni−1 ¯

ψ ¯ ≤ Cλ (λn (η(n) − n)ψ(n)E n (f )1 +

∞ k=n

¯

ψ ¯ λk ψ(k)E k (f )1 ).

392

Approximation by Fourier Sums in Spaces C and L1

Chapter 5

Estimate (24.31) is proved. It is clear that, by analogy, one can obtain estimate (24.31 ). 24.4. Clearly, Theorems 24.1–24.3 remain valid for n = 1. In this case, e.g., relation (24.19) takes the form ∞ ∞ ¯ λk |ρk (f ; x)|p ≤ Cp λk ψ¯p (k)Ekp (f ψ ) ∀p > 0. (24.32) k=1

k=1

This estimate is valid for all ±ψ1 ∈ F and ±ψ2 ∈ F and any sequence λk , ¯ k ∈ N, for which the numbers λk ψ(k) do not increase. Here, the constant Cp depends only on p. Therefore, we can assume that, in this inequality, as well as in (24.15), (24.19), (24.19 ), (24.22), and (24.22 ), the numbers λk depend on one more parameter. In other words, these inequalities remain valid if the number sequence λk is replaced by a sequence of functions λk (u) defined on a certain set U, provided that the sequence λk = λk (u0 ) ∀u0 ∈ U satisfies the conditions of the corresponding theorems. We now set n−1 , k < n, (n) λk = λk = (24.33) 0, k ≥ n. (n)

For every fixed n ∈ N, the sequence λk , k = 1, 2, . . . , does not increase. (n) The same is true for the sequence λk ψ¯p (k) with any p > 0 if ±ψ1 ∈ F and ±ψ2 ∈ F. In this case, it follows from (24.32) that 1 ¯p 1 ¯ ψ (k)Ekp (f ψ ) |ρk (f ; x)|p ≤ Cp n n n−1

n−1

k=1

k=1

Thus, we have established an estimate for the convergence rate of the left-hand ¯ side of (24.8) for every f ∈ C ψ C 0 . 24.5. If a sequence of functions λk (u), k ∈ N, u ∈ U, is given, then the quantities ∞ λk (u)ak α(u) = k=1

are called λ-means of a given sequence ak , k ∈ N. In particular, if λk (u) are defined by (24.33), then α(u) are called Fejér means, or arithmetical means. In this terminology, relation (24.32) means that if, for any uo ∈ U, the numbers ¯ λk (uo )ψ¯p (k) do not increase, then, for any f ∈ C ψ C 0 , ±ψ1 ∈ F, and ±ψ2 ∈ F, the λ-means of the sequence |ρk (f ; x)|p do not exceed the λ-means of the ¯ sequence ψ¯p Ekp (f ψ ) to within a factor Cp .

BIBLIOGRAPHICAL NOTES (PART I) Chapter 1 For more detailed information on the methods of Fejér, Zygmund, de la Valée Poussin, and Riesz, see Chapter 12 and the corresponding bibliographical notes. The Rogosinski method was introduced by Rogosinski in [1]. He considered the polynomials 1 Rn (f ; x; βn ) = [Sn (f ; x + βn ) + Sn (f ; x − βn )], 2

(1)

where Sn (f ; x), as always, is the nth-order partial Fourier sum of the function f (x) and βn are some real numbers. Rogosinski showed that, under certain conditions on these numbers, the following relation holds for any continuous function f ∈ C: lim Rn (f ; x; βn ) − f (x)C = 0. (2) n→∞

Conditions that guarantee equality (2) were considered by Timan and Ganzburg [1], Bernstein [2], and Ganzburg [1]. The polynomials Rn (f ; x; βn ) for βn = π/2n were considered by Stechkin [3] and for βn = π/(2n + 1) by Korneichuk [1]. Most complete results on approximating properties of the polynomials Rn (f ; x; π/2n) can be found in the works of Dzyadyk and Stepanets [2], Dzyadyk, Gavrylyuk, and Stepanets [1, 2], and Stepanets [3, 4]. These results are introduced by Stepanets in [9]. Theorems 1.1 and 1.2 were proved by Nikol’skii [9]. Theorem 2.2 and Lemma 2.1 belong to Nagy [4, 5]. Equality (3.11) for different estimates of the quantities Rn is well known. The estimate |Rn | < 1.8724 was obtained by Galkin [1]. Estimates (3.20), (3.21), and (3.25) can be found in the book by Dzyadyk [5]. Theorem 4.1 and Lemma 4.1 were proved by Stechkin [7], and Theorem 4.2 was proved by Sidon [1]. 393

394

Bibliographical Notes (Part I)

Theorems 5.1 and 5.3 were proved in the work by Karamata and Tomic [1]. Theorems 6.1–6.4 are proved in various manuals on Fourier series (see, e.g., Bari [1]). Theorems 6.5 and 6.6 are due to Telyakovskii [6], and Lemma 6.1 belongs to Boas [1]. Theorem 6.6 and Lemma 6.2 were proved by Telyakovskii [7]. The main results of Sections 7 and 8 were established by Telyakovskii [13]. The first part of Corollary 8.3 belongs to Kolmogorov [1] and the second one belongs to Telyakovskii [7]. Lemma 8.1 is presented in the work by Telyakovskii [10] and Theorem 8.1 can be found in the work by Bausov [1]. There are many works devoted to the problem of summability of functions represented by Fourier series. In this connection, the reader can be referred to the works by Efimov [7–13], Karamata [1], Nagy [1–6], Telyakovskii [1–14], Fomin [1–4], etc. The multidimensional case was introduced in the works by Zaderei [1–8]. Chapter 2 The first results concerning the determination of the order and the classes of saturation of certain linear methods Un (Λ) were first introduced in the works by Alexits [1, 2], Zygmund [2, 3], Zamanski [1, 2], Favard [4, 5], etc. In particular, the notion of “saturation” of linear method was introduced by Favard and Zamanski. Different variations and special cases of Theorem 2.1 were proved by Alexits [1, 2], Zamanski [1, 2], Favard [5], Sunouchi and Watari [1, 2], Kharshiladze [1], etc. The results are presented according to the scheme introduced by Sunouchi [1, 2], Kharshiladze [1], and Butzer and Nessel [1]. Almost all results of this chapter can be found in the survey by Gavrylyuk and Stepanets [3]. Chapter 3 This chapter plays an auxiliary role. The material of Sections 1–10 is taken from the book of Stepanets [21], and, thus, the bibliographical notes to these sections coincide with those to Chapter 1 of that book. Apparently for the first time, the notion of the modulus of continuity was introduced by Lebesgue [2], Jackson [1, 2], and Bernstein [1]. Various properties of moduli of continuity were established by many authors. In particular, the assertion in Subsection 1.5 belongs to Nikol’skii [6], and Lemma 1.1 is due to Stechkin (see Efimov [13]). For more detailed information on moduli of continuity, we refer the reader to Akhiezer [1], Dzyadyk [5], Korneichuk [5, 6], Timan [5], etc.

Bibliographical Notes (Part I)

395

The functional classes Hω and W r Hω , r > 0, were introduced by Nikol’skii [6, 7], and the Weyl–Nagy classes were studied by Nagy [3]. The classes Wβr Hω , r ∈ [0, ∞), β ∈ R1 , were introduced by Efimov [8]. The information on convolutions used in Sections 7 and 8 is taken from the books of Zygmund [4] and Korneichuk [5]. Theorem 8.1 and its Corollary 8.1 ψ are taken from Akhiezer [2]. The classes Lψ β N and Cβ N were introduced by Stepanets [11–17]. Section 10 is based on results obtained by Stepanets [18]. ¯ For the first time, the notion of ψ-integrals in the form considered in Section 11 was introduced by Stepanets in [30, 31]. In these papers, all main assertions of this section were proved. ¯ In the course of the investigation of approximations on the sets of ψ-integrals, ¯ where ψ = (ψ1 , ψ2 ) and ψ1 , ψ2 ∈ M, the following function plays an important role: ∞ Fn (x, ψ) =

ψ(nt + n) dt + t

∞ dt (ψ(n) − ψ(n + n/t)) , t x

1/x

x > 0,

ψ ∈ M.

Namely, the remainders in various asymptotic equalities are expressed in terms of the values of this function. In addition, the parameter x is, as a rule, known. By virtue of the strict monotonicity of the function ψ(t), the derivative of the function Fn (x; ψ) vanishes only at the unique point x = x(n) = n(ψ −1 (

ψ(n) ) − n)−1 , 2

which is its point of minimum. For this reason, the functions η(t) = η(ψ; t) and μ(t) = μ(ψ; t) were introduced. In the first works of Stepanets and his followers, the symbol M∞ denoted the subset of the functions ψ ∈ M that is denoted by M+ ∞ in the present book and is defined by (12.8). Various properties of functions from M0 , MC , and M∞ were established by Stepanets (see, e.g., [11–39]). The main results of Sections 12–16 were published by Stepanets in [36]. Chapter 4 The main assertions of this chapter were established by Stepanets in [30, 31]. The statement of Proposition 1.1 is well known. Its proof can be found, e.g.,

396

Bibliographical Notes (Part I)

in the book of Bochner [2]. In approximation theory, it was used by Nagy [3], Telyakovskii [4], etc. Lemma 3.2 plays the key role in obtaining equality (3.60). For the first time, it was proved by Stepanets in [11]. Chapter 5 The main material of Sections 1 and 2 is taken from the books of Stepanets [21, 29]. Lemma 1.1 is a simplified version of the Korneichuk–Stechkin lemma (see Lemma 1.4) and was first used by Nikol’skii [1]. The proof of the inclusion f∗ ∈ Hω [a, b] from Subsection 1.2 belongs to Efimov [8]. He also proved that the constant 2/3 in (1.2) cannot be greater. For convex moduli of continuity, equality (1.23) was obtained by Lebesgue [2]; for arbitrary moduli of continuity, it was obtained by Efimov [8]. Lemma 1.4 was proved by Korneichuk [4, 5]. Its special case for ω(t) = tα , 0 < α ≤ 1, was published for the first time in the work by Korneichuk [1] as a lemma. In that work, it was also noted that such a statement was independently obtained by Stechkin (see comments in Korneichuk [1]). Later, it was called the Korneichuk–Stechkin lemma. Its multidimensional case can be found in the works by Stepanets [6, 9, 40]. The proofs of statements from Section 2 were published earlier by Stepanets in [21, 29]. The statements of Sections 3–9 were obtained by Stepanets in [32, 33]. Equality (3.8) is due to Kolmogorov [2] for r ∈ N and β = r and Pinkevich [1] for arbitrary r > 0. Theorem 6.1 for θ ∈ [1/2, 1] and Theorem 6.1 were proved by Nikol’skii [6, 7]; in the general case, it is due to Efimov [8]. Theorem 7.1 was proved by Efimov [8]. In the case where ψ(t) = tr , r > 0, and β = r, equality (8.7) is due to Nikol’skii [8] and equality (8.7 ) is due to Demchenko [1]. Lemma 8.2 belongs to Nikol’skii [7], and Theorems 9.1 and 9.3 were proved by Berdyshev [1]. The main results of Sections 10–15 were obtained by Stepanets [30, 31]. Equalities (15.12) and (15.13) are due to Nikol’skii [8] and Berdyshev [1], respectively. Theorem 16.1 with remainder O(1)q n n−1 in (16.3) was proved by Nikol’skii [8]; in the form used in this book, it was proved by Stechkin [10]. Lemma 16.1 is also due to Stechkin [10]. Theorem 17.2 was obtained by Telyakovskii [14]. The results of Section 18 are due to Stepanets [38, 39]. The results of Sections 19 and 20 were obtained by Stepanets and Serdyuk [1, 2, 4, 5]. The main results of

Bibliographical Notes (Part I)

397

Sections 21–24 were obtained by Stepanets [37]. In Subsection 24.1, the works of Hardy and Littlewood [1–3], Marcinkiewicz [2], and Zygmund [1] were mentioned. The history of the problem of strong summability and the corresponding results can be found, e.g., in the book of Alexits [3].

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Babenko, K. I. [1] On the best approximations of classes of analytic functions, Izv. Akad. Nauk SSSR, Ser. Mat., 22, No. 5, 631–640 (1958). [2] On the approximation of periodic functions of many variables by trigonometric polynomials, Dokl. Akad. Nauk SSSR, 132, No. 2, 247–250 (1960). [3] On the approximation of one class of periodic functions of many variables by trigonometric polynomials, Dokl. Akad. Nauk SSSR, 132, No. 5, 982– 985 (1960).

Baiborodov, S. P. [1] Approximation by de la Vallée Poussin sums, Mat. Zametki, 27, No. 1, 33–48 (1980).

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Butzer, P., and Nessel, J. R. [1] Fourier Analysis and Approximation, Birkhäuser, Basel (1971). Carleson, L. [1] On convergence and growth of partial sums of Fourier series, Acta Math., 116, No. 1-2, 135–157 (1966). Cesari, L. [1] Sull cerie di Fourier della fonzioni Lipschitziane di pin variabili, Ann. di Scuola Norm. Sup. di Pisa, 7, 279–295 (1938). Chebyshev, P. I. [1] Problems of least values related to approximate representation of functions, in: Selected Works [in Russian], Vol. 2, Izd. Akad. Nauk SSSR, Moscow– Leningrad (1947), 151–235. Chernykh, N. I. [1] On Jackson inequalities in L2 , Tr. Mat. Inst. Akad. Nauk SSSR, 88, 71–74 (1967). [2] On the best approximation of periodic functions by trigonometric polynomials in L2 , Mat. Zametki, 2, No. 5, 513–522 (1967). Demchenko, A. G. [1] Approximation of functions from classes W r H[ω]L in the mean, Ukr. Mat. Zh., 25, No. 2, 267–276 (1973). Dienes, P. [1] The Taylor series, Dover, New York (1957).

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Suetin, P. K. [1] Series in Faber Polynomials [in Russian], Nauka, Moscow (1984). Sunouchi, G. [1] Direct theorem in the theory of approximation, Acta Math. Acad. Scie. Hung., 20, No. 3-4, 409–420 (1969). [2] Characterization of certain classes of functions, Acta Math. Acad. Scie. Hung., 14, No. 1, 127–134 (1962). Sunouchi, G., and Watari, C. [1] On determination of the class of saturation in the theory of approximation of functions, Proc. Jpn. Acad., 34, 477–481 (1958). [2] On determination of the class of saturation in the theory of approximation of functions, Tohokh Math. J., 11, No. 3, 480–488 (1959). Taikov, L. V. [1] On summation methods for Taylor series, Usp. Mat. Nauk, 17, No. 1, 252– 254 (1962). [2] New conditions for regularity of triangular summation methods, Mat. Zametki, 1, No. 5, 541–547 (1967). Telyakovskii, S. A. [1] Approximation of differentiable functions by de la Vallée Poussin sums, Dokl. Akad. Nauk SSSR, 121, No. 3, 426–429 (1958). [2] Approximation of functions differentiable in the Weyl sense by de la Vallée Poussin sums, Dokl. Akad. Nauk SSSR, 131, No. 2, 259–262 (1960). [3] On approximation of differentiable functions by linear means of their Fourier series, Izv. Akad. Nauk SSSR, Ser. Mat., 24, No. 2, 213–242 (1960).

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[4] On the norms of trigonometric polynomials and approximation of differentiable functions by linear means of their Fourier series. I, Tr. Mat. Inst. Akad. Nauk SSSR, 62, 61–97 (1961). [5] On the norms of trigonometric polynomials and approximation of differentiable functions by linear means of their Fourier series. II, Tr. Mat. Inst. Akad. Nauk SSSR, 27, No. 2, 253–272 (1963). [6] Integrability conditions for trigonometric series and their application to the investigation of linear summation methods for Fourier series, Izv. Akad. Nauk SSSR, Ser. Mat., 28, No. 6, 1209–1236 (1964). [7] Some estimates for trigonometric series with quasiconvex coefficients, Mat. Sb., 63(105), No. 3, 426–444 (1964). [8] Asymptotic estimate for integral of modulus of the function defined by series of sines, Sib. Mat. Zh., 8, No. 6, 1416–1422 (1967). [9] Estimates for trigonometric series and polynomials in connection with the problems of approximation theory of functions, Mat. Zametki, 1, No. 5, 611–623 (1967). [10] On the problem of convergence of Fourier series in the metric of L , Mat. Zametki, 1, No. 1, 91–98 (1967). [11] Approximation of differentiable functions by partial sums of their Fourier series, Mat. Zametki, 4, No. 3, 291–300 (1968). [12] Approximation of functions satisfying the Lipschitz condition by Fejér sums, Ukr. Mat. Zh., 21, No. 3, 334–343 (1969). [13] Estimate for norm of function by its Fourier coefficients which is convenient in problems of approximation theory, Tr. Mat. Inst. Akad. Nauk SSSR, 109, 65–97 (1971). [14] On approximation of functions of high smoothness by Fourier sums, Ukr. Mat. Zh., 41, No. 4, 510–518 (1989). Tikhomirov, V. M. [1] Problems of Approximation Theory [in Russian], Moscow University, Moscow (1976).

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[2] On multidimensional analog of one result by Boas, Ukr. Mat. Zh., 39, No. 3, 380–383 (1987). [3] On the convergence rate of Fourier series of (ψ, β)-differentiable functions of two variables, in: Investigations in the Approximation Theory of Functions [in Russian], Institute of Mathematics, Ukrainian Academy of Sciences, Kiev (1987), pp. 24–34. [4] On asymptotic equalities for norms in the metric of L of functions determined by multiple trigonometric series, Dokl. Akad. Nauk SSSR, Ser. Mat., 302, No. 6, 1305–1308 (1988). [5] On the convergence of Fourier series in the mean, Ukr. Mat. Zh., 41, No. 4, 562–568 (1989). [6] Asymptotic equalities for integrals of modulus of the sum of multiple series of sines, Analysis Mathematica, 15, No. 3, 245–262 (1989). [7] On the conditions of integrability of multiple trigonometric series, Ukr. Mat. Zh., 44, No. 3, 340–365 (1992). [8] Approximation of (ψ, β)-differentiable periodic functions of many variables, Ukr. Mat. Zh., 45, No. 3, 367–377 (1993). Zamanski, M. [1] Classes de saturation de certains procédés d’approximation des séries de Fourier des fonctions continues et application a` quelques problémes d’approximation, Ann. Sci. Ecole Norm. Sup., 66, No. 1, 19–93 (1949). [2] Classes de saturation des procédés de sommation des séries de Fourier et applications des séries trigonométriques, Ann. Sci. Ecole Norm. Sup., 67, 161–198 (1960). Zygmund, A. [1] On the convergence and summability of power series on the circle on convergence, Proc. London Math. Soc., 47, 326–350 (1941). [2] The approximation of functions by typical means of their Fourier series, Duxe Math. J., 12, 695–704 (1945).

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[3] On the degree of approximation of functions by Fejér means, Bull. Amer. Math. Soc., 51, No. 4, 274–278 (1945). [4] Trigonometric Series [Russian translation], Vol. 1, Mir, Moscow (1965). [5] Trigonometric Series [Russian translation], Vol. 2, Mir, Moscow (1965). [6] Trigonometric Series [Russian translation], GONTI, Moscow, Leningrad (1939). [7] Smooth functions, Duke Math. J., 12, 47–76 (1945).

PART II 6. CONVERGENCE RATE OF FOURIER SERIES AND THE BEST APPROXIMATIONS IN THE SPACES Lp 0.

Introduction

0.1. In the present chapter, we consider the values of deviations of Fourier sums π ρn (f ; x)s = f (x)−Sn−1 (f ; x)s , ϕs = (

|ϕ(t)|s dt)1/s , s ≥ 1, (0.1)

−π ¯

in the metric of the spaces Ls for functions from the sets Lψ N, where N is a certain subset in the space Lp . Most frequently, we have N = Sp = {ϕ : ¯ ¯ ϕp ≤ 1} . In this case, we set Lψ Sp = Lψ p. As earlier, we mainly study the upper bounds of deviations in the classes ¯ Lψ N, namely, ¯ En (Lψ N)s = sup ρn (f ; x)s . (0.2) f ∈Lψ¯ N

In addition, we now study the best approximations of the functions from these classes by trigonometric polynomials. This is explained by the following reasons: As earlier, we assume that T2n−1 is a set (subspace) of trigonometric polynomials c0 tn−1 (x) = (ck cos kx + dk sin kx), ck , dk ∈ R1 , + 2 n−1 k=1

429

(0.3)

430

Convergence Rate of Fourier Series and Best Approximations

Chapter 6

of degree not higher than n − 1, X is a set (space) of 2π-periodic functions with norm · X , and N ⊆ X is a fixed subset (class) from X. Then the quantity En (f )X =

inf

tn−1 ∈T2n−1

f (·) − tn−1 (·)X , f ∈ X,

(0.4)

is called the best approximation of a function f, and the quantity En (N)X = sup En (f )X = sup f ∈N

inf

f ∈Ntn−1 ∈T2n−1

f (·) − tn−1 (·)X

(0.5)

is called the best approximation of the class N by trigonometric polynomials of degree not higher than n − 1 (by the subspace T2n−1 ) in the metric of the space X. If X = Lp , then we write En (f )X=Lp = En (f )p and En (N)X=Lp = En (N)p . For p ∈ (1, ∞), the orders of the quantities ρn (f ; x)p and En (f )p , ¯ ψ¯ En (Lψ p )p and En (Lp )p coincide as n → ∞. This explains the necessity of simultaneous analysis of these quantities. The polynomial t∗n−1 ∈ T2n−1 for which the lower bound in (0.4) is attained, i.e., f (·) − t∗n−1 (·)X = En (f )X , (0.6) is called the polynomial of the best approximation of the function f (·) in the space X. General problems encountered in the investigation of functionals of the best approximation En (f )X are discussed in detail in numerous recently published monographs and, thus, there is no need to repeat these discussions in the present book. If necessary, we use facts whose proofs can be found, e.g., in the books by Korneichuk [5, 6]. However, in this chapter, we mainly need almost obvious assertions. Thus, we use the fact of existence of a polynomial of the best approximation t∗n−1 (·) in the space Lp . Its proof is not difficult even in a more general case. Indeed, assume that X is an arbitrary normed space, g1 , . . . , gn are n linearly independent elements from X, λ = (λ1 , . . . , λn ) is a vector with numerical coordinates λi , and ϕx (λ) = x −

n

λi gi X , x ∈ X.

i=1

Let us show that, for any x ∈ X, there exists a vector λ∗ = (λ∗1 , . . . , λ∗n ) such that df e(x) = inf ϕx (λ) = ϕx (λ∗ ). λ

Section 0

Introduction

431

Indeed, for any fixed x ∈ X, ϕx (λ) is continuous as a function of the variables λ1 , . . . , λn , because, by virtue of the triangle inequality for the norm, we have n

|ϕx (λ) − ϕx (λ )| = |x −

λi gi X − x −

n

i=1

≤

n

λi gi X |

i=1

(λi − λi )gi X ≤

i=1

n

|λi − λi |gi X

i=1

≤ max |λk − λk | k

n

gi X .

i=1

Just as ϕx (λ), the function ψ(λ) =

n

λi gi X

i=1

is also continuous in λi . Therefore, in a bounded closed set {λ :

n

|λi |2 = 1},

i=1

this function attains its minimum value m, which is positive (m > 0) due to the linear independence of the vectors gi . Further, if n df ( |λi |2 )1/2 > (e(x) + 1 + xX )m−1 = R, i=1

then ϕx (λ) > ψ(λ) − xX > (

n

|λi |2 )1/2 m − xX > e(x) + 1.

i=1

Thus, in seeking the minimum of the function ϕx (λ), we can restrict ourn selves to the ball |λi |2 ≤ R2 . In this case, it is possible to conclude that the i=1

point at which the minimum value of this function is attained exists. Since the trigonometric system 1, cos x, sin x, . . . , cos nx, sin nx

432

Convergence Rate of Fourier Series and Best Approximations

Chapter 6

is linearly independent, the established assertion implies the existence of a polynomial tn−1 ∈ T2n−1 guaranteeing the best approximation of the function f (·) both in the space Lp and in the space C. ¯

0.2. In the present chapter, the classes Lψ N are understood in a sense of definitions introduced in Section 3.11, i.e., in this case, ψ1 (k) and ψ2 (k) are arbitrary functions of a natural argument satisfying certain conditions. These conditions are often formulated in terms of the quantity ¯ ¯ ψ(k) = (ψ12 (k) + ψ22 (k))1/2 . ν(n) = sup ψ(k),

(0.7)

k≥n

¯ ¯ For a nonincreasing function ψ(k), k ∈ N, , this quantity coincides with ψ(n) . As earlier, K (Ki , i = 1, 2, . . .) denotes absolute constants. If it is necessary to denote constants depending on parameters (e.g., on p, s, q, etc.), then we write Cp , Cp,s , etc.

1.

Approximations in the Space L2

1.1. We first consider a simple but very important case where p = s = 2. and we prove the following assertion: Proposition 1.1. Let f ∈ L2 . Then ρn (f ; x)22 = En2 (f )2 = π

∞

(a2k + b2k ),

(1.1)

k=n

where ak and bk , k ∈ N, are Fourier coefficients of the function f. Proof. If tn−1 (·) is an arbitrary polynomial and has the form (0.3), then, after elementary transformations, we get π f (·) −

tn−1 (·)22

|f (t) − tn−1 (t)|2 dt

= −π

π −π

πa20 π −π (a2k + b2k ) + (c0 − a0 )2 2 2 n−1

f 2 (t)dt −

=

k=1

+π

n−1 k=1

((ck − ak )2 + (dk − bk )2 ).

Approximations in the Space L2

Section 1

433

The right-hand side of this equality takes the minimum value for ck = ak and dk = bk , i.e., in the case where tn−1 (·) coincides with the partial sum of order n − 1 of the Fourier series of the function f (·). Thus, the partial Fourier sums of the function f (·) are polynomials of its best approximation in the space L2 and, therefore, the first part of equality (1.1) is proved. The proof of the second part of (1.1) is also quite simple. indeed, we have π ∞ 2 ρn (f ; x)2 = ( (ak cos kx + bk sin kx))2 dx. −π k=n

It remains to apply the Parseval equality according to which 1 π

π

∞

a2 (g) 2 g (t)dt = 0 (ak (g) + b2k (g)) ∀g ∈ L2 . + 2 2

−π

(1.2)

k=1

1.2. By virtue of equality (1.2), the right-hand side of (1.1) monotonically approaches zero as n → ∞ for all f ∈ L2 . Hence, the same is true for the quantities ρn (f ; x)2 and En (f )2 . 1.3. Now let ψ¯ = (ψ1 , ψ2 ) be a couple of arbitrary number systems ψ1 (k) and ψ2 (k), k = 0, 1, . . . , ψ1 (0) = 1, and ψ2 (0) = 0. Also let ϕ ∈ L2 , let ∞

S[ϕ] =

∞

α0 (αk cos kx + βk sin kx) = Ak (ϕ; x), + 2 k=1

(1.3)

k=0

¯

and let f (x) = J ψ ϕ(x). Then, according to Definition 3.11.1, ∞

S[f ] =

a0 (ak cos kx + bk sin kx) + 2 k=1

=

∞

ψ1 (k)Ak (ϕ; x) + ψ2 (k)A˜k (ϕ; x),

(1.4)

k=0

whence it follows that a0 = α0 ,

ak = ψ1 (k)αk − ψ2 (k)βk ,

bk = ψ1 (k)βk + ψ2 (k)αk , k = 1, 2, . . . ,

(1.5)

434

Convergence Rate of Fourier Series and Best Approximations

Chapter 6

and, consequently, a2k + b2k = ψ¯2 (k)(αk2 + βk2 ), k = 1, 2, . . . .

(1.6)

Therefore, according to (1.2), we have ∞

α02 2 (αk + βk2 ) = π −1 ϕ22 . + 2

(1.7)

k=1

¯ ¯ If we assume that the quantity ψ(k) is bounded, i.e., ψ(k) ≤ K, k ∈ N, then, in view of relations (1.6) and (1.7), we get ∞

a20 2 (ak + b2k ) ≤ K 2 π −1 ϕ22 . + 2

(1.8)

k=1

By virtue of the well-known Fischer–Riesz theorem, this means that f ∈ L2 . Thus, the following assertion is true: ¯ ¯ Proposition 1.2. If ψ(k) ≤ K, k ∈ N, and f ∈ Lψ L2 , then f ∈ L2 , i.e., ⊂ L2 .

¯ Lψ L2

Indeed, the inequality ¯ ψ(k) ≤ K, k ∈ N,

(1.9)

is a sufficient condition for the inclusion ¯

Lψ L2 ⊂ L2 .

(1.10)

On the other hand, for any positive sequence {ck } whose upper bound is ∞ equal to +∞, one can indicate a convergent series ak with positive terms k=1

such that lim sup ak ck = +∞. k→∞

Therefore, if ¯ = ∞, lim sup ψ(k) k→∞

¯

then, by virtue of (1.6), one can indicate a function ϕ∗ ∈ L2 such that J ψ ϕ∗ ∈L2 , i.e., condition (1.9) is also a necessary condition for the validity of inclusion (1.10). Hence, the following assertion is true:

Approximations in the Space L2

Section 1

435

Theorem 1.1. Condition (1.9) is necessary and sufficient for the validity of inclusion (1.10). 1.4. Let us now prove the following auxiliary assertion: ¯

¯

Lemma 1.1. If f ∈ Lψ ∩ L2 and f ψ ∈ L2 , then, for any n ∈ N, ¯

¯

ρn (f ; x)2 = En (f )2 ≤ ν(n)ρn (f ψ ; x)2 = ν(n)En (f ψ )2 .

(1.11)

Indeed, substituting expressions ak = ak (f ) and bk = bk (f ) from relations (1.5) in the right-hand side of (1.1), we get ρn (f ; x)22

=

En2 (f )2

=π

∞

¯ ¯ ψ¯2 (k)(a2k (f ψ ) + b2k (f ψ ))

k=n ¯

¯2

¯

≤ sup ψ (k)ρn (f ψ ; x)22 = ν 2 (n)ρn (f ψ ; x)22 k≥n

¯

= ν 2 (n)En2 (f ψ )2 . 1.5. Approximations by Fourier sums and the best approximations in the ¯ classes Lψ 2 are characterized by the following statement: Theorem 1.2. Let ψ1 (k) and ψ2 (k), k ∈ N, be arbitrary functions satisfying condition (1.9). Then, for any n ∈ N, , ¯

¯

ψ En (Lψ 2 )2 = En (L2 )2 = ν(n).

(1.12) ¯

Proof. By virtue of condition (1.9) and Proposition 1.2, we have Lψ 2 ⊂ L2 . ¯ ψ¯ Therefore, relation (1.11) holds for all f ∈ L2 . In the analyzed case, f ψ ∈ S2 , ¯ i.e., f ψ 2 ≤ 1 and, consequently, ¯

¯

En (f ψ )2 ≤ 1 ∀f ∈ Lψ 2.

(1.13)

Hence, according to (1.11), ¯

ρn (f ; x)2 = En (f )2 ≤ ν(n) ∀f ∈ Lψ 2.

(1.14) ¯

It remains to show that this estimate is exact in the entire class Lψ 2. ¯ Assume that, for a given function ψ(k) and a number n ∈ N, there exists a number kn such that ¯ ¯ n ). ν(n) = sup ψ(k) = ψ(k (1.15) k≥n

436

Convergence Rate of Fourier Series and Best Approximations

Chapter 6

We set fn (x) = π −1/2 (ψ1 (kn ) cos kn x + ψ2 (kn ) sin kn x). The function fn (x) ¯ −1/2 cos k x and, conse¯ belongs to Lψ n 2 because its ψ-derivative is equal to π ¯ quently, fnψ (x)2 = 1. For this function, we have ¯ n ) cos(kn x − θk )2 = ψ(k ¯ n ) = ν(n), ρn (fn ; x)2 = fn (x)2 = π −1/2 ψ(k n i.e., the theorem is proved in the analyzed case. ¯ If, for the function ψ(k) and a number n ∈ N, there is no number kn for ¯ which relation (1.15) is true, then, in view of the boundedness of the set {ψ(k)}, we get df ¯ ¯ ν(n) = sup ψ(k) = sup{ψ(k)} = gn . k≥n

k≥n

Moreover, there exists a sequence ni , i ∈ N, such that ni ≥ n and the numbers ¯ i ) form a nondecreasing sequence approaching gn . We set ψ(n fni (x) = π −1/2 (ψ1 (ni ) cos ni x + ψ2 (ni ) sin ni x) and consider the set Φn = ∪ fni (·), formed by all functions fni (·), i ∈ N. Both i

¯

the function fn (·) and the functions fni (·) belong to Lψ 2 and ¯ i ). ρn (fni ; x)2 = fni (x)2 = ψ(n Hence, in this case, we also have ¯ ψ¯ ¯ En (Lψ 2 )2 = En (L2 )2 ≥ sup ρn (f ; x)2 = sup ψ(ni ) = gn = ν(n). f ∈Φn

i∈N

Theorem 1.2 is proved. ¯ ¯ 1.6. If ψ(k), k ∈ N, is a nonincreasing function, then ν(n) = ψ(n). Therefore, Theorem 1.2 yields the following corollary: Corollary 1.1. Assume that ψ1 (k) and ψ2 (k) are arbitrary sequences for ¯ which ψ(k) is a nonincreasing quantity. Then ¯ ψ¯ ¯ En (Lψ 2 )2 = En (L2 )2 = ψ(n).

(1.16)

ψ1 (k) = ψ(k) cos βπ/2, ψ2 (k) = ψ(k) sin βπ/2,

(1.17)

If where ψ(k), k ∈ N, is a sequence of positive numbers and β is a number from ¯ ψ −r R1 , then, instead of Lψ 2 , we write Lβ,2 . If, in addition, ψ(k) = k , r > 0, r then we set Lψ β,2 = Wβ,2 . In this case, equality (1.16) takes the form r r En (Wβ,2 )2 = En (Wβ,2 )2 = n−r .

(1.16 )

Direct and Inverse Theorems in the Space L2

Section 2

2.

437

Direct and Inverse Theorems in the Space L2

2.1. Equality (1.12) completely characterizes the upper bounds of the best ¯ approximations in the space L2 in the classes Lψ 2 . In this section, we consider ¯ the approximations of individual functions from the sets Lψ L2 . Our principal result can be formulated as follows: ¯

Theorem 2.1. Let f ∈ Lψ L2 and let ψ1 (k) and ψ2 (k) be sequences such that condition (1.9) is satisfied. Then the series ∞

¯

(ψ¯2 (k) − ψ¯2 (k − 1))Ek2 (f ψ )2

(2.1)

k=1

is convergent and , for any n ∈ N, ¯ En2 (f )2 = ψ¯2 (n)En2 (f ψ )2 +

∞

¯ (ψ¯2 (k) − ψ¯2 (k − 1))Ek2 (f ψ )2 .

(2.2)

k=n+1

On the other hand, if f ∈ L2 and the sequences ψ1 (k) and ψ2 (k) are such ψ¯ ¯ that lim ψ(k)E k (f ) = 0, then the inclusion f ∈ L L2 is true if and only if the k→∞

series

∞

(ψ¯−2 (k) − ψ¯−2 (k − 1))Ek2 (f )2

(2.1 )

k=1 ¯

is convergent. In the case where this series is convergent, f ∈ Lψ L2 and, for any n ∈ N, ¯ En2 (f ψ )2

=ψ

−2

(n)En2 (f )

+

∞

(ψ¯−2 (k) − ψ¯−2 (k − 1))Ek2 (f )2 .

(2.2 )

k=n+1

The first part of the theorem, namely, equality (2.2), enables us to make conclusions concerning the rate of vanishing of the quantities En (f )2 or, equiva¯ lently, of the quantities ρn (f ; x)2 by using the data about the ψ-derivative of the function f (·). In the theory of approximation, statements of this sort are called direct theorems. The second part of the theorem is the inverse assertion: the properties of the sequence En (f )2 are used to make conclusions concerning the properties of the function itself and its derivatives.

438

Convergence Rate of Fourier Series and Best Approximations

Chapter 6

We also note that, as follows from equality (2.2), for any f ∈ L2 and n ∈ N, the quantity ¯ ψ¯2 (n)En2 (f ψ )2 +

∞

¯ (ψ¯2 (k) − ψ¯2 (k − 1))Ek2 (f ψ )2

(2.3)

k=n+1

is invariant and independent of the sequences ψ1 (k) and ψ2 (k) provided that it is bounded. The second part of the theorem shows, in particular, that the function f ∈ L2 ¯ possesses the ψ-derivative with finite L2 -norm if and only if the series (2.1 ) is ¯ convergent. This, in particular, implies that, for any function f ∈ L2 , its ψ¯ derivative for which ψ(k) ≡ Ek (f )2 , cannot belong to L2 (in this case, series (2.2 ) divergent).

2.2. To prove the theorem we need the following lemma: Lemma 2.1. Assume that a series ∞

ck

(2.4)

k=1

is convergent and a sequence {λk }∞ k=1 is such that lim λn An+1 = 0, An =

n→∞

∞

(2.4 )

ck .

k=n

Then the series ∞

∞

λk ck and

k=n

(λk − λk−1 )

k=n+1

∞

ci

(2.5)

i=k

converge and diverge simultaneously. Moreover, in the case of their convergence, ∞ k=n

λk ck = λn

∞

ck +

k=n

∞ k=n+1

(λk − λk−1 )

∞

ci .

(2.6)

i=k

Proof. The proof of the lemma is obtained by passing to the limit in the equality ∞

(λk − λk−1 )Ak =

k=n+1

valid for all sufficiently large m.

∞ k=n+1

λk ck − λn An + λm Am+1

Section 3

Extension to the Case of Complete Orthonormal Systems

439

¯ Proof of Theorem 2.1. Assume that f ∈ Lψ L2 . We set λk = ψ¯2 (k) and ¯ ¯ ck = π(a2k (f ψ ) + b2k (f ψ )). Then, by virtue of the Parseval equality (see (1.2)), ∞

ck = π

k=1

∞

¯ (a2k (f ψ )

+

¯ b2k (f ψ ))

=

¯

πa2 (f ψ ) − 0 0. Therefore, n En2 (f )2 ≤ 4

π/n ω22 (f ; t) sin ntdt.

(4.5)

0

By assumption, the function f (·) is not constant. Therefore, ω2 (f ; π/n) > 0 and n 4

π/n π/n n 1 ω22 (f ; t) sin ntdt < ω22 (f ; π/n) sin ntdt = ω22 (f ; π/n). 4 2 0

(4.6)

0

By combining inequalities (4.5) and (4.6), we arrive at estimate (4.2). It remains to show that this estimate cannot be improved. Let us prove that, for any n ∈ N and arbitrarily small δ > 0, there exists a function fn (x) = fn (δ; x) from L2 such that # π−δ (4.7) En (fn )2 = ω2 (fn ; π/n). 2π This implies the required assertion. We fix n and δ ∈ (0, π/n) . Let gn (δ; t) be an even 2π/n-periodic function such that 1 − t/2δ, t ∈ [0, 2δ], gn (δ; t) = 0, t ∈ [2δ; π/n].

Jackson Inequalities in the Space L2

Section 4

447

This function can be expanded in the Fourier series ∞ δ 2 sin knδ 2 gn (δ; t) = + cos knt π πδ kn k=1

and, in addition, gn (δ; 0) = 1 and gn (δ; t) ≥ 0 if t ∈ [0, π/n]. Further, we set ∞

fn (t) = fn (δ; t) = (

2 1/2 sin kπn ) cos knt. πδ kn k=1

By virtue of the Fischer–Riesz theorem, we have fn ∈ L2 and, hence, this function satisfies equality (4.4), which, in the analyzed case, has the form ∞

fn (x + h) − fn (x)22 =

4 sin knδ 2 ( ) (1 − cos knh) δ kn k=1

= 2π[gn (δ; 0) − gn (δ; h)]. The function gn (δ; t) is nonincreasing for t ∈ (0; π/n). Therefore, ω22 (fn ; π/n) = 2π[gn (δ; 0) − gn (δ; π/n)] = 2π. At the same time, since ak (fn ) = bk (fn ) = 0 for k < 0, according to equality (1.1), we obtain ∞

En2 (fn )2

2 sin knδ 2 = ( ) = π[gn (δ; 0) − δ/π] = π − δ δ kn k=1

and ω22 (fn ; π/n) = 2En2 (fn )2

π . π−δ

This yields relation (4.7). Theorem 4.1 is proved. 4.2. Combining Proposition 1.1, Lemma 1.1, and Theorem 4.1, we arrive at the following statement: Theorem 4.2. Let ψ1 (k) and ψ2 (k), k ∈ N, be arbitrary functions such that ¯ ψ(k) = ψ12 (k) + ψ22 (k) ≤ K.

448

Convergence Rate of Fourier Series and Best Approximations

Chapter 6

¯

Then, for any f ∈ Lψ L2 , ¯ ¯ ρn (f ; x)2 = En (f )2 < 2−1/2 ν(n)ω2 (f ψ ; π/n), ν(n) = sup ψ(k).

(4.8)

k≥n ¯

Now let ω(t) be an arbitrary modulus of continuity and let Lψ Hω2 be a class ¯ ¯ of functions f ∈ Lψ such that ω2 (f ψ ; t) ≤ ω(t). By analyzing the upper bounds in relation (4.8) for this class of functions, we arrive at the following assertion: Corollary 4.1. Under the conditions of Theorem 4.2, ¯

¯

En (Lψ Hω2 )2 = En (Lψ Hω2 )2 < 2−1/2 ν(n)ω(π/n).

(4.9)

If ψ1 (k) = ψ(k) cos βπ/2 and ψ2 (k) = ψ(k) sin βπ/2, where ψ(k) is a function of natural argument and β ∈ R1 , then we can write Lψ β Hω2 instead of ¯

¯

Lψ Hω2 . If, in addition, ψ(k) = k −r , r > 0, then we can set Lψ Hω2 = Wβr Hω2 . In the last case, relation (4.9) takes the form En (Wβr Hω2 )2 = En (Wβr Hω2 )2 < 2−1/2 n−r ω(π/n).

(4.9 )

Note that the constant 2−1/2 in inequality (4.2) is unimprovable. At the same time, this cannot be stated for relations (4.8)–(4.9 ). At present, the unimprovable constants for these inequalities are unknown.

5.

Marcinkiewicz, Riesz, and Hardy–Littlewood Theorems

5.1. In what follows, we establish analogs of the principal assertions obtained in Sections 1–4 for the spaces Lp , p ∈ (1, ∞). For this purpose, we extensively use one result obtained by Marcinkiewicz and related to the theory of operators of special form called multiplicators. These operators have already been used in Section 2.3. We refer the reader to the books by Zygmund [4, 5], Stein [1], Butzer and Nessel [1], and Edwards [1, 2] containing a vast amount of data accumulated in this field and comprehensive bibliography. Here, we restrict ourselves to the presentation of necessary definitions, most of which have already been encountered in Section 2.3, and formulate the Marcinkiewicz theorem. Let T be a set of trigonometric series of the form ∞

a0 y(x) = (ak cos kx + bk sin kx) + 2 df

k=1

(5.1)

Section 5

Marcinkiewicz, Riesz, and Hardy–Littlewood Theorems

449

and let μ = μ(k), k = 0, 1, . . . , be a fixed number sequence. Each y ∈ T is associated with an element z ∈ T as follows: df

z(x) =

∞

a0 μ0 μ(k)(ak cos kx + bk sin kx). + 2

(5.2)

k=1

Thus, every sequence μ = μ(k) specifies an operator M acting from T into T and called a multiplicator (from T into T ). Let A and B be subsets of T. By MA,B we denote the collection of multiplicators M acting from A into B. In this case, if A = B, then, by definition, we set MA,A = MA . An important role in our analysis is played by a class of multiplicators converting the Fourier series of every function f (·) from Lp with fixed p ∈ (1, ∞) into trigonometric series which are the Fourier series of certain functions from Ls , s ∈ (1, ∞). The set of multiplicators of this sort is denoted by Mp,s . Thus, if M ∈ Mp,s , then the fact that series (5.1) is S[y] and y ∈ Lp implies that series (5.2) is S[z] and z ∈ Ls . If s = p, then we set Mp,p = Mp . If y and z satisfy relations (5.1) and (5.2) and the series in these equalities are S[y] and S[z], respectively, then we write z(x) = M y(x). Further, if M ∈ Mp , then M p denotes the norm of the multiplicator M as an operator from Lp into Lp , i.e., M p = sup M y(x)p . (5.3) yp ≤1

In this notation, we can formulate the following Marcinkiewicz theorem: Theorem 5.1. Assume that, for a sequence μ = μ(k), the quantities ν0 = ν0 (μ) = sup |μ(k)|, σ0 = σ0 (μ) = sup

m+1 2

m∈N k=2m

k

|μ(k + 1) − μ(k)| (5.4)

are finite. Then the multiplicator M generated by this sequence belongs to Mp for any p ∈ (1, ∞) and, moreover, M p ≤ Cp λ, λ = λ(μ) = max(ν0 , σ0 ), where the constant Cp depends only on p.

(5.5)

450

Convergence Rate of Fourier Series and Best Approximations

Chapter 6

This theorem gives not only sufficient conditions for the validity of the inclusion M ∈ Mp but also, which is extremely important, enables one to estimate the norms M p via the characteristics ν0 (μ) and σ0 (μ) of the sequence μ = μ(k). 5.2. Theorem 5.1 immediately yields the following assertion: Corollary 5.1. Let Sn be an operator associating every function f ∈ L with its partial Fourier sum of order n − 1, i.e., Sn f (x) = Sn−1 (f, x). Then, for any p ∈ (1, ∞), Sn p = sup Sn f (x)p = sup Sn−1 (f ; x)p ≤ Cp , f p ≤1

(5.6)

f p ≤1

where the constant Cp depends only on p. Indeed, Sn is a multiplicator specified by the sequence 1, k < n, μn = μn (k) = 0, k ≥ n, such that ν0 (μn ) = 1 and σ0 (μn ) = 1. Therefore, relation (5.6) follows from (5.5). Relation (5.6), in turn, enables us to establish the following inequality (called Lebesgue inequality in approximation theory): En (f )p ≤ ρn (f ; x)p ≤ Cp En (f )p ∀f ∈ Lp , 1 < p < ∞,

(5.7)

where Cp is a constant independent of n and f. Indeed, let t∗n−1 (·) be the polynomial of the best approximation of a function f (·) in the space Lp (see Subsection 0.1). Then En (f )p ≤ ρn (f ; x)p = f (x) − t∗n−1 (x) + Sn (f − t∗n−1 ; x)p ≤ f (x) − t∗n−1 (x)p + Sn (f − t∗n−1 ; x)p ≤ En (f )p (1 + Sn p ) ≤ Cp En (f )p . Inequality (5.7) shows that, for any fixed p ∈ (1, ∞), the quantities ρn (f ; x)p and En (f )p , for f ∈ Lp , may differ only by a factor uniformly bounded in f and n, i.e., ρn (f ; x)p = O(1)En (f )p .

(5.8)

Section 5

Marcinkiewicz, Riesz, and Hardy–Littlewood Theorems

451

Hence, if N is a subset of Lp , p ∈ (1, ∞), then En (N)p = sup ρn (f ; x)p = O(1)En (N)p = O(1) sup En (f )p , f ∈N

f ∈N

(5.8 )

where O(1) are uniformly bounded in n. 5.3. We also need the Riesz theorem (Riesz [1]) presented below. Its proof can be found, e.g., in Bari [1]. Theorem 5.2. Let U be an operator associating each function f ∈ L with its trigonometrically conjugate function, i.e., U f (x) = f˜(x). Then, for any p ∈ (1, ∞), U p = sup U f (x)p = sup f˜p ≤ Cp , f p ≤1

(5.9)

f p ≤1

where Cp is a constant depending only on p. This theorem implies that if f ∈ Lp and 1 < p < ∞, then f˜ ∈ Lp . Note that similar assertions are not true for the spaces L1 , L∞ , and C. For these spaces, relations (5.6) –(5.8 ) are violated. In what follows, we also use the well-known Hardy–Littlewood theorem [3]. The detailed proof of this theorem can be found in the books by Zygmund [4, 6]. The Hardy–Littlewood theorem can be formulated as follows: Theorem 5.3. Let 1 < p < s < ∞, let α = p−1 − s−1 , and let Dα (t) =

∞

k −α cos kt.

k=1

Then, for any ϕ ∈ Lp , the convolution 1 Φα (x) = π

π ϕ(x + t)Dα (t)dt −π

belongs to Ls and, moreover, Φα s ≤ Cp,s ϕp , where Cp,s is a constant depending only on p and s.

452

Convergence Rate of Fourier Series and Best Approximations If ϕ ∈ Lp and S[ϕ] =

∞

Chapter 6

Ak (ϕ; x), then

k=0

S(Φα ) =

∞

k −α Ak (ϕ; x),

k=1

i.e., Φα (x) = M (α) ϕ(x), where M (α) is a multiplicator specified by the sequence μ(α) (k) = 0, 1−α , 2−α , . . . . Theorem 5.3 implies that M (α) ∈ Mp,s .

6.

¯

Imbedding Theorems for the Sets Lψ Lp

6.1. In what follows, we consider approximations of functions from the sets in the norm of the spaces Ls provided that 1 < p, s < ∞. The number s can be equal to, larger, or smaller than p. Therefore, it is necessary first to ¯ clarify the conditions under which the inclusion f ∈ Lψ Lp implies the inclusion f ∈ Ls . Clearly, for fixed numbers p, this is completely determined by a couple ψ¯ = (ψ1 , ψ2 ). We give the following definition: ¯ Lψ Lp

Definition 6.1. For fixed α ≥ 0, we say that a couple ψ¯ = (ψ1 , ψ2 ) of numerical sequences ψ1 (k) and ψ2 (k), k = 0, 1, . . . , ψ1 (0) = 1, and ψ2 (0) = 0 belongs to the set Pα if the quantities να (ψi ) = sup |ψi (k)|k α

(6.1)

k

and σα (ψi ) = sup

m+1 2

m∈N k=2m

|ψi (k + 1)(k + 1)α − ψi (k)k α |, i = 1, 2,

(6.2)

are bounded. The condition of boundedness of the quantities να (ψi ), can be checked fairly easily. For the quantities σα (ψi ), this problem is more complicated. However, if we somewhat restrict generality and assume that the numbers ψi (k)k α do not increase, then it becomes clear that the quantities να (ψi ) and σα (ψi ) are bounded. In what follows, we always keep this example in mind. 6.2. We now pass to the imbedding theorems and consider the case where p = s.

¯

Imbedding Theorems for the Sets Lψ Lp

Section 6

453

¯ Theorem 6.1. If ψ¯ ∈ P and p ∈ (1, ∞), then Lψ Lp ⊂ Lp . ¯

Proof. In view of equality (1.4), for any f ∈ Lψ Lp , we obtain S[f ] =

∞

∞

Ak (f ; x) =

k=0

k=1

a0 (f ) = + 2 =

a0 (f ) ¯ ¯ (ψ1 (k)Ak (f ψ ; x) + ψ2 (k)A˜k (f ψ ; x)) + 2

∞

ψ¯

μ(k)Ak (f ; x) +

k=1

∞

¯ μ ˜(k)A˜k (f ψ ; x)

k=1

a0 (f ) ¯ ˜ U f ψ¯ (x), + M f ψ (x) + M 2

(6.3)

˜ are the where U is the operator of conjugation, i.e., U ϕ = ϕ, ˜ and M and M multiplicators given by the equalities μ(k) = ψ1 (k), μ ˜(k) = ψ2 (k).

(6.4)

The condition ψ¯ ∈ P0 and Theorem 5.1 imply that the multiplicators M and ¯ ¯ ˜ M belong to Mp . Moreover, the condition f ∈ Lψ Lp gives f ψ ∈ Lp . Hence, ¯ according to Theorem 5.2, we conclude that U f ψ ∈ Lp . Therefore, a0 (f ) ¯ ˜ U f ψ¯ (x)p + M f ψ (x) + M 2 a0 (f ) ˜ p U p )f ψ¯ p ≤ Cp , ≤ p + (M p + M 2

f p =

i.e., f ∈ Lp . It is clear that Theorem 6.1 is also true for p = 2. In this case, its assertion is contained in Proposition 1.2. 6.3. If f ∈ Lp and 1 < s ≤ p < ∞, then, by virtue of the Hölder inequality, π f s = (

|f (t)|s dt)1/s ≤ (2π)(p−s)/ps f p .

−π

Hence, Theorem 6.1 implies the following statement: ¯ Corollary 6.1. If ψ¯ ∈ P0 and 1 < s ≤ p < ∞, then Lψ Lp ⊂ Ls .

6.4. We now consider the case 1 < p < s < ∞.

(6.5)

454

Convergence Rate of Fourier Series and Best Approximations

Chapter 6

Theorem 6.2. Assume that 1 < p < s < ∞, α = p−1 − s−1 , and ψ¯ ∈ Pα . ¯ Then Lψ Lp ⊂ Ls . ˜ (−α) ) the multiplicators generated Proof. Denote by (M M (−α) ) and (M M α α by sequences k μ(k) and k μ ˜(k), where μ(k) and μ ˜(k) are specified by relations (6.4). Then ∞

ψ¯

μ(k)Ak (f ; x) =

k=1

∞

¯

k α μ(k)(k −α Ak (f ψ ; x))

k=1

= (M M

(−α)

)

∞

¯

k −α Ak (f ψ ; x)

k=1

= (M M

(−α)

1 )S[ π

π

¯

f ψ (x + t)Dα (t)dt],

(6.6)

−π ¯

where Dα (t) is the function from in Theorem 5.3. Since f ψ ⊂ Lp , by virtue of Theorem 5.3, the convolution 1 g(x) = π

π

¯

f ψ (x + t)Dα (t)dt

(6.7)

−π

belongs to Ls and, since ψ¯ ∈ Pα , , we have (M M (−α) ) ∈ Mp for any p ∈ (1, ∞) by virtue of Theorem 5.1. Therefore,

∞

¯

μ(k)Ak (f ψ ; x)s = (M M −α) )S[g]s ≤ Cp,s ,

(6.8)

k=1

where Cp,s is a quantity depending only on p and s. Similarly, ∞

¯ ˜ M (−α) ) μ ˜(k)A˜k (f ψ ; x) = (M

k=1

∞

¯ k −α A˜k (f ψ ; x)

k=1

˜M = (M

(−α)

)S[˜ g ].

˜ M (−α) ) ∈ Mp , and, by Theorem 5.2, g˜ ∈ Lp , we conclude that, Since (M according to Theorem 5.3,

∞ k=1

¯ μ ˜(k)A˜k (f ψ ; x)s ≤ Cp,s .

(6.9)

Section 7

¯

Approximations of Functions from Lψ Lp by Fourier Sums

455

Combining relations (6.3), (6.8), and (6.9), we complete the proof of Theorem 6.2.

7.

¯

Approximations of Functions from the Sets Lψ Lp by Fourier Sums

7.1. We now establish estimates of the quantities ρn (f ; x)s and En (f )s ¯ in order for the functions f ∈ Lψ Lp in the case where 1 < p, s < ∞ and the couples ψ¯ belong to the set Pα with α = p−1 − s−1 and satisfy some additional conditions. First, we prove the following auxiliary assertion: Lemma 7.1. Let 1 < p, s < ∞ and let −1

α = (p

(α)

Further, let Mn

−1

−s

)+ =

−1 p − s−1 , p < s,

(7.1)

p ≥ s.

0,

˜ n(α) be multiplicators given by the sequences and M μn(α) = μ(α) n (k) =

and μ ñ(α) = μ ˜(α) n (k) =

0,

k < n,

(7.2)

k α ψ1 (k), k ≥ n, 0,

k ≥ n,

(7.2 )

k α ψ2 (k), k ≥ n, (α)

and belonging to Ms for any n ∈ N, i.e., Mn ¯ f ∈ Lψ Lp then, for any n ∈ N,

˜ n(α) ∈ Ms . If ∈ Ms , and M

¯

¯

(n) (n) En (f )s ≤ ρn (f ; x)s ≤ Kp,s ρn (f ψ ; x)p ≤ Cp Kp,s En (f ψ )p ,

(7.3)

where (n) ˜ (α) s U s ), Kp,s ≤ Cp,s (Mn(α) s + M n

(7.4)

U is an operator of conjugation, i.e., U ϕ = ϕ, ˜ and Cp and Cp,s are constants depending only on p and p and s, respectively.

456

Convergence Rate of Fourier Series and Best Approximations

Chapter 6

Proof. By using equality (1.4), we find En (f )s ≤ ρn (f ; x)s =

∞

Ak (f ; x)s

k=n

≤

∞

¯

μ(k)Ak (f ψ ; x)s +

k=n

∞

¯

μ ˜(k)A˜k (f ψ ; x)s ,

(7.5)

k=n

where μ(k) and μ ˜(k) are the sequences defined by relations (6.4). First, let s > p. In this case, we have (see (6.8)) ∞

ψ¯

μ(k)Ak (f ; x) = (M M

k=n

(−α)

1 )S[ π

π

¯

ρn (f ψ ; x + t)Dα (t)dt], −π

where (M M −α) ) is the multiplicator from Subsection 6.4. Instead of the opera(α) tor (M M (−α) ), we can use the multiplicator Mn . Hence, by virtue of Theorem 5.3,

∞

ψ¯

μ(k)Ak (f ; x)s ≤

1 Mn(α) s

π

k=n

π

¯

ρn (f ψ ; x + t)Dα (t)dts −π ¯

≤ Cp,s Mn(α) s ρn (f ψ ; x)p .

(7.6)

Similarly, by virtue of Theorem 5.2, we find

∞

ψ¯

μ ˜(k)A˜k (f ; x)s =

˜ n(α) U ( 1 M

π

k=n

≤

π

¯

ρn (f ψ ; x + t)Dα (t)dts −π

˜ (α) s U s ρn (f ψ¯ ; x)p . Cp,s M n

(7.6 )

Combining relations (7.5)–(7.6 ), we establish the intermediate estimate in (7.3). Thus, to complete the proof of Lemma 7.1 for s > p, it suffices to apply inequality (5.7). (α) (0) If p ≥ s, then α = 0, and, for k ≥ n, we get μn (k) = μn (k) = μ(k). Therefore,

∞ k=n

¯

¯

¯

μ(k)Ak (f ψ ; x)s = Mn(0) ρn (f ψ ; x)s ≤ Mn(0) s ρn (f ψ ; x)s . (7.7)

Section 7

¯

Approximations of Functions from Lψ Lp by Fourier Sums

457

Similarly, in view of Theorem 5.2, we obtain ∞

¯

¯

(7.7 )

˜ n(0) s U s ρn (f ψ ; x)s . μ ˜(k)A˜k (f ψ ; x)s ≤ M

k=n

By comparing relations (7.5), (7.7), and (7.7 ), we find ˜ (0) s U s )ρn (f ψ¯ ; x)s . En (f )s ≤ ρn (f ; x)s ≤ (Mn(0) s + M n Further, for p = s, inequality (7.3) is obtained directly by applying inequality (5.7). At the same time, for p > s, one must first use inequality (6.5). Lemma 7.1 is proved. (n) 7.2. For any fixed α ≥ 0, by Pα we denote a subset of couples ψ¯ = (ψ1 , ψ2 ) from Pα satisfying the following conditions for any natural n :

να (ψi,n ) = sup |ψi,n (k)|k α ≤ C1 νi (n)nα ,

(7.8)

k

σα (ψi , n) = sup

m+1 2

m∈N k=2m

|ψi,n (k + 1)(k + 1)α − ψi,n (k)k α | ≤ C2 νi (n)nα , (7.9)

where ψi,n =

0,

k < n,

ψi (k), k ≥ n, i = 1, 2,

(7.10)

νi (n) = ν(ψi ; n) = sup|ψi (k)|, and C1 and C2 are constants uniformly k≥n

bounded in n. (n) (α) ˜ n(α) genLemma 7.2. If ψ¯ ∈ Pα , then the multiplicators Mn and M erated by sequences (7.2) and (7.2 ) belong to Mp for any p ∈ (1, ∞) and, moreover,

˜ n(α) p ≤ Cp,α ν2 (n)nα , Mn(α) p ≤ Cp,α ν1 (n)nα , M

(7.11)

where Cp,α is a constant depending only on p and α. Proof. The proof of the lemma follows from Theorem 5.1. Indeed, according to (7.8) and (7.9), we have (α) α α ν0 (μ(α) n ) = sup |μn (k)| = sup |ψ1 (k)|k ≤ C1 ν1 (n)n , k

k≥n

(7.12)

458

Convergence Rate of Fourier Series and Best Approximations

σ0 (μ(α) n )

= sup

2m+1 −1

m∈N k=2m

Chapter 6

|ψ1,n (k + 1)(k + 1)α − ψ1,n (k)k α | (7.12 )

≤ C2 ν1 (n)nα . Since, for any n ∈ N, ν1 (n)nα ≤ sup |ψ1 (k)|k α , k∈N

the quantities ν1 (n)nα are bounded due to the inclusion ψ¯ ∈ P0 . Therefore, (α) (α) Mn ∈ Mp for any p ∈ (1, ∞) and, in view of the fact that λ = λ(μn ) ≤ Cν1 (n)nα in the analyzed case, the first inequality in (7.11) follows from estimate ˜ n(α) . (5.5). It is clear that the same reasoning remains true for the operator M 7.3. Combining the assertions of Lemmas 7.1 and 7.2, we arrive at the following statement: (n) Theorem 7.1. Let 1 < p, s < ∞, let α = (p−1 − s−1 )+ , and let ψ¯ ∈ Pα . ¯ If f ∈ Lψ Lp , then, for any n ∈ N, ¯

(1) ν(n)nα ρn (f ψ ; x)p En (f )s ≤ ρn (f ; x)s ≤ Cp,s ¯

(2) ≤ Cp,s ν(n)nα En (f ψ )p ,

(7.13)

(i) ¯ ¯ ψ(n) = (ψ12 (n) + ψ22 (n))1/2 , and Cp,s , i = 1, 2, are where ν(n) = supψ(n), k≥n

quantities uniformly bounded in n and f. Indeed, substituting (7.4) in (7.3), in view of Theorem 5.2 and estimate (7.11), we find ¯

(1) En (f )s ≤ ρn (f ; x)s ≤ Cp,s (ν1 (n) + ν2 (n))nα ρn (f ψ ; x)p ¯

(2) ≤ Cp,s (ν1 (n) + ν2 (n))nα En (f ψ )p

and, to get (7.13), it suffices to note that, for any n ∈ N, 1 (ν1 (n) + ν2 (n)) ≤ ν(n) ≤ ν1 (n) + ν2 (n). 2

(7.14)

Consider a special case of Theorem 7.1. If the numbers |ψi (k)|k α , i = 1, 2, (n) α ≥ 0, do not increase, then the couple ψ¯ belongs to Pα and, moreover, να (ψi,n ) = |ψi (n)|nα , i = 1, 2,

(7.15)

Section 7

¯

Approximations of Functions from Lψ Lp by Fourier Sums

and

459

(7.15 )

σα (ψi,n ) ≤ 2|ψi (n)|nα .

¯ Therefore, ν1 (n) = |ψ1 (n)|, ν2 (n) = |ψ2 (n)|, and ν(n) = ψ(n). Hence, the following assertion is true: Theorem 7.1. Let 1 < p, s < ∞, let α = (p−1 −s−1 )+ , and let the number ¯ sequence |ψi (k)|k α , i = 1, 2, k ∈ N, be nonincreasing. If f ∈ Lψ Lp , then, for any n ∈ N, ¯

(1) ¯ En (f )s ≤ ρn (f ; x)s ≤ Cp,s ψ(n)nα ρn (f ψ ; x)p ¯

(7.13 )

(2) ¯ ψ(n)nα En (f ψ )p , ≤ Cp,s (1)

(2)

where Cp,s and Cp,s are constants independent of n and f. ¯

¯

ψ 7.4. If f ∈ Lψ p , then, by definition, f ≤ 1 and, therefore, ¯

¯

¯

En (f ψ )p ≤ f ψ (x) − 0p ≤ f ψ (x)p ≤ 1. Hence, Theorems 7.1 and 7.1 yield the following assertion: (n) Theorem 7.2. Let 1 < p, s < ∞, let α = (p−1 − s−1 )+ , and let ψ¯ ∈ Pα . Then, for any n ∈ N, ¯

¯

α ψ En (Lψ p )s ≤ En (Lp )s ≤ Cp,s ν(n)n .

(7.16)

Moreover, in a special case of nonincreasing sequences |ψi (k)|k α , i = 1, 2, k ∈ N, , ¯ α ψ¯ ¯ En (Lψ (7.16 ) p )s ≤ En (Lp )s ≤ Cp,s ψ(n)n , where Cp,s is a constant depending only on p and s. 7.5. Let us show that, for α = 0, estimates (7.16) and (7.16 ) are exact in order and that the same assertion is true for any α > 0 provided that the functions |ψi (k)| decrease not too rapidly,. (n) First, we set p = s and, hence, α = 0. Let us show that if ψ¯ ∈ P0 , then ¯

En (Lψ p )p ≥ Cp ν(n),

(7.17)

where Cp is a constant depending only on p. For this purpose, it suffices to repeat the arguments presented in Subsection 1.5.

460

Convergence Rate of Fourier Series and Best Approximations

Chapter 6

¯ Indeed, assume that, for a given function ψ(k) and a number n ∈ N, there exists a number kn such that relation (1.15) is true. We consider a function fn (x) = a−1 (ψ1 (kn ) cos kn x + ψ2 (kn ) sin kn x), a = cos tp .

(7.18)

¯ The function fn (x) is the ψ-integral of the function a−1 cos kn x and, thus, beψ¯ longs to Lp . Moreover, we have ¯ n ) cos(kn x − θk )p = ψ(k ¯ n ) = ν(n). ρn (fn ; x)p = fn (x)p = a−1 ψ(k n Hence, relation (1.15) yields (7.17). Following the reasoning presented in Subsection 1.5, we conclude that inequality (7.17) holds in the general case. It is clear that the role of the functions fni (x) is, in this case, played by the functions fni (x) = a−1 (ψ1 (ni ) cos ni x + ψ2 (ni ) sin ni x).

(7.19)

7.6. To prove that inequalities (7.16) and (7.16 ) cannot be improved in order for s < p, it is necessary show that ¯

En (Lψ p )s ≥ Cp,s ν(n)

(n)

∀ψ¯ ∈ P0 .

(7.20)

This inequality is also established by repeating the arguments presented in Subsection 1.5. In this case, we set fn (x) = b−1 (ψ1 (kn ) cos kn x + ψ2 (kn ) sin kn x), b = (2π)s−p)/ps cos tp , and, therefore, fni (x) = b−1 (ψ1 (ni ) cos ni x + ψ2 (ni ) sin ni x). These results are summarized in the following assertion: (n) Theorem 7.3. Let 1 < s ≤ p < ∞ and let ψ¯ ∈ P0 . Then, for any n ∈ N, ¯

¯

(1) (2) ψ ν(n) ≤ En (Lψ Cp,s p )s ≤ En (Lp )s ≤ Cp,s ν(n).

(7.21)

¯ In particular, if the sequence ψ(k) is nonincreasing, then ¯ (1) ¯ (2) ¯ ψ¯ Cp,s ψ(n) ≤ En (Lψ p )s ≤ En (Lp )s ≤ Cp,s ψ(n), (1)

(2)

(7.21 )

where Cp,s and Cp,s are positive constants which may depend only on p and s.

¯

Approximations of Functions from Lψ Lp by Fourier Sums

Section 7

461

7.7. We now select a set of couples ψ¯ for which estimates (7.16) and (7.16 ) are exact in order for any α > 0. We denote this set by PB and assume that it contains all couples ψ¯ = (ψ1 , ψ2 ) for which there exists a constant K such that ν(n) ¯ ≤ K ∀n ∈ N, ν(n) = sup ψ(k) ¯ n≤k≤2n ψ(k) k≥n max

(7.22)

and the following conditions are satisfied: sup

m+1 2

m∈N k=2m

where

|hi (k + 1) − hi (k)| ≤ K, i = 1, 2,

⎧ 0, ⎪ ⎪ ⎨ hi (k) = hi (k, n) =

(7.23)

0 ≤ k ≤ n − 1, k > 2n,

ν(n)ψi (k) ⎪ ⎪ , n ≤ k ≤ 2n. ⎩ ¯2 ψ (k)

(7.24)

(n) 7.8. Let ψ¯ ∈ Pα ∩ PB . We now show that, in this case, there exists a ¯ function f ∗ (·) in the class Lψ p such that, for any n ∈ N,

En (f ∗ )s ≥ Cp,s ν(n)nα , α = p−1 − s−1 , α > 0.

(7.25)

To do this, for fixed n ∈ N, we consider a function fn (t) = ν(n)gn (t), gn (t) =

2n

cos kt

(7.26)

k=n

¯ and find its ψ-derivative. We have ¯ fnψ (t)

2n ψ1 (k) ψ2 (k) = ν(n) ( ¯2 cos kt − ¯2 sin kt) ψ (k) ψ (k) k=n

=

(Hn(1)

− Hn(2) U )gn (t),

(7.27)

(i)

where Hn , i = 1, 2, are multiplicators generated by sequences (7.24) and U, as above, is the operator of conjugation. By virtue of conditions (7.22) and (7.23) and Theorems 5.1 and 5.2, we conclude that, for any p ∈ (1, ∞), Hn(1) − Hn(2) U p ≤ Cp ,

(7.28)

462

Convergence Rate of Fourier Series and Best Approximations

Chapter 6

where Cp is a constant depending only on p. Therefore, relation (7.27) implies that ¯ fnψ p ≤ Cp gn p . (7.29) In what follows, we need the following assertion: Lemma 7.3. Let n, q ∈ N, q > n, and let p ∈ (1, ∞). Then Cp(1) (q − n)(p−1)/p ≤

q

cos ktp ≤ Cp(2) (q − n)(p−1)/p ,

(7.30)

k=n (1)

where Cp

(2)

and Cp

are positive constants depending only on p.

Proof. As usual, the Dirichlet kernel of order m is denoted by Dm (t). In view of relation (1.3.1), we get q

cos kt = Dq (t)−Dn−1 (t) =

k=n

sin((q − n + 1)t/2) cos((q + n)t/2) . (7.31) sin t/2

For t ∈ (0, π/2), we have sin t > 2t/π. Therefore,

q

π cos ktp ≤ (2

k=n

|

sin((q − n + 1)t/2) p 1/p | dt) sin t/2

0

≤(

(q−n+1)π/2

πp

(q − n + 1)p−1

22(p−1)

|

sint p 1/p | dt) t

0

≤ Cp (q − n + 1)

(p−1)/p

≤

Cp(2) (q

− n)(p−1)/p ,

and the second inequality in (7.30) is proved. Let us now establish the lower bound. Since q > n , by using the inequality sin t ≤ t and relation (7.31), we find

q

(q−n+1)π/2

cos ktp ≥ (4(q − n + 1)

|

1−p

k=n

0

≥ Cp (q − n)

(1−p)/p

π/2 ( | cos 0

sin t p q+n | | cos t|p dt)1/p t q−n+1

q+n t|p dt)1/p . q−n+1

¯

Approximations of Functions from Lψ Lp by Fourier Sums

Section 7

463

This immediately implies the first inequality in (7.30) because, for any a > 1, aπ/2 π/2 π/2 | cos at|p dt = a−1 | cos t|p dt ≥ a−1 [a] cosp tdt > Cp , 0

0

0

where [a] is the integral part of the number a and Cp is a constant depending only on p. Lemma 7.3 is proved. 7.9. By setting q = 2n in inequalities (7.30), in view of (7.29), we conclude that

¯

fnψ (·)p ≤ Cp n(p−1)/p .

(7.32)

This means that the function f ∗ (t) = fn∗ (t) = Cp−1 ν(n)n(1−p)/p gn (t), ¯

where Cp is the same as in (7.32), belongs to the set Lψ p. ¯ In the analyzed case, according to Theorem 6.2, we have Lψ p ⊂ Ls . Hence, by virtue of inequalities (5.7) and (7.30), we obtain En (f ∗ )s ≥ Cs ρn (f ∗ ; x)s −1 −s−1

= Cs Cp−1 ν(n)n(1−p)/p gn (x)s ≥ Cp,s ν(n)np

.

This proves inequality (7.25). Combining relations (7.16) and (7.25), we arrive at the following assertion: Theorem 7.4. Let 1 < p < s < ∞, let α = p−1 − s−1 , and let ψ¯ ∈ (n) Pα ∩ PB . Then, for any n ∈ N, ¯

¯

(1) (2) α ψ Cp,s ν(n)nα ≤ En (Lψ p )s ≤ En (Lp )s ≤ Cp,s ν(n)n .

(7.33)

α is nonincreasing and ψ ¯ ∈ PB , then ¯ In particular, if the sequence ψ(k)k ¯ (1) ¯ (2) ¯ α ψ¯ ψ(n)nα ≤ En (Lψ Cp,s p )s ≤ En (Lp )s ≤ Cp,s ψ(n)n , (1)

(7.33 )

(2)

where Cp,s and Cp,s are positive constants which may depend only on p and s. 7.10. As indicated in Subsection 7.3, any couple ψ¯ = (ψ1 , ψ2 ) in which the (n) sequences |ψi (k)|k α , i = 1, 2, are nonincreasing belongs to Pα . At the same

464

Convergence Rate of Fourier Series and Best Approximations

Chapter 6

time, the condition ψ¯ ∈ PB is not necessarily satisfied for all pairs of this sort. One can easily see that inequality (7.22) cannot be true, e.g., for the functions ψ1 (k) = ψ2 (k) = exp(−k), k ∈ N. If we restrict ourselves to the case of convex (down) sequences, then we can indicate simple sufficient conditions for ψi (k) guaranteeing the validity of the inclusion ψ¯ ∈ PB . Assume that ψi (k) are the values of some functions ψi (v) of continuous argument for v = k. Then the following assertion is true in the notation of Section 3.12: Proposition 7.1. Let ψ1 ∈ M0 and let ψ2 = λψ1 , where λ is an arbitrary real number. Then ψ¯ = (ψ1 , ψ2 ) ∈ PB . Proof. It is necessary to show that inequalities (7.22) and (7.23) hold under the conditions of the proposition. Since ψ1 ∈ M0 , the function ¯ ψ(k) = (ψ12 (k) + λ2 ψ12 (k))1/2 = ψ1 (k) 1 + λ2 is convex down and satisfies relation (3.16.1). Hence, for any n ∈ N, ¯ ψ(n) ν(n) = ≤ K, ¯ ¯ n≤k≤2n ψ(k) ψ(2n) max

i.e., inequality (7.22) holds. The proof of relation (7.23) is also quite simple. Since ν(n) = ψ1 (n) 1 + λ2 and ψ¯2 (k) = ψ12 (k)(1 + λ2 ), we have ⎧ 0 ≤ k ≤ n − 1, k > 2n, ⎪ ⎨0, h1 (k) = ψ1 (n) 1 ⎪ , n ≤ k ≤ 2n, ⎩√ 2 ψ 1 + λ 1 (k) and

⎧ 0 ≤ k ≤ n − 1, k > 2n, ⎪ ⎨0, h2 (k) = λψ1 (n) 1 ⎪ , n ≤ k ≤ 2n. ⎩√ 1 + λ2 ψ1 (k)

Section 7

¯

Approximations of Functions from Lψ Lp by Fourier Sums

465

Therefore, for any m ∈ N, by virtue of relation (3.16.1), we get m+1 2

|h1 (k + 1) − h1 (k)|

k=2m

≤ h1 (n) + h1 (2n) +

2n

|h1 (k + 1) − h1 (k)|

k=n 2n

1 ψ1 (n) 1 2 ( | =√ − |+1+ ) 2 ψ1 (2n) 1 + λ k=n ψ1 (k + 1) ψ1 (k) ≤

4ψ1 (n) ≤ K. ψ1 (2n)

It is clear that the same inequality also holds for h2 (k). 7.11. Assume that ψ1 (t) = ψ(t) cos βπ/2, ψ2 (t) = ψ(t) sin βπ/2 and ψ ∈ M0 . Then, by virtue of Proposition 7.1, ψ¯ = (ψ1 , ψ2 ) ∈ PB . Further, (n) if ψ(t) = t−r , r > 0, and, in addition, r ≥ α ≥ 0, then ψ¯ ∈ Pα , i.e., in this (n) case, ψ¯ ∈ Pα ∩ PB . Thus, Theorems 7.3 and 7.4 imply the following assertion: Corollary 7.1. If 1 < p, s < ∞, α = (p−1 − s−1 )+ , and r ≥ α, then (1) α−r r r (2) α−r n ≤ En (Wβ,p )s ≤ En (Wβ,p )s ≤ Cp,s n , Cp,s (1)

(7.34)

(2)

where Cp,s and Cp,s are positive constants depending only on p and s and r is the class Lψ¯ for ψ (k) = k −r cos βπ/2 and ψ (k) = k −r sin βπ/2. Wβ,p p 1 2 Theorem 7.4 and Proposition 7.1 give the exact order of the quantities En (Lψ β,p )s also in the case where ψ(v) is an arbitrary function from M0 provided that the product ψ(v)v α is nonincreasing. However, as indicated in the next section, if the functions ψ1 (k) and ψ2 (k) decrease faster than any power function, then inequalities (7.33) are impossible and, thus, should be revised.

466

8.

Convergence Rate of Fourier Series and Best Approximations

Chapter 6

Best Approximations of Infinitely Differentiable Functions ¯

8.1. In this section, we establish the exact orders of the quantities En (Lψ p )s ¯ ψ¯ ψ and En (Lp )s , 1 ≤ p, s ≤ ∞, as n → ∞ in the case where the sets L consist of infinitely differentiable functions (see Subsection 5.14.2). First, we prove the following statement: Lemma 8.1. Assume that a series ∞

(ψ1 (k) cos kx + ψ2 (k) sin kx)

(8.1)

k=1

is the Fourier series of a bounded function Ψ(x). If 1 ≤ p, s ≤ ∞, then, for any ¯ f ∈ Lψ Lp , ∞ ¯ ¯ ρn (f ; x)s ≤ 4En (f ψ )s ψ(k). (8.2) k=n

Proof. The proof of the lemma is obtained by repeating the proof of Theorem 5.15.1. Denote by M∞ the subset of functions ψ ∈ M∞ for which the quantity η(ψ; t) − t is bounded from above, namely, M∞ = {ψ ∈ M∞ : η(ψ; t) − t ≤ K ∀t ≥ 1}.

(8.3)

We now prove the following assertion: Proposition 8.1. If ±ψ1 , ±ψ2 ∈ M∞ , then the couple ψ¯ = (ψ1 , ψ2 ) satisfies the conditions of Lemma 8.1 and, in addition, for any n ∈ N, ∞

¯ ¯ ψ(k) ≤ K ψ(n),

(8.4)

k=n

where K is an absolute constant. Proof. It suffices to establish relation (8.4). By setting n0 = n, ni+1 = ¯ η(ψ; ni ), i = 0, 1, . . . , we find ∞ k=n

¯ ¯ ψ(k) ≤ ψ(n) +

∞ n i+1 i=0 k=ni +1

¯ ψ(k).

(8.5)

Section 8

Best Approximations of Infinitely Differentiable Functions

467

¯ t)], t ≥ 1, the value of the function ψ(·) ¯ decreases In every interval [t, η(ψ; ¯ ¯ exactly twofold. Therefore, if k ∈ [ni + 1, ni+1 ], then ψ(k) ≤ ψ(n)2−i . Moreover, according to Proposition 5.18.1, the inclusion ±ψi ∈ M∞ , i = 1, 2, implies ¯ t) − t ≤ K for any t ≥ 1 and, hence, the estimate η(ψ; ∞

¯ ¯ ψ(k) ≤ K1 ψ(n)

k=n

∞

¯ 2−i ≤ K ψ(n).

i=0

Proposition 8.1 is proved. Combining this proposition with Lemma 8.1, we arrive at the following statement: Theorem 8.1. Assume that 1 ≤ p ≤ s ≤ ∞ and ±ψi ∈ M∞ , i = 1, 2. If ¯ f ∈ Lψ Lp , then, for any n ∈ N, ¯

ψ ¯ En (f )s ≤ ρn (f ; x)s ≤ K ψ(n)E n (f )p ,

(8.6)

where K is an absolute constant. The functions ψr (v) = exp(−αv r ), α > 0, with various values of r ≥ 1 (in this case, η(ψr ; t) = (α−1 ln 2 + tr )1/r ) are typical representatives of the set M∞ . Therefore, Theorem 8.1 yields the following assertion: ¯

ψ ψ Corollary 8.1. Let 1 ≤ p ≤ s ≤ ∞. If f ∈ Lψ β Lp , where Lβ Lp = L Lp for ψ1 (v) = exp(−αtr ) cos βπ/2 and ψ2 (v) = exp(−αtr ) sin βπ/2, α > 0, r ≥ 1, then, for any n ∈ N,

En (f )s ≤ ρn (f ; x)s ≤ K exp(−αnr )En (fβψr )p ,

(8.7)

where K is an absolute constant. 8.2. By analyzing the upper bounds of all parts of relation (8.6) in the class we find ¯ ψ¯ ¯ (8.9) En (Lψ p )s ≤ En (Lp )s ≤ K ψ(n),

¯ Lψ p ,

1 ≤ p ≤ s ≤ ∞,

±ψi ∈ M∞ ,

i = 1, 2.

These estimates are exact in order. To prove this, it suffices to consider the function fn (x) = a−1 p (ψ1 (n) cos nx + ψ2 (n) sin nx), ap = cos tp .

(8.10)

468

Convergence Rate of Fourier Series and Best Approximations

Chapter 6

For this function, we have ¯

fnψ (x)p = a−1 p cos nxp = 1 and

−1 ¯ ¯ ρn (fn ; x)s = a−1 p ψ(n) cos(nx + θn )s = ap as ψ(n).

¯ where Cs is a quantity Therefore, in view of (5.7), En (fn )s ≥ Cs ψ(n), depending only on s. Hence, the following assertion is true by virtue of Theorem 7.3: Theorem 8.2. If ±ψi ∈ M∞ , then, for 1 ≤ p, s ≤ ∞, there exist positive (1) (2) quantities Cp,s and Cp,s depending possibly only on p and s such that, for any n ∈ N, ¯ ¯ ¯ ¯ ≤ En (Lψ )s ≤ En (Lψ )s ≤ C (2) ψ(n). (8.11) C (1) ψ(n) p,s

p

p

p,s

¯

8.3. Recall that, in Section 5.15, we consider the quantities En (Lψ p )s in the case where ±ψi , i = 1, 2, belong to the set F0 . In this case, we establish the ¯ ψ¯ asymptotic equalities for the quantities En (Lψ p )s and, hence, for En (Lp )s (see Theorem 5.15.2). Clearly, the set F0 is contained in M∞ . ¯

8.4. We now restrict ourselves to the analysis of the quantities En (Lψ p )s and ψ¯ ¯ En (Lp )s for the couples ψ = (ψ1 , ψ2 ) in which ψi , i = 1, 2, are chosen from the set M. In this case, as in Subsection 7.11, Theorem 7.4 and Proposition 7.1 give the exact order of these quantities as n → ∞ if ψ1 ∈ M0 and ψ2 (t) = λψ1 (t) provided that the product ψ1 (k)k α , α = (p−1 −s−1 )+ , is nonincreasing. Theorem 8.2 establishes the exact order of these quantities for ±ψi ∈ M∞ . The ) = M \ (M0 ∪ M ) consists of the functions ψ ∈ M for remaining subset M ∞ which the quantities η(t) − t and t/(η(t) − t) are unbounded from above. For this reason, we consider the set M∞ formed by the functions ψ ∈ M∞ with the following properties: (a) η(ψ; t) − t ≥ K1 ∀t ≥ 1,

(8.12)

ψ(v)dv ≤ K2 ψ(t)(η(ψ; t) − t) ∀t ≥ 1,

(8.13)

∞ (b) t

where K1 and K2 are positive constants.

Section 8

Best Approximations of Infinitely Differentiable Functions

469

As typical representatives of the set M∞ , we can mention the functions ψr (v) = exp(−αv r ), α > 0, for any r ∈ (0, 1). In this case, as already indicated, η(ψr ; t) = (α−1 ln 2 + tr )1/r , and the validity of conditions (8.12) and (8.13) is checked fairly easily. In connection with condition (8.13), we note that, for any ψ ∈ M and t ≥ 1, ∞

η(ψ;t)

ψ(v)dv > ψ(η(ψ; t))(η(ψ; t) − t)

ψ(v)dv > t

t

1 = ψ(t)(η(ψ; t) − t), 2

(8.14)

and, hence, the indicated condition is satisfied for η (ψ; t) ≤ q < 2 ∀t ≥ 1.

(8.15)

Indeed, by setting t0 = t, and ti = η(ψ; ti−1 ), i = 1, 2, . . . , we obtain ∞ t

t ∞ i+1 ∞ ψ(v)dv = ψ(v)dv ≤ ψ(ti )(ti+1 − ti ). i=0 t i

i=0

At the same time, ψ(ti ) ≤ 2−i ψ(t) and, in view of (8.15), ti+1 − ti ≤ q i (t1 − t0 ) = q i (η(ψ; t) − t). Consequently, ∞ t

∞ q ψ(v)dv ≤ ψ(t)(η(ψ; t) − t) ( )i ≤ Kψ(t)(η(ψ; t) − t), 2 i=0

i.e., condition (8.15) implies inequality (8.13). The following statement is also true: Proposition 8.2. If a function ψ(·) belongs to the set M+ ∞ , then it satisfies condition (8.13). Consequently, a part of the set M+ whose elements satisfy ∞ inequality (8.12) belongs to M∞ . Proof. Recall that, in Section 3.12, the subset of functions ψ ∈ M∞ for which the quantities t μ(ψ; t) = η(ψ; t) − t

470

Convergence Rate of Fourier Series and Best Approximations

Chapter 6

monotonically and infinitely increase is denoted by M+ ∞ (as typical representa+ tives of the set M∞ , we can mention the functions ψr (t) = exp(−αtr ) for any α > 0 and r > 0 ). Thus, let ψ ∈ M+ ∞ . In view of (3.12.24), for any ψ ∈ M, we have ψ(t) ≤ −2ψ (t)(η(t) − t) and

η(t) = η(ψ; t).

Therefore, for any ψ ∈ M+ ∞ , we get ∞

∞ ψ(v)dv ≤ −2

i(t) = t

ψ (t)(η(v) − v)dv

t

∞ = 2ψ(v)(η(v) −

v)|t∞

+2

ψ(v)(η (v) − 1)dv

t

∞ ≤ 2ψ(t)(η(t) − t) + 2

ψ(v)(η (t) − 1)dv.

(8.16)

t

Since the function μ(ψ, t) monotonically and infinitely increases, we conclude that η(t)−t = tα(t), where α(t) is a function monotonically approaching zero as t → ∞. Therefore, η (t) − 1 < α(t) and, hence, in view of (8.16), i(t) < 2ψ(t)(η(t) − t) + 2α(t)i(t) or (1 − 2α(t))i(t) < 2ψ(t)(η(t) − t), which yields (8.13). 8.4. Let us now prove the following assertion: Theorem 8.3. Let ±ψ1 , ±ψ2 ∈ M∞ , let 1 < p, s < ∞, and let α = (p−1 − s−1 )+ . Assume that the following condition is satisfied: 0 < K1 ≤

η(ψ1 ; t) − t ≤ K2 < ∞ ∀t ≥ 1. η(ψ2 ; t) − t

(8.17)

¯

Then, for any f ∈ Lψ Lp and n ∈ N, ¯ ¯ − n)α En (f ψ )p , En (f )s ≤ ρn (f ; x)s ≤ Cp,s ψ(n)(η(n)

(8.18)

Section 8

Best Approximations of Infinitely Differentiable Functions

471

¯ where ψ(n) = ψ12 (n) + ψ22 (n), η(n) is either η(ψ1 ; n) , or η(ψ2 ; n), or ¯ η(ψ; n), and Cp,s is a quantity which may depend only on p and s. If, in addition, |ψ2 (t)|(η(ψ2 ; t) − t) ≤ K|ψ1 (t)|(η(ψ1 ; t) − t) ∀t ≥ 1,

(8.19)

¯

then, for any f ∈ Lψ Lp and n ∈ N, ¯

En (f )s ≤ ρn (f ; x)s ≤ Cp,s |ψ1 (n)|(η(ψ1 ; n) − n)α En (f ψ )p ,

(8.18 )

where Cp,s is a quantity which may depend only on p and s. Proof. If ±ψi ∈ M∞ , i = 1, 2, then the couple ψ¯ = (ψ1 , ψ2 ) belongs to Thus, for p ≥ s, inequalities (8.18) and (8.18 ) follow from Theorem 7.1. Hence, it suffices to prove the theorem for α > 0, i.e., for s > p. Condition (8.13) guarantees the imbedding M∞ ⊂ M and, hence, for the representation of the quantity ρn (f ; x) , one can use Corollary 4.3.1 according to which the ¯ following equality holds almost everywhere for all f ∈ Lψ : (n) P0 .

∞ Δ(x; t)(J2 (ψ1 ; n; t)0 + J2 (ψ2 ; n; t)1 )dt

ρn (f ; x) = −∞

1 + 2

π Δ(x; t)(ψ1 (n) cos nt + ψ2 (n)) sin ntdt, −π

where 1 J2 (ψ1 ; n; t)0 = π J2 (ψ1 ; n; t)1 =

1 π

∞ ψ1 (v) cos vtdv, n ∞

ψ2 (v) sin vtdv, n

and

¯

Δ(x; t) = f ψ (x − t) − tn−1 (x − t), where, in turn, tn−1 (·) is an arbitrary trigonometric polynomial of order n − 1. As in Section 5.11, we now set ∞ ρn (f ; x)1 = −∞

ψ1 (n) Δ(x, t)J2 (ψ1 ; n; t)0 dt + 2π

π Δ(x; t) cos ntdt, (8.20) −π

472

Convergence Rate of Fourier Series and Best Approximations ∞

ρn (f ; x)2 = −∞

ψ2 (n) Δ(x, t)J2 (ψ2 ; n; t)1 dt + 2π

π

Chapter 6

Δ(x; t) sin ntdt. (8.20 )

−π

Therefore, ρn (f ; x) = ρn (f ; x)1 + ρn (f ; x)2 .

(8.21)

Let us show that if ±ψi ∈ M∞ , i = 1, 2, and 1 < ρ < s < ∞, then, for ¯ any f ∈ Lψ Lp , ¯

(i) |ψi (n)|(η(ψi ; n) − n)α En (f ψ )p , i = 1, 2. ρn (f ; x)i s ≤ Cp,s

(8.22)

Assume that α = α(n) = η(ψ1 ; n) − n. Then (see Subsection 5.11.1) we get ρn (f ; x)1 = R0 (f ; x) + R1 (f ; x) + R2 (f ; x) + R3 (f ; x), where

ψ1 (n) R0 (f ; x) = − π

Δ(x, t)

(8.23)

sin nt dt, t

(8.24)

|t|≥α(n)

R1 (f ; x) =

Δ(x, t)J2 (ψ1 ; n; t)0 dt,

(8.25)

|t|≤α(n)

1 R2 (f ; x) = − π

Δ(x, t)J3 (ψ1 ; t)0 dt,

(8.26)

|t|≥α(n)

1 J3 (ψ1 ; t)0 = t

∞

ψ1 (v) cos vtdv,

n

and ψ1 (n) R3 (f ; x) = 2π

π Δ(x; t) cos ntdt.

(8.27)

−π ¯

We now estimate the quantities Ri (f ; x)s , i = 0, 3, for f ∈ Lψ Lp . The required estimate is especially simple for R3 (f ; x)s . Indeed, by applying the Young inequality for convolutions (see Subsection 5.15.1), in view of the fact that ¯ Δp ≤ En (f ψ )p , we find |ψ1 (n)| R3 (f ; x)s ≤ 2π

π Δ(x; t) cos ntdts

−π

≤

|ψ1 (n)| ¯ Δp cos q ≤ |ψ1 (n)|En (f ψ )p . 2π

(8.28)

Section 8

Best Approximations of Infinitely Differentiable Functions

473

¯

Further, we show that, for any f ∈ Lψ Lp and n ∈ N, ¯

R0 (f ; x)s ≤ Cp,s |ψ1 (n)|(η(ψ1 ; n) − n)α En (f ψ )p ,

(8.29)

α = (p−1 − s−1 )+ . We choose a natural number k from the condition 2(k − 1)π < α(n) ≤ 2k π. Then, according to (8.24),

|ψ1 (n)| R0 (f ; x)s = π

Δ(x; t)

(8.30)

sin nt dts t

|t|≥α(n)

2|ψ1 (n)| ≤ π

2k π

|Δ(x; t)|

dt s t

α(n) ∞

ψ1 (n) + ( π k=k

2(k+1)π

Δ(x, t) 2kπ

−2kπ

Δ(x, t)

+

sin nt dt t

sin nt dt)s . t

(8.31)

−2(k+1)π

By virtue of the Young inequality, the first term on the right-hand side of this relation satisfies the inequality 2|ψ1 (n)| Δp ( π

2k π

dt 1/q −1 ¯ ) ≤ Cq |ψ1 (n)|(η(ψ1 ; n) − n)1−q En (f ψ )p . (8.32) q t

α(n)

At the same time, for the second term, we can write ∞

ψ1 (n) π

2π |Δ(x; t)|

k=k 0

|ψ1 (n)| ≤ 2π 2

2|t − π| dts (t + 2kπ)(2(k + 1) − t)

2π |Δ(x; t)| 0

∞

k −2 dts

k=k ¯

≤ Cq |ψ1 (n)|Δp ≤ Cq |ψ1 (n)|En (f ψ )p .

(8.33)

474

Convergence Rate of Fourier Series and Best Approximations

Chapter 6

In view of the equality 1 − q −1 = α, relations (8.31)–(8.33) imply estimate (8.29). ¯ Further, we prove that, for any f ∈ Lψ Lp and n ∈ N, ¯

R1 (f ; x)s ≤ Cp,s |ψ1 (n)|(η(ψ1 ; n) − n)α En (f ψ )p .

(8.34)

We have R1 (f ; x)s α(n)

≤

0

Δ(x; t)J2 (ψ1 ; n; t)0 dts + −α(n)

0 (1)

Δ(x; t)J2 (ψ1 ; n; t)0 dts

(2)

= R1 + R1 .

(8.35)

In view of inequality 5.11.15, for any n ∈ N and t > 0, 1 |J2 (ψ1 ; n; t)0 | ≤ π

n+2π/t

df

|ψ1 (v)|dv = ϕn (t)

(8.36)

n

and, therefore, α(n)

(1)

R1 =

Δ(x; t)J2 (ψ1 ; n; t)0 dts 0

≤

−2 k

2(k+1)π

k=0

|Δ(x; t)|ϕn (t)dts 2kπ α(n)

+

|Δ(x; t)|ϕn (t)dts .

(8.37)

2(k −1)π

where k is the number from condition (8.30). Clearly, we assume that k > 1. If k = 1, then the sum on the right-hand side of (8.37) is absent. By applying the Young inequality, we obtain (1) R1

−2 k

≤ Δp (

k=0 ¯

2(k+1)π

ϕqn (t)dt)1/q

( 2kπ

≤ k En (f ψ )p Qn ,

α(n)

+(

ϕqn (t)dt)1/q )

2(k −1)π

(8.38)

Section 8

Best Approximations of Infinitely Differentiable Functions

475

where, in view of the notation made in (8.36), α(n)

1 = ( π

ϕqn (t)dt)1/q

Qn = (

α(n) n+2π/t

0

=

|ψ1 (v)|dv)q dt)1/q

( n

0

∞

(2π)1/q π

n+t

2π(η(ψ1 ;n)−n)

∞ ≤ Cq (

|ψ1 (v)|dv)q

(

dt 1/q ) t2

n

t ( |ψ1 (v)|dv)q

η(ψ1 ;n) n

dt )1/q . (t − n)2

Hence, by using (8.13), we find ∞ Qn ≤ Cq |ψ1 (n)|(η(ψ1 ; n) − n)(

dt )1/q (t − n)2

η(ψ1 ;n)

≤ Cq(1) |ψ1 (n)|(η(ψ1 ; n) − n)1−1/q .

(8.39)

We substitute this estimate in (8.38) and take into account the fact that, in view of inequality (8.12), the number k is uniformly bounded in n. This gives (1) R1 ≤ Cp,s |ψ1 (n)|(η(ψ1 ; n) − n)α . (2) It is clear that the same estimate is also true for R1 . Thus, relation (8.34) is proved. ¯ Finally, we prove that, for any f ∈ Lψ Lp and n ∈ N, R2 (t, x)s ≤ Cp,s |ψ1 (n)|(η(ψ1 ; n) − n)α En (f ψ )p .

(8.40)

We have 1 R2 (f ; x)s ≤ π

∞ Δ(x, t)J3 (ψ1 ; t)0 dts

α(n)

1 + π df

=

(1) R2

−α(n)

Δ(x, t)J3 (ψ1 ; t)0 dts −∞ (2)

+ R2 .

(8.41)

476

Convergence Rate of Fourier Series and Best Approximations

Chapter 6

We now set x0 = α(n) and xk = x0 + 2kπ, k ∈ N. By using inequality (4.3.29), we obtain (1) R2

1 ≤ π

x1

1 |Δ(x; t)||J3 (ψ1 ; t)0 |dts + π

x0

1 ≤ π

∞ |Δ(x; t)||J3 (ψ1 ; t)0 |dts x1

x1 |Δ(x; t)|(ψ1 (n) − ψ(n + 2π/t))tdts x0

+

2|ψ1 (n)|

∞ |Δ(x; t)|

dt s t2

x1

|ψ1 (n)| ≤ π

x1 |Δ(x; t)|

dt s t

x0

x1 |Δ(x; t)|

+ 2|ψ (n)|

∞

(t + 2kπ)−2 dts .

k=1

x0

Further, in view of the Young inequality, we get

(1) R2

x1 x ∞ |ψ1 (n)| 2|ψ1 (n)| −q 1/q ≤ Δp ( t dt) + Δp ( ( (t + 2kπ)−2 )q )1/q dt π π x0

x0

k=1

¯

¯

≤ Cq |ψ1 (n)|(η(ψ1 ; n) − n)α En (f ψ )p + Cq(1) |ψ1 (n)|En (f ψ )p .

(8.42)

At the same time, by virtue of inequality (8.12), for any t ≥ 1, there exists a number K > 0 such that 1 |ψ1 (η(ψ1 ; t))| = |ψ1 (t)| = 2

η(ψ 1 ;t)

|ψ (t)|dt

t

≥

|ψ1 (η(ψ1 ; t))|(η(ψ1 ; t)

− t) ≥ K|ψ (η; t)|.

This means that the inequality |ψ1 (n)| ≤ K1 |ψ1 (n)| holds for all n ∈ N, . Substituting this estimate in (8.42), we conclude that (1)

¯

R2 ≤ Cp,s |ψ1 (n)|(η(ψ1 ; n) − n)α En (f ψ )p .

(8.43)

Section 8

Best Approximations of Infinitely Differentiable Functions

477

(2)

It is clear that the same estimate is true for the quantity R2 and, hence, inequality (8.40) is proved. Combining relations (8.23)–(8.29), (8.34), and (8.40), we arrive at relation (8.22) for i = 1. Acting in the same way (see Subsection 5.11.5), we also establish (8.22) for i = 2. Thus, relation (8.22) is proved. Further, if condition (8.17) is satisfied, then, in view of (8.21) and (8.22), relation (8.18) is also true for η(n) = η(ψ1 ; n) or ¯ n) η(n) = η(ψ2 ; n) . Since, according to Proposition 5.18.1, the value of η(ψ; lies between the values of η(ψ1 ; n) and η(ψ2 ; n), relation (8.18) remains true ¯ n). Finally, if condition (8.19) is satisfied, then (8.22) implies for η(n) = η(ψ; estimate (8.18 ). Theorem 8.3 is proved. Remark 8.1. Theorem 8.3 remains valid and its proof does not change if 1 ≤ p < s < ∞, i.e., the number p can take the value 1 but in the case where s > p. 8.4. Theorem 8.1 yields the following assertion:

¯

Corollary 8.2. Let 1 < p < s < ∞, let α = p−1 − s−1 , and let Lψ β Lp =

Lψ Lp for ψ1 (t) = ψ(t) cos βπ/2 and ψ2 = ψ(t) sin βπ/2. If, in addition, f ∈ Lψ β Lp , ψ ∈ M∞ , and β is an arbitrary real number, then, for any n ∈ N, En (f )s ≤ ρn (f ; x)s ≤ Cp,s ψ(n)(η(ψ; n) − n)−α .

(8.44)

Moreover, if, in particular, ψ2 (t) = exp(−γtr ), γ > 0, 0 < r ≤ 1, then En (f )s ≤ ρn (f ; x)s ≤ Cp,s exp(−γnr )n(1−r)α .

(8.44 )

In relations (8.44) and (8.44 ), Cp,s is a quantity which may depend only on p and s. Indeed, relation (8.44) follows from (8.18) because, in this case, inequalities (8.17) are true. Inequalities (8.44 ) follow from (8.44) because, for ψr (t) = exp(−γtr ), we can write η(ψr ; t) − t = t1−r (

ln 2 + O(1)), rγ

(8.45)

where O(1) is a quantity uniformly bounded in t ≥ 1. 8.5. We now establish analogs of Theorems 7.3, 7.4, and 8.2 for the case where ±ψi ∈ M0 , i = 1, 2. To do this, we find the upper bounds of all parts

478

Convergence Rate of Fourier Series and Best Approximations

Chapter 6

¯

of inequalities (8.18) and (8.18 ) in the class Lψ Lp and take into account the fact ¯ ¯ that the inequality En (f ψ )p ≤ 1, n = 1, 2, . . . , holds for all f ∈ Lψ Lp . As a result, we arrive at the following assertion: Theorem 8.4. Let ±ψ1 , ±ψ2 ∈ M∞ , let 1 < p, s < ∞, and let α = (p−1 − s−1 )+ . If, in addition, condition (8.17) is satisfied, then ¯

¯

(i) ¯ α ψ (8.46) En (Lψ p )s ≤ En (Lp )s ≤ Cp,s ψ(n)(η(n) − n) , ¯ where ψ(n) = ψ12 (n) + ψ22 (n), η(n) is either η(ψ1 ; n), or η(ψ2 ; n), or (i) ¯ n), and Cp,s η(ψ; , i = 1, 2, are quantities which may depend only on p and s. Furthermore, if condition (8.19) is satisfied, then ¯

¯

α ψ En (Lψ p )s ≤ En (Lp )s ≤ Cp,s |ψ1 (n)|(η(ψ1 ; n) − n) .

(8.46 )

8.6. For ±ψi ∈ M∞ , i = 1, 2, inequalities (8.46) and (8.46 ) cannot be improved in order. More precisely, the following theorem is true: Theorem 8.5. Let ψ1 ∈ M∞ and ψ2 (t) = λψ1 (t), where λ is an arbitrary real number. Further, let 1 < p, s < ∞ and α = (p−1 − s−1 )+ . Then ¯ α ¯ En (Lψ p )s ≥ Cp,s ψ(n)(η(n) − n) ,

(8.47)

where the quantities Cp,s and η(n) have the same meaning as in Theorem 8.4. Proof. If p ≥ s, then the assertion of the theorem follows from Theorem 7.3. Thus, it remains to prove inequality (8.47) in the case where p < s, i.e., for α > 0. It is clear that, in order to prove inequality (8.47), it suffices to indicate ¯ ∗ a function f ∗ ∈ Lψ p for which there exists a positive quantity Cp,s , which may depend only on p and s such that ∗ ¯ ψ(n)(η(n) − n)α . ρn (f ∗ ; x)s ≥ Cp,s

(8.48)

¯ n)], where [α] We act as in Subsection 7.8. For fixed n, we set (n) = [η(ψ; is the integral part of α, and consider the function ¯ fn (t) = ψ(n)g (n) (t), g(n) (t) =

(n) k=n

cos kt.

(8.49)

Section 8

Best Approximations of Infinitely Differentiable Functions

479

It is clear that, without loss of generality, we can assume that the number ¯ η(ψ; n) is greater, e.g., than n + 2 because, otherwise, relation (8.48) takes the form (8.11). For the function fn (t), we get (see Subsection 7.8) ¯ fnψ (t)

¯ = ψ(n)

(n) ψ2 (k) ψ1 (k) ( ¯2 cos kt − ¯2 sin kt) ψ (k) ψ (k)

k=n (1)

(2)

= (H(n) − H(n) U )g(n) (t),

(8.50)

(i)

where U is the operator of conjugation and H(n) , i = 1, 2, are multiplicators given by the sequences hi (k) = h(k, (n)) ⎧ 0, 0 ≤ k ≤ n − 1, k > (n), ⎪ ⎪ ⎨ = ¯ ψ(n)ψi (k) λi−1 ψ1 (n) ⎪ ⎪ √ = , n ≤ k ≤ (n), i = 1, 2. ⎩ ¯2 ψ (k) ψ1 (k) 1 + λ2

(8.51)

Let us show that, for any p ∈ (1, ∞), (1)

(2)

H(n) − H(n) U p ≤ Cp .

(8.52)

To this end, in view of Theorems 5.1 and 5.2, it suffices to show that the quantities ν0 (hi ) and σ0 (hi ), i = 1, 2, specified by relations (5.4) are bounded uniformly in n. We have λ λi−1 ψ1 (n) ψ1 (n) √ ≤√ ≤ 2. 2 ψ n≤k≤(n) ψ1 (k) 1 + λ2 1 + λ 1 (η(n))

ν0 (hi ) = max

(8.53)

(Here we take into account the fact that, in the analyzed case, η(ψ1 ; t) = ¯ t).) η(ψ2 ; t) = η(ψ; σ0 (hi ) = max m∈N

m+1 2

|hi (k + 1) − hi (k)|

k=2m

≤ hi (n) + hi ((n)) +

(n) k=n

|hi (k + 1) − hi (k)|

480

Convergence Rate of Fourier Series and Best Approximations

Chapter 6

(n)−1 1 λi−1 ψ1 (n) 1 1 2 ( ( = √ − )+ + ) ψ1 (n) ψ1 ((n)) 1 + λ2 k=n ψ1 (k + 1) ψ1 (k)

4ψ1 (n) ≤ 8. ψ1 (η(n))

≤

(8.54)

Therefore, inequality (8.52) is proved. However, in this case, by virtue of (8.50) and Lemma 7.3, we have ¯

(1)

(2)

fnψ (x)s ≤ H(n) − H(n) U p gn (x)p ≤ Kp ((n) − n)(p−1)/p . Hence, the function ¯ f ∗ (t) = fn∗ (t) = Kp−1 ((n) − n)(1−p)/p fn (t) = Kp−1 ((n) − n)(1−p)/p ψ(n)g n (t) ¯

belongs to Lψ p and, according to Lemma 7.3, satisfies the inequality ¯ − n)α ρn (f ∗ ; x)s = f ∗ (x)s ≥ Cp,s ψ(n)((n) ((n) − n)α ¯ − n)α = Cp,s ψ(n)(η(n) . (η(n) − n)α Thus, to prove (8.48), it remains to show that, for any α > 0, there exists a constant Kα which may depend only on α such that ((n) − n)α > Kα . (η(n) − n)α This is obvious. Indeed, since η(n) − n > 2, we have ((n) − n)α ([η(n)] − n)α η(n) − 1 − n α = ≥( ) > 2α . α α (η(n) − n) (η(n) − n) η(n) − n Theorem 8.5 is proved. 8.7. If ψ1 (v) = ψ(v) cos βπ/2, ψ2 (v) = ψ(v) sin βπ/2, ψ ∈ M∞ , and β ∈ R1 , then the functions ψ1 (·) and ψ2 (·) satisfy all assumptions of Theorems 8.3 and 8.4. Hence, the following assertion is true: Theorem 8.6. Let 1 < p, s < ∞, let α = (p−1 − s−1 )+ , and let ψ ∈ M∞ . (1) (2) Then there exist positive constants Cp,s and Cp,s depending only on p and s such that, for any n ∈ N and β ∈ R1 , ψ (1) ψ(n)(η(n) − n)α ≤ En (Lψ Cp,s β,p )s ≤ En (Lβ,p )s

Section 9

Jackson Inequalities in the Spaces C and Lp

481

(2) ≤ Cp,s ψ(n)(η(n) − n)α ,

(8.55)

where η(n) = η(ψ; n) = ψ −1 ( 12 ψ(n)). In view of the fact that the function ψr (t) = exp(−γtr ), γ > 0, r > 0, satisfies equality (8.45), by using Theorems 8.6 and 8.2, we arrive at the following assertion: Corollary 8.3. Let ψr (t) = exp(−γtr ), γ > 0, r > 0, let 1 < p, s < ∞, (1) (2) and let α = (p−1 − s−1 )+ . Then there exist positive quantities Cp,s and Cp,s independent of n and such that, for any n ∈ N, ψr (1) r (1 + nα(1−r) ) exp(−γnr ) ≤ En (Lψ Cp,s β,p )s ≤ En (Lβ,p )s (2) ≤ Cp,s (1 + nα(1−r) ) exp(−γnr ).

9.

(8.55 )

Jackson Inequalities in the Spaces C and Lp

9.1. Let En (f ) and En (f )p be the values of the best approximations by trigonometric polynomials tn−1 (·) of degree not higher than n−1 for continuous functions f ∈ C in the uniform metric and functions f ∈ Lp in the metric of the space Lp , respectively, namely, En (f ) =

inf

f (x) − tn−1 (x)C

inf

f (x) − tn−1 (x)p .

tn−1 ∈T2n−1

and En (f )p =

tn−1 ∈T2n−1

The following assertion is true for the quantities En (f ) and En (f )p : Theorem 9.1. There exists an absolute constant K such that, for any n ∈ N,

and

En (f ) ≤ Kω(f ; n−1 ) ∀f ∈ C

(9.1)

En (f )p ≤ Kωp (f ; n−1 ) ∀f ∈ Lp , 1 ≤ p < ∞.

(9.2)

Proof. Clearly, to prove the theorem, it suffices to indicate specific polynomials Rn (f ; x) of degree not higher than n − 1 satisfying the inequalities f (x) − Rn (f ; x)C ≤ Kω(f ; n−1 ) ∀f ∈ C

(9.1 )

482

Convergence Rate of Fourier Series and Best Approximations

Chapter 6

and f (x) − Rn (f ; x)p ≤ Kωp (f ; n−1 ) ∀f ∈ Lp , 1 ≤ p < ∞.

(9.2 )

As polynomials of this sort, we use the Rogosinski sums 1 Rn (f ; x) = [Sn (f ; x + π/2n) + Sn (f ; x − π/2n)]. 2 Representing the partial Fourier sums in terms of convolutions with Dirichlet kernel Dn (t), we get 1 Rn (f ; x) = π

π f (x + t)Rn (t)dt, −π

where 1 sin π/2n · cos nt Rn (t) = [Dn (t − π/2n) + Dn (t + π/2n)] = 2 2(cos t − cos π/2n)

(9.3)

is a trigonometric polynomial of degree n−1 called the Rogosinski kernel. Since π Rn (t)dt = 1, we have −π

1 rn (f ; x) = f (x) − Rn (f ; x) = π df

π [f (x) − f (x + t)]Rn (t)dt.

(9.4)

−π

This equality is true for all x if f ∈ C and for almost all x in a period if f ∈ L. In the interval (π/2n, π), the function (cos t − cos π/2n)−1 is nondecreasing. Therefore, the function π x

sin π/2n Rn (t)dt = 2

∞ x

cos nt dt cos t − cos π/2n

possesses a unique simple zero xk in each interval [(2k+1)π/2n, (2k+3)π/2n], k = 0, 1, . . . , n − 2, and, hence, π/2n ≤ x0 < 3π/2n < x1 < . . . < xn−2 < (2n − 1)π/2n.

(9.5)

Jackson Inequalities in the Spaces C and Lp

Section 9

483

df

If, in addition, we set xn−1 = π, then xk+1

Rn (t)dt = 0, k = 0, n − 1.

(9.6)

xk

In view of these results and the fact that the kernel Rn (t) is even, by virtue of (9.4), we obtain 1 rn (f ; x) ≤ π

x0 (f (x) − f (x + t))Rn (t)dt

−x0

1 + π

n−2

xk+1

[f (x + t) + f (x − t)]Rn (t)dt, (9.7)

k=0 x k

where · denotes the norm in the space C or Lp . If f ∈ C, then 1 π

x0

−x0

2 [f (x) − f (x + t)]Rn (t)dtC ≤ π

x0 ω(f ; t)|Rn (t)|dt 0

2 ≤ ω(f ; x0 ) π

x0 |Rn (t)|dt.

(9.8)

0

At the same time, if f ∈ Lp , 1 ≤ p < ∞, then, by using the Minkowski inequality b d d b p 1/p ( | f (x, y)dy| dx) ≤ ( |f (x, y)|p dx)1/p dy, 1 ≤ p < ∞, a

c

c

(9.9)

a

we get 1 π

x0

−x0

1 [f (x) − f (x + t)]Rn (t)dtp ≤ π

x0 f (x) − f (x + t)p |Rn (t)|dt −x0

2 ≤ ωp (f ; x0 ) π

x0 |Rn (t)|dt. 0

(9.8 )

484

Convergence Rate of Fourier Series and Best Approximations

Chapter 6

We now establish similar estimates for each term of the sum in (9.7). In each interval [xk , xk+1 ], k = 0, n − 2, by virtue of (9.6), the function Rn (t) satisfies the requirements imposed on the function ϕ(t) in Lemma 5.2.2 for c = ck = (2k + 3) π/2n. Therefore, the functions ρk (x) defined by the equalities x

ρk (x)

Rn (t)dt, xk ≤ x ≤ ck ≤ ρk (x) ≤ xk+1

Rn (t)dt = xk

xk

are strictly monotone and absolutely continuous (see Subsection 5.2.2). Moreover, Rn (ρn (x))ρ (x) = Rn (t) and ρk (xk ) = xk+1 , ρk (ck ) = ck , almost everywhere in (xk , ck ). Hence, by setting t = ρ(x), we can write xk+1

[f (x + t) + f (x − t)]Rn (t)dt ck

ck = − {f [x + ρk (t)] + f [x − ρk (t)]}Rn (t)dt. xk

and, consequently, xk+1

[f (x + t) + f (x − t)]Rn (t)dt

xk

ck = ({f (x + t) − f [x + ρk (t)]} xk

+ {f (x − t) − f [x − ρk (t)]})Rn (t)dt. Thus, if f ∈ C, then xk+1

ck

[f (x + t) + f (x − t)]Rn (t)dtC ≤ 2 xk

ω[ρk (t) − t]|Rn (t)|dt.

(9.10)

xk

At the same time, if f ∈ Lp , 1 ≤ p < ∞, then, in view of inequality (9.9), we get a similar estimate, namely,

Jackson Inequalities in the Spaces C and Lp

Section 9 xk+1

ck

[f (x + t) + f (x − t)]Rn (t)dtp ≤ 2

xk

485

ωp [ρk (t) − t]|Rn (t)|dt. (9.10 )

xk

Note that, by virtue of (9.5), for t ∈ [xk , ck ], we get ρk (t)−t ≤ xk+1 −xk < 2π/n and x0 < 3π/2n and, by virtue of (9.6), we find ck

1 |Rn (t)|dt = 2

xk

xk+1

|Rn (t)|dt. xk

Thus, it follows from inequalities (9.7)–(9.10 ) that x0 π 2 2π 1 rn (f ; x)C < ω( )( |Rn (t)|dt + |Rn (t)|dt) π n 2 x0

0

= ω(

2π 1 )( n π

π

1 |Rn (t)|dt + ). 2

0

The same inequality is also true for rn (f ; x)p but with ωp (2π/n) instead of ω(2π/n). Thus, if we use the fact that 2 π

π |Rn (t)|dt = sup Rn (f ; x)C = Ln (R), |f |≤1

0

where Ln (Rn ) are Lebesgue constants of the Rogosinski method (as shown in Subsection 1.3.4, the collection of these constants is uniformly bounded), then, in order to prove estimates (9.1 ) and (9.2 ), it remains to use the inequalities ω(2π/n) ≤ (2π + 1)ω(n−1 ), which remain valid for the moduli ωp (f ; t) (see Subsections 3.1.3 and 3.3.1). 9.2. To characterize the properties of smoothness of the functions, parallel with the moduli of continuity ω(f ; x) and ωp (f ; x) introduced in Sections 3.1 and 3.3, one can also use the higher-order moduli of continuity, which are usually called smoothness moduli. Let X be either C or Lp , 1 ≤ p < ∞, and let k ∈ N. The quantity Δkh (f ; x)

=

k ν=1

k−ν

(−1)

ν k

f (x + νh)

486

Convergence Rate of Fourier Series and Best Approximations

Chapter 6

is called the difference of a function f ∈ X of order k with step h ∈ R1 . The smoothness modulus of order k is defined as a function ω (k) (f ; t) = sup Δkh (f ; x)C |h|≤t

for f ∈ C or as a function ωp(k) (f ; t) = sup Δkh (f ; x)p |h|≤t

for f ∈ Lp . It is clear that, for k = 1, we have ω (1) (f ; t) = ω(f ; t) and (1) ωp (f ; t) = ωp (f ; t). Note that Theorem 9.1 admits the following strengthening: Theorem 9.2. Let k ∈ N. Then there exists a constant Ck depending only on k such that, for any n ∈ N,

and

En (f ) ≤ Ck ω (k) (f ; n−1 ) ∀f ∈ C

(9.11)

En (f )p ≤ Ck ωp(k) (f ; n−1 ) ∀f ∈ Lp , 1 ≤ p < ∞.

(9.12)

The detailed proofs of inequalities (9.11) and (9.12) can be found, e.g., in the books by Dzyadyk [5] and Timan [5]. ¯

9.3. Substituting estimates (9.2) and (9.12) instead of the quantities En (f ψ )p in relations (7.13), (7.13 ), (8.6), (8.7), (8.18), and (8.18 ), we obtain Jackson-type inequalities for the corresponding classes of functions. We now formulate basic results for the first modulus of continuity. (n) Theorem 9.3. Let 1 < p, s < ∞, let α = (p−1 − s−1 )+ , and let ψ¯ ∈ Pα . ¯ ψ If f ∈ L Lp , then, for any n ∈ N, ¯

En (f )s ≤ ρn (f ; x)s ≤ Cp,s ν(n)nα ωp (f ψ ; n−1 ),

(9.13)

where Cp,s is a constant depending only on p and s. Theorem 9.3. Let 1 < p, s < ∞, let α = (p−1 −s−1 )+ , and let |ψi (k)|k α , ¯ i = 1, 2, k ∈ N, be nonincreasing sequences of numbers. If f ∈ Lψ Lp , then, for any n ∈ N, ¯ α ¯ En (f )s ≤ ρn (f ; x)s ≤ Cp,s ψ(n)n ωp (f ψ ; n−1 ).

(9.14)

Jackson Inequalities in the Spaces C and Lp

Section 9

487

Theorem 9.4. Let 1 ≤ p ≤ s ≤ ∞ and let ±ψi ∈ M∞ , i = 1, 2. If ¯ f ∈ Lψ Lp , then, for any n ∈ N, ψ¯ −1 ¯ En (f )s ≤ ρn (f ; x)s ≤ K ψ(n)ω p (f ; n ),

(9.15)

where K is an absolute constant. Theorem 9.5. Let ±ψ1 , ±ψ2 ∈ M∞ , let 1 < p, s < ∞, and let α = ¯ (p−1 − s−1 )+ . Assume that condition (8.17) is satisfied. Then, for any f ∈ Lψ Lp and n ∈ N, ¯

¯ − n)α ωp (f ψ ; n−1 ). En (f )s ≤ ρn (f ; x)s ≤ Cp,s ψ(n)(η(n)

(9.16) ¯

Furthermore, if inequality (8.19) is satisfied, then, for any f ∈ Lψ Lp and n ∈ N, ¯

En (f )s ≤ ρn (f ; x)s ≤ Cp,s |ψ1 (n)|(η(ψ1 ; n) − n)α ωp (f ψ ; n−1 ).

(9.17)

The quantities Cp,s and η(n) have the same meaning as in Theorem 8.3. 9.4. The constants Cp,s in relations (9.13)–(9.17) are uniformly bounded in ¯ n ∈ N and f ∈ Lψ Lp . Thus, under corresponding conditions, relations (9.13)– ¯ (9.17) yield the following estimates for the upper bounds of En (Lψ Hωp )s in the ¯ ¯ classes Lψ Hωp = {f : f ψ ∈ Hωp = (ϕ : ωp (ϕ; t) ≤ ω(t))} : ¯

¯

En (Lψ Hωp )s ≤ En (Lψ Hωp )s ≤ Cp,s ν(n)nα ω(1/n),

(9.13 )

¯ ¯ α ¯ ω(1/n), En (Lψ Hωp )s ≤ En (Lψ Hωp )s ≤ Cp,s ψ(n)n

(9.14 )

¯ ¯ ¯ En (Lψ Hωp )s ≤ En (Lψ Hωp )s ≤ K ψ(n)ω(1/n), ¯

¯

¯ − n)α ω(1/n), En (Lψ Hωp )s ≤ En (Lψ Hωp )s ≤ Cp,s ψ(n)(η(n)

(9.15 ) (9.16 )

and ¯

¯

En (Lψ Hωp )s ≤ En (Lψ Hωp )s ≤ Cp,s |ψ1 (n)|(η(ψ1 ; n) − n)α ω(1/n). (9.17 )

7. BEST APPROXIMATIONS IN THE SPACES C AND L 0.1. In the present chapter, we consider the best approximations in the spaces C and L, i.e., En (f )X =

inf

Tn−1 ∈Tn−1

f (·) − Tn−1 (·)X ,

(0.1)

where X is either C or L, in more details. In these spaces, unlike the spaces Lp , 1 < p < ∞, the approximations given by Fourier sums are, in general, not the best approximations (in order). Moreover, as already indicated, there are continuous functions whose Fourier series are divergent. ∗ (·) for which The existence of a polynomial Tn−1 ∗ f (·) − Tn−1 (·)X = En (f )X ,

(0.2)

i.e., the polynomial of best approximation in the space X, follows from Subsection 5.0.1. However, the problem of finding this polynomial t∗n−1 (·) for a given function f (·) and evaluation of expression (0.1) is, as a rule, quite complicated and, most often, absolutely hopeless. At the same time, it is often possible to indicate the exact order as n → ∞ and, in some important cases, find the exact value of the quantities En (N)X = sup En (f )X , (0.3) f ∈N

where N is a class of functions from X. The key problem encountered in this case is connected with the analysis of the characteristic properties of a polynomial delivering, for a given function f (·), the approximation En (f )X or, at least, an approximation of the same order as En (f )X . For the space C, this problem was completely solved by Chebyshev in his famous theorem on the criterion of polynomial of the best uniform approximation. The proof of this theorem can be found, in fact, in almost all books devoted to approximation theory and, hence, we present it without proof. 489

490

Best Approximations in the Spaces C and L

Chapter 7

Later, a criterion of polynomial of the best approximation was also obtained for the space L (and, clearly, for Lp , 1 < p < ∞ ). On the basis of these criteria, we determine the orders of the best approximations in the spaces C and L for the sets of functions Cβψ and Lψ β . As in Chapter 5, we assume that the sequences ψ(k) are generated by the functions ψ(v) of continuous argument v ≥ 1 from the set M of convex (down) functions.

1.

Chebyshev and de la Vallée Poussin Theorems

1.1. A criterion of polynomial of the best approximation in the space C is given by the following Chebyshev theorem [1]: ∗ (t) deliver the Theorem 1.1. In order that a trigonometric polynomial Tn−1 best uniform approximation of a function f ∈ C in the set of all trigonometric polynomials Tn−1 (t) of degree not higher than n − 1, it is necessary and sufficient that its period 0 ≤ t < 2π contain 2n points tk : 0 ≤ t1 < t2 < . . . < ∗ (t) reaches t2n < 2π, where the modulus of the difference Δ(t) = f (t) − Tn−1 its maximum value Δ = ΔC with alternating signs, i.e., either

Δ(tk ) = (−1)k Δ or Δ(tk ) = (−1)k+1 Δ, k = 1, 2, . . . . 1.2. The Chebyshev theorem immediately implies the uniqueness of the polynomial of the best uniform approximation of fixed degree for each function (1) (2) f ∈ C. Indeed, let Tn−1 (t) and Tn−1 (t) be such that (1)

(2)

En (f )C = f (·) − Tn−1 (·)C = f (·) − Tn−1 (·)C . (1)

(2)

Then the polynomial [Tn−1 (·) + Tn−1 (·)]/2 also delivers the best approximation of the function f, i.e., (1)

(2)

En (f )C ≤ f (·) − [Tn−1 (·) + Tn−1 (·)]/2 1 (1) (2) = [f (·) − Tn−1 (·)] + [f (·) − Tn−1 (·)]C ≤ En (f )C 2 and, by virtue of Theorem 1.1, one can find 2n points tk , k = 1, 2n, in the period such that (1)

(2)

[f (tk ) − Tn−1 (tk )] + [f (tk ) − Tn−1 (tk )] = ±2En (f )C .

(1.1)

Section 1

Chebyshev and de la Vallée Poussin Theorems

491

The absolute value of each term on the left-hand side does not exceed En (f )C . Therefore, equalities in (1.1) are true only in the case where (1)

(2)

f (tk ) − Tn−1 (tk ) = f (tk ) − Tn−1 (tk ) = ±En (f )C , i.e., where

(1)

(2)

Tn−1 (tk ) = Tn−1 (tk ), k = 1, 2, . . . , 2n. Thus, to obtain the required statement it suffices to apply the well-known fact that if two trigonometric polynomials T1 (x) and T2 (x) of degree n coincide at more than 2n points in a period [a, a + 2π), then they are identically equal. 1.3. To estimate the value of the best uniform approximation from below, it is customary to use the following de la Vallée Poussin theorem [1]: Theorem 1.2. Assume that f ∈ C and, for a polynomial Tn−1 ∈ T2n−1 , there exist 2n points tk , 0 ≤ t1 < t2 < . . . < t2n < 2π, in the period [0, 2π), where the difference Δ(t) = f (t) − Tn−1 (t) takes values with alternating signs, i.e., (1.2) sgn Δ(tk ) = −sgn (tk+1 ), k = 1, 2n. Then En (f )C ≥ min |Δ(tk )|. 1≤k≤2n

(1.3)

Proof. We denote the value of the right-hand side of (1.3) by η and assume ∗ (·) is the contrary, i.e., that En (f )C < η. This enables us to conclude that if Tn−1 the polynomial of the best approximation for f (·), then the following inequality is satisfied: ∗ |f (tk ) − Tn−1 (tk )| < η ≤ |f (tk ) − Tn−1 (tk )|, k = 1, 2n.

(1.4)

∗ ∗ (t) − tn−1 (t) = Δ(t) − (f (t) − Tn−1 (t)). rn−1 (t) = Tn−1

(1.5)

Let

Then it follows from relations (1.4) and (1.5) that sgn rn−1 (tk ) = sgn Δ(tk ), k = 1, 2n, and, hence, by virtue of the condition of the theorem, the trigonometric polynomial rn−1 (·) of degree n − 1 must have at least, 2n − 1 zeros in the interval [0, 2π), which is impossible because, in view of (1.4), rn−1 (t) ≡ 0.

Best Approximations in the Spaces C and L

492

2.

Chapter 7

Polynomial of the Best Approximation in the Space L

2.1. The following assertion characterizes the polynomial of the best approximation in the space L : ∗ (t) deliver Theorem 2.1. In order that the trigonometric polynomial Tn−1 the best approximation of a function f ∈ L in the metric of the space L, i.e., ∗ f (t) − Tn−1 (t)1 =

inf

Tn−1 ∈T2n−1

f (t) − Tn−1 (t)1 = En (f )1 ,

(2.1)

it is sufficient and, in the case where ∗ (t)} = 0, mes{t : f (t) = Tn−1

(2.2)

it is necessary that 2π

∗ Qn−1 (t)sgn (f (t) − Tn−1 (t))dt = 0,

(2.3)

0

for any trigonometric polynomial Qn−1 (t) of degree n − 1. ∗ (t). Proof. Assume that the conditions of the theorem are satisfied for Tn−1 Then, by virtue of (2.3), for any Qn−1 ∈ T2n−1 , we get

2π

2π ∗ |f (t) − Qn−1 (t)|dt ≥ (f (t) − Qn−1 (t))sgn (f (t) − Tn−1 (t))dt

0

0

2π =

∗ |f (t) − Tn−1 |dt.

0

Since the polynomial of the best approximation exists (see Subsection 6.0.1), we conclude that 2π |f (t) − 0

∗ Tn−1 (t)|dt

2π =

|f (t) − Qn−1 |dt = En (f )1 ,

inf

Qn−1 ∈T2n−1

0

which proves that the conditions of the theorem are sufficient.

Section 2

Polynomial of the Best Approximation in the Space L

493

∗ (t) satisfies relations (2.1) and We now assume that the polynomial Tn−1 (2.2). Let us show that, in this case, relation (2.3) is also true. Assume the contrary, i.e., that relation (2.3) is not satisfied. Then, by virtue of (2.2), there exist a polynomial Q∗n−1 ∈ T2n−1 and a number ε sufficiently small in the absolute value such that

2π ε

Q∗n−1 (t) sgn [Δ(t) − εQ∗n−1 (t)]dt > 0,

∗ Δ(t) = f (t) − Tn−1 (t).

0

However, in this case, 2π

2π |Δ(t)|dt =

0

2π Δ(t) sgn Δ(t)dt ≥

0

0

2π |Δ(t) −

=

Δ(t) sgn [Δt − εQ∗n−1 (t)]dt

εQ∗n (t)|dt

2π +ε

0

Q∗n−1 sgn [Δ(t) − εQ∗n−1 (t)]dt,

0

i.e., 2π |f (t) −

∗ Tn−1 (t)|dt

0

2π >

∗ |f (t) − Tn−1 (t) − εQ∗n−1 (t)|dt.

(2.4)

0

∗ (t) − εQ∗ Since the sum Tn−1 n−1 (t) is a polynomial of degree not higher ∗ (t) is not the polynomial of the best than n − 1, relation (2.4) means that Tn−1 approximation, which contradicts condition (2.1). n we denote the set of functions ϕ ∈ 2.2. For every n = 0, 1, 2, . . . , by HM SM , i.e., such that ϕM ≤ 1 , orthogonal for (n ≥ 1) to all trigonometric polynomials of degree n − 1, namely,

π

π ϕ(t) cos ktdt =

−π

ϕ(t) sin ktdt = 0,

k = 0, 1, . . . , n − 1.

(2.5)

−π

We now prove the following statement: Proposition 2.1. If a function f ∈ L is periodic with period 2π/n, then π

π f (t) cos ktdt =

−π

f (t) sin ktdt = 0, −π

k = 1, 2, . . . , n − 1.

(2.6)

Best Approximations in the Spaces C and L

494

Chapter 7

In addition, if f ⊥ 1, i.e., π f (t)dt = 0, −π

then f (·) is orthogonal to all trigonometric polynomials of degree n − 1. Indeed, by setting x = t + 2π/n, we get π

π ikx

e

2ikπ/n

f (x)dx = e

−π

eikt f (t)dt,

−π

or

π (1 − e

2ikπ/n

eikt f (t)dt = 0.

) −π

However, for k = 1, n − 1, we have 1 − e2ikπ/n = 1 − cos 2kπ/n − i sin 2kπ/n = 0 and, hence, π

π ikt

e −π

f (t)dt =

f (t)(cos kt + i sin kt)dt = 0, −π

which yields relation (2.6). The second assertion is obvious. Corollary 2.1. If a function ϕ ∈ SM , ϕ ⊥ 1, is periodic with period 2π/n, n . then ϕ ∈ HM ∗ (t) be the polynomial of the best approximation of a function f ∈ L Let Tn−1 ∗ (t). According to (2.3), the function in the space L and let Δ(t) = f (t) − Tn−1 ∗ n h (t) = sgn Δ(t) belongs to HM and, therefore,

π [f (t) −

En (f )1 = −π

∗ Tn−1 (t)]h∗ (t)dt

π

π

∗

f (t)h (t)dt ≤ sup

= −π

n h∈HM

f (t)h(t)dt. −π

Section 3 General Facts on the Approximations of Classes of Convolutions 495 On the other hand, π sup

n h∈HM

π f (t)h(t)dt = sup

n h∈HM

−π

π ≤

∗ [f (t) − Tn−1 (t)]h(t)dt

−π

∗ |f (t) − Tn−1 (t)|dt = En (f )1

−π

and, hence, π En (f )1 =

inf

Tn−1 ∈T2n−1

f (t) − Tn−1 (t)1 = sup

n h∈HM

f (t)h(t)dt.

(2.7)

−π

This equality enables us to reduce the problem of finding the best approximations in the space L to the evaluation of the upper bound in the dual space L∗ of linear functionals given on L. These equalities are called duality relations.

3.

General Facts on the Approximations of Classes of Convolutions 3.1. If ψ ∈ F, then, by virtue of Theorems 3.13.4, 3.7.2 and 3.7.3, a series ∞

ψ(k) cos(kt + βπ/2)

(3.1)

k=1

is the Fourier series of a summable function Dψ,β (t) for any β ∈ R1 . If ψ ∈ M0 , then, by Theorem 3.7.2, this series is also the Fourier series of a summable function Dψ,0 (t). Therefore, if ψ ∈ F, then, by virtue of Propositions 3.7.1 and 3.7.2, for any β ∈ R1 , the set Lψ β coincides with the set of functions representable in the form a0 (f ) 1 f (x) = + 2 π

π ϕ(x + t)Dψ,β (t)dt = −π

a0 (f ) + ϕ ∗ Dψ,β . 2

(3.2)

The function ϕ(·) coincides with fβψ (·) almost everywhere. If ψ ∈ M0 , then the same conclusion is true at least for β = 0. This yields the following assertion:

496

Best Approximations in the Spaces C and L

Chapter 7

Proposition 3.1. Assume that ψ ∈ F and β ∈ R1 or ψ ∈ M0 and β = 0. Also let N ⊂ L. Then the class Lψ β N coincides with the class of functions f (·) representable in the form of equality (3.2) with ϕ ∈ N and the class Cβψ N coincides with the class of continuous functions representable in the same form. 3.2. Numerous general procedures are developed for the investigation of expressions of the form (0.1) in the sets of functions representable in the form of convolutions (3.2). In what follows, we present some of these procedures. Let K(t) be a summable 2π-periodic function, let ∞

α0 (αk cos kt + βk sin kt) + 2

(3.3)

k=1

be its Fourier series, and let N be a subset of functions ϕ from L. By N∗K we denote the set of functions f (x) sourcewise representable via the kernel K(t), i.e., functions of the form a0 1 f (x) = + 2 π

π ϕ(x + t)K(t)dt = −π

a0 + ϕ ∗ K(t), 2

ϕ ∈ N.

(3.4)

Let us now prove the following assertion: Proposition 3.2. If f ∈ M ∗ K, then πEn (f )C ≤ En (ϕ)M En (K)1

(3.5)

πEn (f )C ≤ ϕM En (K)1 .

(3.6)

πEn (f )1 ≤ En (ϕ)1 En (K)1

(3.7)

πEn (f )1 ≤ ϕ1 En (K)1 .

(3.7 )

and, in particular, If f ∈ L ∗ K, then

and, in particular,

Proof. For any polynomials Tn−1 (·) and tn−1 (t) of degree n − 1, we have (ϕ − Tn−1 ) ∗ (K − tn−1 ) = f (·) − τn−1 (·),

(3.8)

Section 3 General Facts on the Approximations of Classes of Convolutions 497 where τn−1 (·) is a polynomial of degree n − 1. Therefore, 1 En (f )C ≤ π

π [ϕ(x + t) − τn−1 (x + t)][K(t) − tn−1 (t)]dtC

−π

≤

1 ϕ(x) − Tn−1 (x)M · K(t) − tn−1 (t)1 , π

which immediately yields (3.5). Similarly, by virtue of (3.8), we get 1 En (f )1 ≤ π

π π | [ϕ(t) − Tn−1 (t)][K(t − x) − tn−1 (t − x)]dt|dx −π −π

1 ≤ sup π |g|≤1 1 ≤ π ≤

π

π g(x)

−π

[ϕ(t) − Tn−1 (t)][K(t − x) − tn−1 (t − x)]dtdx

−π

π

π |ϕ(t) − Tn−1 (t)| sup |

−π

|g|≤1

g(t − x)[K(x) − tn−1 (x)]dx|dt

−π

1 ||ϕ(x) − Tn−1 (x)1 K(t) − tn−1 (t)1 , π

which proves (3.7). Relations (3.6) and (3.7 ) follow from (3.5) and (3.7) because En (ϕ) ≤ ϕX . 3.3. In the general case, estimates (3.5)–(3.7 ) are not exact. At the same time, under certain conditions imposed on the function K(t) (fairly general, as becomes clear in what follows), they are unimprovable. Following Nikol’skii, we say that the kernel K(t) of the form (3.3) satisfies the condition A∗n if, for any fixed n ∈ N, there exist a trigonometric polynomial t∗n−1 (t) of degree not higher than n − 1 and a natural number n∗ > n such that the function ϕ∗ (t) = sgn K∗ (t),

K∗ (t) = K(t) − t∗n−1 (t),

(3.9)

satisfies the equality ϕ∗ (t + αn ) = −ϕ∗ (t),

αn = π/n∗ ,

(3.10)

498

Best Approximations in the Spaces C and L

Chapter 7

almost everywhere in t . The polynomial t∗n−1 (t) appearing in the condition A∗n realizes the best approximation of the kernel K(t) in the metric of L, i.e., the following assertion is true: Proposition 3.3. If K(t) satisfies the condition A∗n , then En (K)1 = K(t) − t∗n−1 (t)1 .

(3.11)

To prove this, we first note that, by virtue of (3.10), the function ϕ∗ (t) is periodic with period 2αn and, therefore, 2π

2π ϕ∗ (t)dt = −

0

ϕ∗ (t + αn )dt 0 2π+α n

2π

ϕ∗ (t)dt = −

=− αn

ϕ∗ (t)dt = 0.

(3.12)

0

In this case, according to Proposition 2.1, the function ϕ∗ (t) is orthogonal to all polynomials Qn−1 (·) from T2n−1 . Therefore, for any Qn−1 ∈ T2n−1 , we get π [K(t) − Qn−1 (t)]sgn [K(t − Qn−1 (t)]dt

K(t) − Qn−1 (t)1 = −π

π [K(t) − Qn−1 (t)]Y∗ (t)dt

≥ −π

π =

[K(t) − t∗n−1 (t)]Y∗ (t)dt = K(t) − t∗n−1 (t)1 ,

−π

which yields (3.11). Thus, if the kernel K(t) satisfies the condition A∗n , then the polynomial of its best approximation in the space L interpolates K(t) at at least 2n equidistant points t∗i of the period so that the sign of the difference K∗ (t) in the intervals (ti , ti+1 ) is alternating. This fact is decisive in finding the quantities En (K)1 used, in turn, to determine the quantities characterizing the best approximations for numerous classes of functions.

Section 3 General Facts on the Approximations of Classes of Convolutions 499 3.4. Let us now show that estimates (3.6) and (3.7 ) cannot be improved under the condition A∗n . Theorem 3.1. If the kernel K(t) satisfies the condition A∗n , then df

En (SM ∗ K)C = sup En (f )C = ϕM ≤1

and df

En (S1 ∗ K)1 = sup En (f )1 = ϕ1 ≤1

1 En (K1 ) π

(3.13)

1 En (K)1 , π

(3.14)

where f (·) and ϕ(·) satisfy relation (3.4). Proof. By virtue of (3.6) and (3.7 ), it suffices to show that 1 En (K)1 π

(3.13 )

1 En (K)1 . π

(3.14 )

En (SM ∗ K)C ≥ and En (S1 ∗ K)1 ≥ To prove (3.13 ), we consider the function 1 f∗ (x) = π

π −π

1 ϕ∗ (x + t)K(t)dt = π

π

ϕ∗ (x + t)[K(t) − t∗n−1 (t)]dt. (3.15)

−π

Since ϕ∗ ∈ SM , the function f∗ (t) is continuous, belongs to SM ∗ K, and, in view of (3.11), 1 f∗ C ≤ π

π

|K(t) − t∗n−1 (t)|dt =

−π

1 En (K)1 . π

Furthermore, since the function ϕ∗ (t) is 2αn -periodic, for i = 0, ±1, ±2, . . . , we can write 1 f∗ (2iαn ) = f∗ (0) = π

π −π

1 f∗ (αn + 2iαn ) = f∗ (αn ) = π

1 ϕ∗ (t)[K(t) − t∗n−1 (t)]dt = − En (K)1 , π

π −π

1 ϕ∗ (t + αn )[K(t) − t∗n−1 (t)]dt = − En (K)1 . π

500

Best Approximations in the Spaces C and L

Chapter 7

Thus, at the points iαn , i = 0, ±1, ±2, . . . , the function f∗ (x) takes the maximum absolute values with alternating signs and, in view of the fact that αn ≤ π/n , the number of points of this sort in a period is not smaller than 2n. According to Theorem 1.1, this yields 1 En (K)1 , π

En (f∗ )C = f C =

which is equivalent to (3.13). We now prove (3.14). Assume that i = 0, ±1, ±2, . . . . For each natural m > 2/αn , we set ti = iαn , (m)

Δi

= (ti − 1/m, ti + 1/m),

ϕn (t) =

(m)

σ = ∪ Δi i

(m) (−1)i m/4n∗ , t ∈ Δi , 0,

1 fm (x) = π

,

(3.16)

t∈σm , π K(t + x)ϕm (−t)dt.

−π

The functions ϕm (t) are 2π/n-periodic and even; moreover, as a result of simple calculations, we conclude that π ϕm (t)dt = 0 and ϕm 1 = 1.

(3.17)

−π

Hence, these functions are orthogonal to any polynomial Qn−1 ∈ T2n−1 and, therefore, π 1 fm (x) = K∗ (t + x)ϕm (−t)dt, π −π

where K∗ (t) is the function defined in (3.9). According to (2.7), we have π En (fm )1 = sup

n h∈HM

fm (t)h(t)dt. −π

Section 3 General Facts on the Approximations of Classes of Convolutions 501 n , In view of the fact that the function ϕ∗ (t) in relation (3.9) belongs to HM we get

π En (fm )1 ≥

π fm (t)ϕ∗ (t)dt =

−π

π = −π

−π

1 ϕm (τ )[ π

1 ϕ∗ (t)[ π

π K(τ + t)ϕm (−τ )dτ ]dt −π

π

π ϕ∗ (t)K(t − τ )dt]dτ =

−π

ϕm (t)f∗ (t)dt, −π

where f∗ (t) is the function from relation (3.15). Hence, by using the fact that f∗ (ti ) = (−1)i En (K)1 /π and relation (3.16), we obtain π En (fm )1 ≥

1/m

ϕm (t)f∗ (t)dt = 2n∗ −π

ϕm (t)f∗ (t)dt.

−1/m

Since the function f∗ (t) is continuous, for any ε > 0, one can find a number mε such that the inequality (m)

|f∗ (t)| > f∗ C − ε, t ∈ Δi

, i = 0, ±1, ±2, . . . ,

holds for all m > mε . Therefore, En (fm )1 > 2n∗

2 1 m (f∗ C − ε) = f∗ C − ε = En (K)1 − ε 4n∗ m π

for m > mε and, in view of the fact that fm (t) is the convolution of a function ϕm (·) from S1 and with kernel K(t) , it is clear that sup En (f )1 >

ϕ∈S1

1 En (K)1 − ε. π

Since ε is arbitrary, this yields (3.14 ) and, hence, the proof of Theorem 3.1 is completed. 3.5. By virtue of (3.12) and (3.17), the functions ϕ∗ (t) and ϕm (t) are orthogonal to any constant, i.e., ϕ∗ ⊥ 1 and ϕm ⊥ 1 for any m ∈ N. Therefore, Theorem 3.1, in fact, implies that if the kernel K(t) satisfies the condition A∗n , then 1 sup En (f )C = sup En (f )C = En (K)1 (3.18) π ϕM ≤1 ϕ1 ≤1 ϕ⊥1

502

Best Approximations in the Spaces C and L

and sup En (f )1 = sup En (f )C =

ϕM ≤1

ϕ1 ≤1 ϕ⊥1

1 En (K)1 , π

Chapter 7

(3.19)

where, as earlier, the functions f (·) and ϕ(·) satisfy relation (3.4). If ψ ∈ F and β ∈ R or ψ ∈ M0 and β = 0, then, according to Proposiψ tion 3.1, the classes Lψ β N and Cβ N coincide with the corresponding classes of functions of the form (3.4) provided that ϕ ⊥ 1. Therefore, equalities (3.18) and (3.19) yield the following assertion: Proposition 3.4. Let ϕ ∈ F and β ∈ R or ψ ∈ M0 and β = 0. Assume that the function ∞ ψ(k) cos(kt + βπ/2) Dψ,β = k=1

satisfies the condition

A∗n .

Then 1 En (Dψ,β )1 π

(3.20)

1 En (Dψ,β )1 , π

(3.21)

ψ ) = sup En (f )C = En (Cβ,∞ ψ f ∈Cβ,∞

and En (Lψ β,1 )1 = sup En (f1 ) = f ∈Lψ β,1

where ψ = {f ; f ∈ Cβψ ; fβψ M ≤ 1} Cβ,∞

ψ and Lψ β,1 = {f ; f ∈ Lβ ; f 1 ≤ 1}.

3.6. We now present several general facts concerning the approximations of classes of convolutions by polynomials generated by the linear processes of summation of Fourier series. (n) Assume that f ∈ L , Λ = λk , k = 0, 1, . . . , n − 1, n = 0, 1, . . . , is an (0) arbitrary infinite triangular number matrix, λ0 = 1, a0 (n) λk (ak cos kx + bk sin kx), Un (f ; x; Λ) = + 2 n−1

(3.22)

k=1

ak and bk are the Fourier coefficients of the function f (·), 1 (n) λk cos kx Un (x; Λ) = + 2 n−1 k=1

(3.23)

Section 3 General Facts on the Approximations of Classes of Convolutions 503 is the kernel of the analyzed Λ-method, and En (B; Λ)X = sup f (t) − Un (f ; t; Λ)X , f ∈B

(3.24)

where B is a class of functions in the space X (X is either C or L). Also let En (B; Λ)C = En (B; Λ)

and

En (B; Λ)L = En (B; Λ)1 .

Then 1 Un (f ; x; Λ) = π

π f (x + t)Un (t, Λ)dt = f ∗ Un (·; Λ).

(3.25)

−π

Therefore, if B = N ∗ K, then, in view of the fact that the function Un (t; Λ) is even, we easily find that, for any f ∈ B, Un (f ; x; Λ) = (ϕ ∗ K) ∗ Un (·; Λ) = ϕ ∗ (K ∗ Un (·; Λ))

(3.25 )

and, hence, f (x)−Un (f ; x; Λ) = ϕ∗K −ϕ∗[K ∗Un (·; Λ)] = ϕ∗[K −K ∗Un (·; Λ)], (3.26) where the functions f (·) and ϕ(·) satisfy relation (3.4). df

Since the function Kn (t) =[K ∗ Un (·; Λ)](t) is a trigonometric polynomial of degree n − 1, we have K(t) − [K ∗ Un (·; Λ)](t)1 ≥

inf

Tn−1 ∈T2n−1

K(t) − Tn−1 (t)1 = En (K)1 . (3.27)

On the other hand, by virtue of relation (3.26) and the fact that, in view of the (0) equality λ0 = 1 , we always have Kn ⊥ 1, we can write sup ϕM ≤1, ϕ⊥1

f (t) − Un (f ; x; Λ)C =

=

sup ϕM ≤1, ϕ⊥1

ϕ ∗ [K − K ∗ Un (·; Λ)]C

sup ϕ ∗ Kn C .

(3.28)

ϕM ≤1

Clearly, sup ϕ ∗ Kn C =

ϕ≤1

1 1 Kn (t)1 = K(t) − [K ∗ Un (·; Λ)](t)1 . π π

(3.29)

Best Approximations in the Spaces C and L

504

Chapter 7

Therefore, by combining relations (3.27)–(3.29), we arrive at the following assertion: Proposition 3.5. For any polynomial of the form (3.25), sup ϕM ≤1, ϕ⊥1

f (t) − Un (f ; t; Λ)C ≥

1 En (K)1 , π

(3.30)

where the functions f (·) and ϕ(·) satisfy relation (3.4). In view of Proposition 3.1, this yields the following corollary: Corollary 3.1. Assume that ψ ∈ F and β ∈ R or ψ ∈ M0 and β = 0. Then 1 ψ En (Cβ,∞ ; Λ) ≥ En (Dψ,β )1 , (3.31) π where Dψ,β (·) is the function from relation (3.20). If, in estimate (3.5), we take the upper bounds in the class Cβψ N, then we get πEn (Cβψ N) = π sup ≤ sup En (ϕ)C En (Dψ,β )1 f ∈Cβψ N

ϕ∈N ϕ⊥1

= En (N)En (Dψ,β )1 ,

(3.32)

where En (N) = sup{En (ϕ)C : ϕ ∈ N; ϕ ⊥ 1}.

(3.33)

By comparing relations (3.32) and (3.31), we conclude that the following assertion is true: Corollary 3.2. Assume that ψ ∈ F and β ∈ R or ψ ∈ M0 and β = 0. Then, for any N ⊆ M, ψ ; Λ) En (Cβψ N) ≤ En (N)En (Cβ,∞

(3.34)

ψ ψ ) ≤ En (Cβ,∞ ; Λ). En (Cβ,∞

(3.34 )

and, in particular, Further, in view of the Jackson inequality (see Section 6.9), ψ En (Cβψ Hω ) ≤ En (Hω )En (Cβ,∞ ; Λ).

(3.34 )

Section 4

Orders of the Best Approximations

505

Similarly, if, in estimate (3.7), we take the upper bounds in the classes Lψ β N, then we get the following statement: Corollary 3.3. Assume that ψ ∈ F and β ∈ R or ψ ∈ M0 and β = 0. Then, for any N ⊂ L, ψ En (Lψ β N) ≤ En (N)1 En (Cβ,∞ ; Λ),

(3.35)

where En (N)1 = sup{En (ϕ)1 : ϕ ∈ N, ϕ ⊥ 1}. In particular,

for N = Hω1 and

ψ En (Lψ β Hω1 )1 ≤ En (Hω1 )1 E(Cβ,∞ ; Λ)

(3.35 )

ψ En (Lψ β,1 )1 ≤ En (Cβ,∞ ; Λ)

(3.35 )

ψ for N = S1 , i.e., in the case where Lψ β N = Lβ,1 ,

4.

Orders of the Best Approximations

4.1. In the present section, we establish exact (in order) estimates of the quanψ ψ tities En (Cβ,∞ ), En (Cβψ Hω ), En (Lψ β,1 )1 , and En (Lβ Hω1 )1 in the case where ψ ∈ F and β ∈ R or ψ ∈ M0 and β = 0. To obtain the required upper bounds, we use inequalities (3.34 ) and (3.34 ). The right-hand sides of these relations contain the quantity ψ En (Cβ,∞ ; Λ) = sup f (x) − Un (f ; x; Λ)C , ψ f ∈Cβ,∞

and polynomials [of the form (3.25)] can be chosen arbitrarily. As Un (f ; x; Λ), [η(m)] we take the polynomials Vm (f ; x) used in Section 5.22. Hence, by virtue of Theorems 5.22.2, 5.22.3, 5.23.4, and 5.23.5, we get ψ En (Cβ,∞ ) ≤ Kψ(n),

En (Lψ β,1 )1 ≤ Kψ(n), where K is an absolute constant.

En (Cβψ Hω ) ≤ Kψ(n)ω(1/n),

(4.1)

En (Lψ β Hω1 )1 ≤ Kψ(n)ω(1/n),

(4.2)

Best Approximations in the Spaces C and L

506

Chapter 7

4.2. Let us show that estimates (4.1) and (4.2) cannot be improved in order. First, let 1 Φ1 (x) = Φ1 (n, x) = (ϕ1 ∗ Dψ,β )X = π

π ϕ1 (x + t)Dψ,β (t)dt, −π

where ϕ1 (t) = sgn sin nt. Since |ϕ1 (t)| ≤ 1, we conclude that, by virtue of Proposition 3.8.2, the function Φ1 (x) is continuous and its (ψ, β)-derivative coψ incides with the function ϕ1 (x) almost everywhere, i.e., Φ1 ∈ Cβ,∞ . Let us now prove that, for any n ∈ N, E1 (Φ1 ) ≥ K1 ψ(n),

(4.3)

where K1 is an absolute positive constant. Indeed, since ϕ1 (x) =

∞ 4 sin(2k − 1)nx , π 2k − 1 k=1

we have S[Φ1 ; x] =

∞ 4 ψ((2k − 1)n) sin((2k − 1)nx − βπ/2). π 2k − 1

(4.4)

k=1

If ψ ∈ F and β ∈ R or ψ ∈ M0 and β = 0, then (see Subsection 3.1) the series ∞ 4 ψ[(2k − 1)n] sin((2k − 1)nx − (β + 1)π/2) (4.5) π k=1

(1)

is the Fourier series of a summable function Φ1 (x). Series (4.4) is obtained as a result of term-by-term integration of series (4.5). Therefore, according to Theorem 3.4.2, its sum is equal to the value of the periodic (1) integral of Φ1 (x), i.e., series (4.4) converges to a continuous function and, since Φ1 (x) is continuous, we always have S[Φ1 ; x] = Φ1 (x) . This means that Φ1 (x) is periodic with period 2π/n and, moreover, ∞

Φ1 (pπ/n) =

(−1)p 4 sin βπ/2 ψ((2k − 1)n) , p = 0, ±1, ±2, . . . , (4.4 ) π 2k − 1 k=1

and

Section 4

Orders of the Best Approximations ∞

Φ1 (

ψ((2k − 1)n) pπ π (−1)p+1 4 cos βπ/2 (−1)k + )= , n 2n π 2k − 1

507 (4.4 )

k=1

p = 0, ±1, ±2, . . . . By using the de la Vallée Poussin theorem (Theorem 1.2) and choosing the polynomial identically equal to zero as Tn−1 (t), in view of relations (4.4 ) and (4.4 ), we conclude that En (Φ1 ) ≥ max{|Φ1 (π/n)|; |Φ1 (3π/2n)|} 4 sin βπ/2 4 cos βπ/2 ψ(3n) ψ(n); [ψ(n) − ]}. π π 3 In view of the monotonicity of the function ψ(·), this immediately implies inequality (4.3). Thus, ≥ max{

ψ En (Cβ,∞ ) ≥ En (Φ1 ) ≥ K1 ψ(n),

(4.6)

which proves the unimprovability of the first estimate in (4.1) in order. 4.3. To prove the unimprovability of the second estimate in (4.1), we consider the function π 1 Φ2 (x) = Φ2 (n, x) = ϕ2 (x + t)Dψ,β (t)dt, π −π

where ϕ2 (t) is an odd 2π/n-periodic function defined on [0; π/n] by the equalities nt ϕ2 (t) = ω(π/n), t ∈ [0, π/2n], 2π and ϕ2 (t) = ϕ(π/n − t), t ∈ [π/2n, π/n]. The function ϕ2 (t) belongs to the class Hω , i.e., for any t ∈ R and h ∈ R, |ϕ2 (t) − ϕ2 (t + h)| ≤ ω(|h|). Since ϕ2 (·) is odd and 2π/n-periodic, it suffices to check this fact for −π/2n ≤ t, t + h ≤ π/2n. In this case, (see Subsection 3.1.4), n|h| ω(π/n) 2π n|h| π 1 n|h| = ω( |h|) ≤ ( + )ω(|h|) ≤ ω(|h|). (4.7) 2π n|h| 2 2π

|ϕ2 (t) − ϕ2 (t + h)| =

Best Approximations in the Spaces C and L

508

Chapter 7

Thus, ϕ2 ∈ Hω and, hence, Φ2 ∈ Cβψ Hω . Further, by direct verification, we conclude that the Fourier expansion of the function ϕ2 (·) has the form ϕ2 (x) =

∞

b(2k−1)n sin(2k − 1)nx,

k=1

where b(2k−1)n = (−1)k+1 2ω(π/n)/π 2 (2k + 1)2 . Consequently, ∞

S[Φ2 ; x] =

ψ[(2k − 1)n] 2ω(π/n) (−1)k+1 sin[(2k − 1)nx − βπ/2]. (4.8) 2 π (2k − 1)2 k=1

This series is uniformly convergent and, therefore, S[Φ2 ; x] = Φ2 (x). In particular, ∞

Φ2 (pπ/n) =

ψ[(2k − 1)n] (−1)p+1 2ω(π/n) sin βπ/2 (−1)k+1 , 2 π (2k − 1)2 k=1

p = 0, ±1, ±2, . . . , and

∞

Φ2 (

ψ[(2k − 1)n] pπ π (−1)p 2ω(π/n) cos βπ/2 , + )= 2 n 2n π (2k − 1)2 k=1

p = 0, ±1, ±2, . . . . By using the de la Vallée Poussin theorem once again, we conclude that En (Cβψ Hω ) ≥ En (Φ2 ) 2ω(π/n) 2ω(π/n) ψ(n) cos βπ/2; ψ(n) sin βπ/2} π2 π2 (4.9) ≥ K1 ψ(n)ω(1/n). ≥ max{

Combining relations (4.1), (4.6), and (4.9), we arrive at the following assertion: Theorem 4.1. Assume that ψ ∈ F and β ∈ R or ψ ∈ M0 and β = 0 . Also let ω = ω(t) be an arbitrary modulus of continuity. Then there exist absolute positive constants K1 and K2 such that, for any n ∈ N, ψ K1 ψ(n) ≤ En (Cβ,∞ ) ≤ K2 ψ(n),

(4.10)

Section 4

Orders of the Best Approximations K1 ψ(n)ω(1/n) ≤ En (Cβψ Hω ) ≤ K2 ψ(n)ω(1/n).

509 (4.10 )

In particular, for ψ(v) = v −r , r > 0, K1 n−r ≤ En (Wβr ) ≤ K2 n−r ,

(4.11)

K1 n−r ω(1/n) ≤ En (Wβr Hω ) ≤ K2 n−r ω(1/n).

(4.11 )

4.4. Inequalities (4.2) are also exact in order. Indeed, the function 1 Φ3 (x) = Φ3 (n, x) = ψ(n) sin(nt − βπ/2) 4 belongs to Lψ β,1 and is orthogonal to all trigonometric polynomials up to degree n − 1. Therefore, according to Theorem 2.1, in the family of all polynomials of degree n − 1, the best approximation of this function in the space L is realized by the polynomial identically equal to zero. Hence, En (Lψ β,1 )1 ≥ En (Φ3 )1 = Φ3 (x)1 = ψ(n).

(4.12)

To check the unimprovability of the second estimate in (4.2), we set 1 Φ4 (x) = Φ4 (n; x) = π

π ϕ4 (x + t)Dψ,β (t)dt, −π

where ϕ4 (t) is an odd 2π/n-periodic function defined on [0, π/n] by the equalities ϕ4 (t) =

nt ω(π/n), t ∈ [0, π/2n], 4π 2

ϕ4 (t) = ϕ4 (π/n − t), t ∈ [π/2; π/n],

so that

1 1 ϕ2 (t) and Φ4 (x) = Φ2 (x). 2π 2π According to (4.7), we have ϕ2 ∈ Hω . Hence, for any x and h ≥ 0, ϕ4 (t) =

|ϕ4 (x + h) − ϕ4 (x)| ≤

1 ω(h). 2π

Therefore, ω1 (ϕ1 ; t) = sup ϕ4 (x + h) − ϕ(x)1 ≤ ω(t), |h|≤t

(4.13)

Best Approximations in the Spaces C and L

510

Chapter 7

i.e., ϕ4 ∈ Hω1 and, hence, Φ4 ∈ Lψ β Hω1 . Let us now estimate the quantity En (Φ4 )1 from below. By virtue of (2.7), (4.13), and (4.8), we find π En (Φ4 )1 = sup

n h∈HM

π ≥ −π

Φ4 (t)h(t)dt −π

1 Φ4 (t) sin(nt − βπ/2)dt = 2π

ω(π/n) ψ(n) = π3

π Φ2 (t) sin(nt − βπ/2)dt −π

π sin2 (nt − βπ/2)dt =

−π

ω(π/n) ψ(n) π2

and, thus, En (Lψ β Hω1 )1 ≥ En (Φ4 )1 ≥

ψ(n) π ψ(n) ω( ) ≥ ω(1/n). π2 n π2

(4.14)

By comparing relations (4.2), (4.12), and (4.14), we arrive at the following analog of Theorem 4.1: Theorem 4.2. Assume that ψ ∈ F and β ∈ R or ψ ∈ M0 and β = 0 . Let ω = ω(t) be an arbitrary modulus of continuity. Then there exist absolute positive constants K1 and K2 such that, for any n ∈ N, K1 ψ(n) ≤ En (Lψ β,1 )1 ≤ K2 ψ(n),

(4.15)

K1 ψ(n)ω(1/n) ≤ En (Lψ β Hω1 )1 ≤ K2 ψ(n)ω(1/n).

(4.15 )

In particular, if ψ(v) = v −r , r > 0, then

5.

r )1 ≤ K2 n−r , K1 n−r ≤ En (Wβ,1

(4.16)

K1 n−r ω(1/n) ≤ En (Wβr Hω1 )1 ≤ K2 n−r ω(1/n).

(4.16 )

Exact Values of the Upper Bounds of Best Approximations

5.1. Theorems 4.1 and 4.2 give exact orders of the upper bounds of the best approximations in the corresponding classes of functions. At present, even the

Section 5

Exact Values of the Upper Bounds of Best Approximations

511

values of these upper bounds are known for many important classes of functions and, in particular, for the classes Wβr with any r > 0 and β ∈ R. If r is not an integer number, then the well-known methods used for the evaluation of En (Wβr ) require quite complicated special investigations. For integer r and β, this problem is much simpler. In this case, the decisive role is played by the fact that the functions from Wβr admit representations in the form of convolutions (3.4), where the kernels K(·) are even or odd functions, and the explicit form of the functions ψ(·) (ψ(k) = k −r ) is insignificant. In what follows, we deψ termine En (Cβ,∞ ) and En (Lψ β,1 )1 in the case where ψ(k) satisfies additional requirements of monotonicity and β is an integer number. As a starting point, we use Proposition 3.4 according to which, for the solution of the posed problem, it suffices to show that the kernel Dψ,β (t) satisfies the condition A∗n and find the quantity En (Dψ,β )1 . 5.2. First, we indicate the conditions under which functions cψ (x) =

∞

ψ(k) cos kx

(5.1)

ψ(k) sin kx

(5.2)

k=1

and sψ (x) =

∞ k=1

A∗n .

In analyzing the functions cψ (x), we assume that ψ(k) satisfy the condition is a three-times monotone infinitesimal sequence such that lim ψ(k) = 0,

k→∞

ψ(k) ≥ ψ(k + 1),

Δ2 ψ(k) = ψ(k) − 2ψ(k + 1) + ψ(k + 2) ≥ 0

(5.3)

and Δ3 ψ(k) = ψ(k) − 3ψ(k + 1) + 3ψ(k + 2) − ψ(k + 3) ≥ 0,

(5.4)

k = 1, 2, . . . . In analyzing the functions sψ (x), we assume that ψ(k) satisfies condition (5.3) and, in addition, ∞ ψ(k) < ∞, (5.5) k k=1

i.e., ψ ∈ F0 (see Subsection 3.7.6).

Best Approximations in the Spaces C and L

512

Chapter 7

Note that, under these conditions, by virtue of Theorems 3.7.2 and 3.7.3, series (5.1) and (5.2) are the Fourier series of summable functions, namely, of their sums. In what follows, we need the following simple strengthening of Theorem 3.7.2: Proposition 5.1. Assume that a sequence c0 , c1 , . . . is such that lim ck = 0 k→∞

and the inequality Δ2 ck = ck − 2ck+1 + ck+2 ≥ 0

(5.6)

holds for all k = 0, 1, 2, . . . . Then the series ∞

c0 ck cos kx + 2

(5.7)

k=1

converges everywhere except, possibly, the point x = 0 to a summable function c(x), plays the role of its Fourier series, and c(x) ≥ 0. Proof. In view of Theorem 3.7.2, it remains to show that c(x) ≥ 0. If we denote a partial sum of series (5.7) by cn (x) and apply (twice) the Abel transformation n n−1 k αk βk = (αk − αk+1 )Bk + αn Bn , Bk = βi , (5.8) k=0

i=0

k=0

then we get cn (x) =

n−2

(k + 1)Δ2 ck Fk (x) + nFn−1 (x)(cn−2 − cn−1 ) + Dn (x)cn , (5.9)

k=0

where Dn (x) is a Dirichlet kernel and Fk (x) are Fejér kernels, 1 sin2 kt/2 Di (x) = . 2 x 2k sin t/2 i=0 k−1

Fk (x) =

(5.10)

If x = 0, then the last two terms in (5.9) approach zero as n → ∞. Therefore, c(x) = lim cn (x) = n→∞

∞

(k + 1)Δ2 ck Fk (x).

k=0

Finally, in view of (5.6) and (5.10), we conclude that c(x) ≥ 0.

Section 5

Exact Values of the Upper Bounds of Best Approximations

513

5.3. The following assertion is true for the functions cψ (x) and sψ (x) : Theorem 5.1. Assume that ψ(k) is a sequence satisfying conditions (5.3) and (5.4). Then, for any n ∈ N, one can find a trigonometric polynomial tcn−1 (x) of degree n − 1 such that sgn (cψ (x) − tcn−1 (x)) = sgn cos nt.

(5.11)

If ψ(k) satisfies conditions (5.3) and (5.5), then there exists a polynomial of degree n − 1 such that

tsn−1 (x)

sgn (sψ (x) − tsn−1 (x)) = sgn sin nt.

(5.12)

Moreover, tcn−1 (x) = −an +

n−1

[ψ(k) − (an−k + an+k )] cos kt,

(5.13)

k=1

where ak =

∞

(−1)ν ψ[(2ν + 1)n + k],

k = 0, 1, . . . ,

(5.14)

ν=0

tsn−1 (x)

=

n−1

bk = ψ(k) −

bk sin kt,

∞

[ψ(2νn + k) − ψ(2νn − k)]. (5.15)

ν=1

k=1

Proof. To prove equality (5.11), it suffices to show that, for any n ∈ N, cψ (x) − tcn−1 (x) = 2g(x) cos nt, where

(5.16)

∞

g(x) =

a0 ak cos kt, + 2 k=1

because g(x) ≥ 0. Indeed, Δ2 ak = ak − 2ak+1 + ak+2 =

∞

(−1)ν Δ2 ψ[(2ν + 1) + k].

(5.17)

ν=0

It follows from condition (5.3) that Δ2 ψ[(2ν + 1) + k] ≥ 0 for all ν = 0, 1, . . . . Condition (5.4) means that the second differences Δ2 ψ(m) monotonically decrease as m increases. Therefore, for fixed k, the quantities

514

Best Approximations in the Spaces C and L

Chapter 7

Δ2 [(2ν + 1) + k] decrease as ν increases. Hence, the series in (5.17) is alternating and the absolute values of its terms decrease. Therefore, its sum has the sign of the first term, i.e., Δ2 ak ≥ 0. In this case, it follows from Proposition 5.1 that g(x) ≥ 0. We now prove equality (5.16). After elementary transformations, we get

2g(x) cos nx = a0 cos nx +

∞

[cos(n + k)x + cos(n − k)x]

k=1

= an + (a0 + a2n ) cos nx +

n−1

(an−k + an+k ) cos kx

k=1 ∞

+

(ak−n + ak+n ) cos kx.

k=n+1

If k = n, n + 1, n + 2, . . . , then ak−n + ak+n = ψ(k). Therefore,

2g(x) cos nx = an +

n−1

(an−k + an+k ) cos kx +

k=1

∞

ψ(k) cos kx

k=n

= cψ (x) − tcn−1 (x). Thus, the first part of the theorem is proved. The proof of the second part is more complicated. First, for any natural n and m, we consider the series ∞

ψ(k) sin kt

k=n

and ∞

[ψ(k) − ψ(2mn + k)] sin kt.

k=n

By virtue of conditions (5.3) and (5.5) and Theorem 3.7.2, we conclude that these series are the Fourier series of summable functions denoted by zn (x) and zn,m (x), respectively. Since

Section 5

Exact Values of the Upper Bounds of Best Approximations

zn (x) − zn,m (x) =

∞

∞

ψ(2mn + k) sin kx =

k=n

515

ψ(k) sin(k − 2mn)

k=(2m+1)n ∞

= cos 2mnx

ψ(k) sin kx

k=(2m+1)n ∞

− sin 2nmx

ψ(k) cos kx

k=(2m+1)n

and the series in the last equality are remainders of convergent series, we conclude that (5.18) lim [zn (x) − zn,m (x)] = 0. m→∞

Further, we set αjm =

m−1

ψ[(2ν + 1)n + j],

m ∈ N, j = 0, 1, . . . ,

ν=0 m m − αn+k , Bm,k = αn−k

Bk = lim Bm,k = m→∞

∞

k = 0, 1, . . . , n − 1,

(5.19)

[ψ(2νn − k) − ψ(2νn + k)],

ν=1

and Pn−1 (x) =

n−1

Bk sin kx.

(5.20)

k=1

If k ≥ n, then

m m ψ(k) − ψ(2mn + k) = αk−n − αk+n .

Therefore, zn,m (x) +

n−1

Bm,k sin kx

k=1

=

∞

m m (α|n−k| − αn+k ) sin kx

k=0

= lim

p→∞

p k=−p

m α|n−k| sin kx = lim

p→∞

n+p s=n−p

m α|s| sin(n − s)x

516

Best Approximations in the Spaces C and L

=

α0m sin nx

+ lim

p→∞

p−n

αkm [sin(n − k)x + sin(n + k)x]

k=1 p+n

+ lim

p→∞

Chapter 7

αkm sin(n − k)x.

(5.21)

k=p−n+1

Since, for fixed m, αkm → 0 as k → ∞, the last term in (5.21) is equal to zero and, hence, zn,m (x) +

n−1

Bm,k sin kx

k=1

= α0m sin nx +

∞

αkm [sin(n − k)x + sin(n + k)x]

k=1

= 2 sin nx(

α0m 2

+

∞

αkm cos kx).

(5.22)

k=1

By virtue of conditions (5.3) and Proposition 5.1, for any fixed ν ≥ 0, the series ∞ 1 ψ[(2ν + 1)n + k] cos kx ψ[(2ν + 1)n] + 2 k=1

specifies a nonnegative function fν (x) in the interval (0, 2π). Therefore, by setting m−1 fν (x), Φm (x) = ν=0

we find 0 < Φ1 (x) ≤ Φ2 (x) ≤ . . . ≤ Φm (x) ≤ . . . . At the same time,

(5.23)

∞

1 Φm (x) = α0m + αkm cos kx. 2

(5.24)

k=1

Thus, it follows from (5.22)–(5.24) that, for any m ∈ N, sgn (zn,m (x) +

n−1 k=1

Bm,k sin kx) = sgn sin nx.

(5.25)

Section 5

Exact Values of the Upper Bounds of Best Approximations

517

In this relation, we pass to the limit as m → ∞ and take into account equalities (5.18)–(5.20). As a result, for any n ∈ N, we find sgn [zn (x) − Pn−1 (x)] = sgn [sψ (x) − tsn−1 (x)] = sgn sin nx. Theorem 5.1 is proved. 5.4. It follows from Theorem 5.1 that the functions cψ (x) and sψ (x) satisfy the condition A∗n provided that relations (5.3), (5.4) and (5.3), (5.5), respectively, are satisfied. Moreover, as t∗n−1 (x), one can use the polynomial tcn−1 (x) for cψ (x) and the polynomial tsn−1 (x) for sψ (x). In this case, according to Proposition 3.3 and equalities (5.11) and (5.12), we get π En (cψ )1 = cψ (x) −

tcn−1 (x)1

|cψ (x) − tcn−1 (x)|dx

= −π

π [cψ (x) − tcn−1 (x)]sgn cos nxdx,

=

(5.26)

−π

π [sψ (x) − tsn−1 (x)]sgn sin nxdx.

En (sψ )1 =

(5.27)

−π

The Fourier expansions of the functions sgn cos nx and sgn sin nx have the form ∞ 4 (−1)k sgn cos nx = cos(2k + 1)nx, π 2k + 1 k=0

∞ 4 sin(2k + 1)nx sgn sin nx = . π 2k + 1 k=0

Therefore, by using Theorem 3.7.1, we obtain π En (cψ )1 =

cψ (x)sgn cos nxdx = 4

∞ (−1)k ψ[(k + 1)n] k=0

−π

π En (sψ )1 =

sψ (x)sgn sin nxdx = 4 −π

2k + 1

∞ ψ[(2k + 1)n] k=0

2k + 1

,

,

(5.28)

(5.28 )

518

Best Approximations in the Spaces C and L

Chapter 7

whence, in view of the fact that En (−f )1 = En (f )1 , we arrive at the following assertion: Theorem 5.2. Assume that β is an integer number and that Dψ,β (t) =

∞

ψ(k) cos(kt + βπ/2),

k=1

where ψ(k) is a function satisfying relations (5.3) and (5.4) for even β and relations (5.3) and (5.5) for odd β . Then Dψ,β (t) satisfies the condition A∗n and, for any n ∈ N, En (Dψ,2p )1 = En (cψ )1 = 4

∞ (−1)k ψ[(2k + 1)n]

2k + 1

k=0

En (Dψ,2p+1 )1 = En (sψ )1 = 4

∞ ψ[(2k + 1)n] k=0

2k + 1

.

,

(5.29)

(5.29 )

Combining this theorem with Proposition 3.4, we conclude that the following assertion is true: Corollary 5.1. Assume that the conditions of Theorem 5.2 are satisfied for the quantities ψ(k) and β. Then, for any n ∈ N, ψ En (C2p,∞ ) = En (Lψ 2p,1 )1 =

∞ 4 (−1)k ψ[(2k + 1)n] , π 2k + 1

(5.30)

k=0

ψ En (C2p+1,∞ ) = En (Lψ 2p+1,1 )1 =

∞ 4 ψ[(2k + 1)n] , π 2k + 1

(5.30 )

k=0

p = 0, ±1, ±2, . . . . 5.5. If ψ(k) = k −r , r > 0, and β = r, then the functions Dψ,β (t) turn into the Bernoulli functions Dr (t) (see Subsections 3.7.7 and 3.7.8): Dr (t) =

∞

k −r cos(kt + rπ/2).

k=1

For integer r, the convolutions ϕ ∗ Dr (t), ϕ ∈ N, generate the classes W r N of r-times differentiable functions f (x) whose rth derivatives belong

Section 5

Exact Values of the Upper Bounds of Best Approximations

519

df

r and W r S = W r , we conclude that to the class N. By setting W r SM = W∞ 1 1 Corollary 5.1 implies the following assertion:

Corollary 5.2. If r ∈ N, then, for any n ∈ N, r ) En (W∞

=

En (W1r )1

∞ 4 (−1)k(r+1) df Kr = = . πnr (2k + 1)r+1 nr

(5.31)

k=0

The quantities Kr =

∞ 4 (−1)k(r−1) , r = 0, 1, . . . , π (2k + 1)r+1

(5.32)

k=0

are known as Favard constants. It is easy to see that K0 = 1, K1 = π/2, K2 = π 2 /8, K3 = π 3 /24, . . . .

(5.33)

As the number r increases, the constants Kr increase for even r and decrease for odd r , i.e., 1 = K0 < K2 < K4 < . . .
0 , let β ∈ R1 , and let θ be a quantity taking the following values: (a) θ = 1 for r ∈ (0, 1] and β ∈ [r, 2r]; (b) θ is a root of the equation ∞ cos((2k + 1)θπ − k=0

(2k + 1)r

βπ 2 )

= 0,

θ ∈ (0, 1],

either for r ∈ (0, 1] and β ∈ [0, 2] \ [r, 2r] or for r > 1 and β ∈ R1 . Then, for any n ∈ N, the polynomial of the best approximation of the func∗ (t) of degree n − 1 intertion Dr,β (t) in the space L1 is the polynomial Tn−1 polating the function at the points π (θ + ν) , ν = 0, 1, . . . , 2n − 2, n and such that En (Dr,β )1 =

∗ Dr,β (t) − Tn−1 (t)1

2π =|

Dr,β (t)sgn sin(nt − θπ)dt =

πMr,β , nr

0

where

∞

Mr,β =

4 sin((2k + 1)θπ − | π (2k + 1)r+1

βπ 2 )

|.

k=0

Consequently, the following equalities are true for all r > 0, β ∈ R1 , and n∈N : r r )C = En (Wβ,1 )1 = En (Wβ,∞

1 En (Dr,β )1 = Mr,β n−r . π

(6.1)

Note that, for r ∈ N and β = r, the constants Mr,β coincide with the Favard constants, namely, Mr,r = Kr ,

r ∈ N.

524

Best Approximations in the Spaces C and L

Chapter 7

6.2. The problem of finding analogs of equalities (6.1) for the classes Wβr Hω turns out to be even more complicated than the problem of generalization of equalr . Up to now, the only results in this field are the ities (6.30) to the classes Wβ,∞ well-known Korneichuk theorems obtained for the case of natural values of r in [4, 5]. To formulate these results, we set ⎧ ⎨ 1 , 0 ≤ x < π, Φ0 (x) = 2 ⎩0, x ≥ π, ⎧ π−x ⎪ ⎪ ⎨1 Φr−1 (t)dt, 0 ≤ x < π, Φr (x) = 2 ⎪ ⎪ ⎩ 0 0, x ≥ π, r = 1, 2, . . . .

Theorem 6.2. The following equalities are true for any convex (upward) modulus of continuity ω(t), any n ∈ N , and any r = 0, 1, . . . : En (Wrr Hω )C

=

π

1 nr+1

Φr (t)ω (t/n)dt,

(6.2)

0

En (Wrr Hω )L

=

4

π

nr+1

2 Φr−1 (t)ω (t/n)dt = r n

0

π

t Φr (π − t)ω( )dt. (6.3) n

0

We also present the special cases of equalities (6.2) and (6.3) π

1 π En (Hω )C = ω( ), 2 n

En (Hω )L =

t ω( )dt, n

0

En (W11 Hω )C

1 = 4n

π

t ω( )dt, n

En (W11 Hω )L

0

En (Wrr H α )C =

1 2nr+α

1 = 2n

π

t tω( )dt, n

0

π tα Φr−1 (π − t)dt, 0

En (Wrr H α )L

=

2

π tα Φr (π − t)dt.

nr+α 0

The detailed proofs of Theorems 6.1 and 6.2 can be found in the cited works.

Section 7

7.

Serdyuk Theorem

525

Serdyuk Theorem ψ 7.1. In this section, we establish the exact values of the quantities En (Cβ,∞ )C

and En (Lψ β,1 )1 in the case where the functions ψ(k) are sufficiently rapidly decreasing. In this case, the principal role is played by Theorem 7.1 proved by Serdyuk [5]. Definition 7.1. We say that a sequence ψ = ψ(k), k ∈ N, satisfies the condition Sn,ρ , n ∈ N, ρ ∈ (0, 1) (and write ψ ∈ Sn,ρ ) if this sequence satisfies the following conditions beginning with a certain number n ∈ N : (i) Δ2 ψ(k) = ψ(k) − 2ψ(k + 1) + ψ(k + 2) ≥ 0, (ii)

ψ(k + 1) ≤ ρ, ψ(k)

(iii)

Δ2 ψ(n) (1 + 3ρ)ρ2n , > ψ(n) (1 − ρ) 1 − 2ρ2n

(7.1) (7.2)

k = n, n + 1, . . . .

(7.3)

The following assertion is true: Theorem 7.1. Assume that ψ ∈ Sn,ρ for given n ∈ N and ρ ∈ (0, 1), Then, for any β ∈ R1 , the function Ψβ (t) =

∞

ψ(k) cos(kt −

k=1

βπ ) 2

(7.4)

satisfies the condition A∗n and ψ )C = En (Lψ En (Cβ,∞ β,1 )1 =

1 En (Ψβ )1 = Ψβ ∗ sgn sin(·)C π

∞

4 ψ((2k + 1)n) βπ = | sin((2k + 1)θn π − )|, π 2k + 1 2

(7.5)

k=0

where θn is the unique root of the equation ∞

ψ((2k + 1)n) cos((2k + 1)θn π −

k=0

in the interval [0, π).

βπ )=0 2

(7.6)

Best Approximations in the Spaces C and L

526

Chapter 7

The proof of this theorem is based on the following assertion: Lemma 7.1. Under the conditions of Theorem 7.1, there exists a trigonomet∗ (·) of degree n − 1 such that ric polynomial Tn−1 ∗ (t)) = −sgn sin(nt + θn π). sgn (Ψβ (t) − Tn−1

(7.7)

∗ (t) is uniquely determined by the conditions In addition, the polynomial Tn−1 ∗ ( Tn−1

θn π + kπ θn π + kπ ) = Ψβ ( ), n n

k = 0, 1, . . . , 2n − 2.

(7.8)

∗ (t) be a Proof. Let tk = (θn π +kπ)/n, k = 0, 1, . . . , 2n−2, and let Tn−1 trigonometric polynomial of degree n−1 interpolating the function Ψβ (t) at the (n) points tk . This polynomial exists and is unique in the set T2n−1 (see Section ∗ (t) interpolates the function ψ (t) also at the point 8.1). The polynomial Tn−1 β (n)

(n)

t2n−1 = (θn π + (2n − 1)π)/n. Indeed, by virtue of the equality 2n−1

ν ik( νπ + nt ) n

(−1) e

=

e

ikt n

(1 − ei2kπ )

1+e

ν=0

ikπ n

0, =

k = (2j + 1)n, j = 0, 1, 2, . . . ,

2nei(2j+1)t , k = (2j + 1)n, j = 0, 1, 2, . . . ,

(7.9)

where k and n are arbitrary natural numbers, we get 2n−1

(−1)ν Tn−1 (α +

ν=0

νπ ) = 0 for any α ∈ R n

(7.10)

and any polynomial Tn−1 (t) of degree n − 1. Hence, by virtue of (7.6), (7.8), and (7.10), we get Ψβ (

θn π (2n − 1)π θn π (2n − 1)π ∗ ( + ) − Tn−1 + ) n n n n 2n−1 θn π νπ θn π νπ ∗ (−1)ν (Ψβ ( ( − ) − Tn−1 − )) =− n n n n ν=0

=−

2n−1

(−1)ν Ψβ (

ν=0 ∞

= −2n

ν=0

θn π νπ − ) n n

ψ((2ν + 1)n) cos((2ν + 1)θπ −

βπ ) = 0. 2

(7.11)

Section 7

Serdyuk Theorem

527

∗ (t) has 2n zeros in a period. This means that the difference Ψβ (t) − Tn−1 Let us show that the number zeros in the interval [0, 2π) does not exceed 2n . To this end, we set u = t − θn π/n. Then ∞

θn π kθn π βπ Ψβ (t) = Ψβ (u + ψ(k) cos(ku + )= − ) n n 2 k=1

=

∞

ak cos ku +

k=1

∞

bk sin ku,

(7.12)

k=1

where ak = ψ(k) cos(

kθn π βπ − ) and n 2

(1)

bk = −ψ(k) sin(

kθn π βπ − ). n 2

(2)

Assume that Tn−1 (u) and Tn−1 (u) are trigonometric polynomials of degree ∞ ∞ ak cos ku and bk sin ku, respectively, n − 1 interpolating the functions k=n

k=n

at the zeros of the function sin nu. In what follows, we need the relations ∞

sin nt(

α0 αk cos kt) + 2 k=1

=

sin nt

∞

n−1

∞

j=1

j=n

1 1 (αn−j − αn+j ) sin jt + (αj−n − αj+n ) sin jt, (7.13) 2 2

βk sin kt

k=1 n−1

∞

j=1

j=n

1 1 (βn−j + βn+j ) cos jt + (βj+n − βj−n ) cos jt (7.14) = (βn + 2 2 valid for all αk and βk approaching zero as k → ∞ at any point t where the ∞ ∞ a0 αk cos kt and βk sin kt are convergent. series + 2 k=1 k=1 As N → ∞, equalities (7.13) and (7.14) follow from the identities

Best Approximations in the Spaces C and L

528

Chapter 7

∞

sin nt(

α0 αk cos kt) + 2 k=1

1 α0 = ( [sin nt − sin(−nt)] + αk [sin(k + n)t − sin(k − n)t]) 2 2 N

k=1

1 = 2

N +n j=n

1 αj−n sin jt + 2

n−1 j=1

N −n 1 αn−j sin jt − αj+n sin jt 2 j=1

1 (αn−j − αn+j ) sin jt 2 n−1

=

j=1

N −n 1 1 + (αj−n − αj+n ) sin jt + 2 2 j=n

N +n

αj−n sin jt

j=N −n+1

and sin nt

N k=1

1 βk sin kt = βk [cos(k − n)t − cos(k + n)t] 2 N

k=1

N +n n−1 1 1 =− βj−n cos jt + βn−j cos jt 2 2 j=n+1

+

j=0

N −n 1 βn+j cos jt 2 j=1

1 = 2

n−1

N −n 1 + βn+j ) cos jt + (βj+n − βj−n ) cos jt 2

(βn−j

j=0

j=n

−

1 2

N +n

βj−n cos jt.

j=N −n+1

Further, since (see Section 8.1) (1) Tn−1 (u)

=

∞

a2nν +

ν=1 (2)

Tn−1 (u) = −

n−1

{

∞

(a2nν−j + a2nν+j )} cos ju,

(7.15)

j=1 ν=1 n−1

{

∞

j=1 ν=1

(b2nν−j − b2nν+j )} sin ju,

(7.15 )

Section 7

Serdyuk Theorem ∞

by setting t = u and αk =

529

b(2ν−1)n+k in (7.13) and t = u and βk =

ν=1

∞ − a(2ν−1)n+k in (7.14), we get ν=1 ∞

(2)

bj sin ju − Tn−1 (u)

j=n

=

∞

bj sin ju +

n−1 j =1

j=n

= 2 sin nu(

∞

{

∞

j=1 ν=1

{

∞

(b2nν−j − b2nν+j )} sin ju

ν=1 ∞

1 b(2ν−1)n+j } cos ju + b(2ν−1)n ) 2

(7.16)

ν=1

and ∞

(1)

aj cos ju − Tn−1 (u)

j=n

=

∞

∞ n−1 ∞ aj cos ju − ( a2nν + { (a2nν−j + a2nν+j )} cos ju) ν=1

j=n

= 2 sin nu

∞ j=1

{−

∞

j=1 ν=1

a(2ν−1)n+j } sin ju.

(7.17)

ν=1

Combining relations (7.16) and (7.17), after elementary transformations, we find ∞ ∞ (1) (2) ak cos ku + bk sin ku − (Tn−1 (u) + Tn−1 (u)) k=n

k=n

= −2 sin nuWn (t), where

∞

(n) co (n) (cj cos jt + dj sin jt), + 2 (n)

Wn (t) =

(7.18)

(7.19)

j=1

(n) cj

=

∞ ν=0

ψ((2ν + 1)n + j) sin((2ν + 1)θn π −

βπ ), 2

(7.20)

Best Approximations in the Spaces C and L

530 (n)

dj

=

∞

ψ((2ν + 1)n + j) cos((2ν + 1)θn π −

ν=0

Chapter 7 βπ ). 2

(7.21)

Equalities (7.12) and (7.18) imply that ∗ (t) = Tn−1 (t − θn π/n) + Tn−1 (t − θn π/n) + Tn−1 (1)

(2)

n−1

ψ(k) cos(kt −

k=1

βπ ) 2

and, therefore, ∗ (t) = −2 sin n(t − Ψβ (t) − Tn−1

θn π )Wn (t). n

(7.22)

Thus, we see that the validity of the relation Wn (t) = 0, t ∈ [0, 2π],

(7.23)

yields equality (7.7). We now prove (7.23). (n) (n) Representing the coefficients cj and dj in the form (n)

cj

(n)

dj where (1) df rn,j =

(2) df

rn,j =

∞ ν=1 ∞

βπ (1) ) + rn,j , 2 βπ (2) = ψ(n + j) cos(θn π − ) + rn,j , 2 = ψ(n + j) sin(θn π −

(7.24) (7.25)

ψ((2ν + 1)n + j) sin((2ν + 1)θn π −

βπ ), 2

(7.26)

ψ((2ν + 1)n + j) cos((2ν + 1)θn π −

βπ ), 2

(7.27)

ν=1

by virtue of relation (7.19), we get ∞

ψ(n) βπ ψ(n + j) cos jt) sin(θn π − + ) + Rn (t), Wn (t) = ( 2 2

(7.28)

j=1

where ∞ βπ df ψ(n + j) sin jt) cos(θn π − Rn (t) = ( ) 2 j=1

(1)

+

rn,0 2

+

∞ j=1

(1)

(2)

(rn,j cos jt + rn,j sin jt). (7.29)

Section 7

Serdyuk Theorem

531

In view of inequality (7.2), ∞

ψ((2ν + 1)n + j) ≤ ψ(3n + j)

ν=1

∞

ρ2nν ≤ ψ(n + j)

ν=0

ρ2n , 1 − ρ2n

(7.30)

j = 0, 1, 2, . . . . Therefore, by using (7.26), (7.27), and (7.6), we get (1)

|rn,0 | ≤

∞

ψ((2ν + 1)n) ≤ ψ(n)

ν=1

|

∞

(1)

ρ2n , 1 − ρ2n

(7.31)

(2)

(rn,j cos jt + rn,j sin jt)|

j=1

=|

∞ ∞

ψ((2ν + 1)n + j) sin((2ν + 1)θn π −

j=1 ν=1

≤

∞ ∞

βπ + jt)| 2

ψ((2ν + 1)n + j)

j=1 ν=1

≤

∞ ρ2n ρ ρ2n ψ(n + j) ≤ ψ(n), 1 − ρ2n 1 − ρ 1 − ρ2n

(7.32)

j=1

and ∞

| cos(θn π −

βπ βπ 1 ψ((2ν + 1)n) cos((2ν + 1)θn π − )| = | )| 2 ψ(n) 2 ν=1 ∞

≤

1 ρ2n ψ((2ν + 1)n) ≤ . ψ(n) 1 − ρ2n

(7.33)

ν=1

Combining relations (7.31)–(7.33), we obtain |Rn (t)| ≤ |

∞

ψ(n + j) sin jt|

j=1

≤ ψ(n)

ρ2n ρ2n 1 ρ + ψ(n) ( + ) 1 − ρ2n 1 − ρ2n 2 1 − ρ

1 + 3ρ ρ2n . 2(1 − ρ) 1 − ρ2n

(7.34)

Best Approximations in the Spaces C and L

532

Chapter 7

By applying (twice) the Abel transformation, we get ∞

∞

j=1

ν=0

ψ(n) Φn (t) = ψ(n + j) cos jt = (ν + 1)Δ2 ψ(n + ν)Fν (t), (7.35) + 2 where 2 Fν (t) = ν+1

sin(ν + 1)t/2 2 sin t/2

2 ,

ν = 0, 1, 2, . . . ,

are the Fejér kernels of degree ν. In view of relation (7.1), this yields 1 Φn (t) ≥ Δ2 ψ(n)F0 (t) = Δ2 ψ(n) ∀t ∈ R. 2

(7.36)

Note that, by virtue of (7.33), for all n such that ρ2n ≤ 1/2, we have * βπ | sin(θn π − )| ≥ 2

ρ4n 1− = (1 − ρ2n )2

1 − 2ρ2n . 1 − ρ2n

(7.37)

Thus, in view of relations (7.1), (7.3), and (7.34)–(7.37), we find βπ 1 − 2ρ2n 1 2 |Φn (t) sin(θn π − )| ≥ Δ ψ(n) 2 2 1 − ρ2n > ψ(n)

1 + 3ρ ρ2n ≥ |Rn (t)|. 2(1 − ρ) 1 − ρ2n

(7.38)

Combining relations (7.28) and (7.38), we obtain sgn Wn (t) = sgn (Φn (t) sin(θn π −

βπ )) = ε0 sgn P hin (t), ε0 = ±1. (7.39) 2

By virtue of relations (7.1), (7.3), and (7.36), the function Φn (t) is strictly positive and, hence, inequality (7.23) is true. This means that relation (7.7) is proved and, therefore, according to Theorem ∗ (·) is the polynomial of the best approximation of the 2.1, the polynomial Tn−1 function Ψβ (·) in the space L1 . Moreover, since the function is continuous, this polynomial is unique. Thus, we conclude that there are no points θn ∈ [0, 1) satisfying equations (7.8) other than θn . All assertions of the lemma are proved. We now return to the proof of the theorem.

Section 7

Serdyuk Theorem

533

By virtue of Lemma 7.1, we conclude that if, for some n ∈ N and ρ ∈ (0, 1), we have ψ ∈ Sn,ρ , then the function Ψβ (·) satisfies the condition A∗n . Hence, by using Proposition 3.3 and equality (7.7), we get En (Ψβ )L = Ψβ (t) −

∗ Tn−1 (t)1

π =

∗ |Ψβ (t) − Tn−1 (t)|dt

−π

π =|

∗ (Ψβ (t) − Tn−1 (t))sgn sin(nt − θn π)dt|

−π

π =|

Ψβ (t)sgn sin(nt − θn π)dt|.

(7.40)

−π

Further, by using the equality ∞ 4 sin((2ν + 1)(nt − θn π)) sgn sin(nt − θn π) = , π 2ν + 1 ν=0

the Parseval equality, and relation (7.40), we conclude that En (Ψβ )L = 4|

∞ ψ((2ν + 1)n) ν=0

2ν + 1

sin((2ν + 1)θn π −

βπ )|. 2

(7.41)

Combining equality (7.41) with Proposition 3.4, we complete the proof of the theorem. 7.2. Let us now present some simple corollaries of Theorem 7.1. Note that if ψ(k) is such that, for any k ∈ N, ψ(k + 1) < ρ∗ , ψ(k)

(7.42)

where the number ρ∗ is chosen from the condition 1 − 2ρ∗ =

(1 + 3ρ∗ )ρ2∗ (1 − ρ∗ ) 1 − 2ρ2∗

(7.43)

(here, ρ∗ = 0.3253678 . . .), then ψ ∈ S1,ρ for any ρ ∈ (0, ρ∗ ). Indeed, in this case, we have Δ2 ψ(k) = ψ(k)(1 − 2

ψ(k + 1) ψ(k + 2) + ) > ψ(k)(1 − 2ρ∗ ) > 0 ψ(k) ψ(k)

Best Approximations in the Spaces C and L

534

Chapter 7

and, for any n ∈ N, Δ2 ψ(k) (1 + 3ρ∗ )ρ2n ψ(k + 1) ∗ . >1−2 > 1 − 2ρ∗ > ψ(k) ψ(k) (1 − ρ∗ ) 1 − 2ρ2n ∗ Thus, we arrive at the following statement: Corollary 7.1. If the sequence ψ(k) satisfies condition (7.42), then equalities (7.5) are true for any n ∈ N. 7.3. If ψ(k) is such that lim

k→∞

ψ(k + 1) = 0, ψ(k)

i.e., ψ ∈ D0 in the terminology of Section 5.20, then Corollary 7.1 implies that there exists a number n0 such that equalities (7.5) hold for all n ≥ n0 . The following assertion shows that this fact remains true also in the case where ψ ∈ Dq , q ∈ [0, 1), i.e., whenever ψ(k + 1) = q, q ∈ [0, 1). k→∞ ψ(k) lim

(7.44)

Theorem 7.2. If a sequence satisfies condition (7.44), then there exists a number n0 such that equalities (7.5) are true for all n ≥ n0 . Proof. Assume that δk =

ψ(k + 1) −q ψ(q)

and εn = sup |δk |. k≥n

The sequence εn monotonically tends to zero and, thus, Δ2 ψ(n) ψ(n + 1) ψ(n + 2) ψ(n + 1) =1−2 + ψ(n) ψ(n) ψ(n + 1) ψ(n) = 1 − 2(q + δn ) + (q + δn+1 )(q + δn ) = 1 − 2q + q 2 + {(q − 2)δn + qδn+1 + δn δn+1 } = (1 − q)2 + ξn ,

(7.45)

where |ξn | ≤ (2−q)εn +qεn+1 +εn εn+1 ≤ (2−q)εn +qεn +ε2n = εn (2+εn ). (7.46)

Section 7

Serdyuk Theorem

535

Let n1 be the least natural number such that εn (2 + εn ) < (1 − q)2 ,

n = n1 , n1 + 1, . . . .

(7.47)

Relations (7.45)–(7.47) imply that the sequence ψ(k) is strictly convex beginning with n1 , namely, Δ2 ψ(n) > 0,

n = n1 , n1 + 1, n1 + 2, . . . .

Moreover, in view of (7.47), we have ψ(n + 1) (1 − q)2 ≤ q + εn < q + ψ(n) 1 + (1 − q)2 + 1 n1 ) . Let n0 be the least natural number such that (1 − q)2 ≥

5 + 3q 2 (1 + q 2 )2n + εn (2 + εn ) 1 − q 2 22n 1 − (1 + q 2 )2n /22n−1

(7.48)

for n = n0 , n0 + 1, . . . . The number n0 exists in view of the fact that the right-hand side of (7.48) monotonically approaches zero as n → ∞. By using the inequality ρ=q+

1 + q2 1 + (1 − q)2 − 1 < , 2

we obtain 1 + 3ρ (1 + q 2 )2n 5 + 3q 2 ρ2n > . 1 − q 2 22n 1 − (1 + q 2 )2n /22n−1 1 − ρ 1 − 2ρ2n

(7.49)

Combining inequalities (7.48) and (7.49), we see that (1 − q)2 >

(1 + 3ρ)ρ2n + εn (2 + εn ), (1 − ρ) 1 − ρ2n

(7.50)

Best Approximations in the Spaces C and L

536

Chapter 7

whence, in view of relations (7.45) and (7.46), we conclude that condition (7.3) is satisfied. 7.4. As the most well-known kernels Ψβ (t) whose coefficients ψ(k) satisfy condition (7.4), we can mention the Poisson kernels Pβq (t) =

∞

q k cos(kt −

k=1

βπ ), 2

q ∈ (0, 1), β ∈ R.

(7.51)

For these kernels, the following analogs of Lemma 7.1 and Theorem 7.2 are true: Lemma 7.2. Let q ∈ (0, 1) and let β ∈ R. Then, for any n ∈ N, there ∗ (x) of degree n − 1 such that exists a trigonometric polynomial Tn−1 ∗ sgn [Pβq (t) − Tn−1 (t)] = −sgn sin(nt − θn π),

(7.52)

where θn is the unique root of the equation ∞

q (2ν+1)n cos((2ν + 1)θn π −

ν=0

βπ )=0 2

(7.53)

∗ is uniquely determined from the conditions of interon [0, 1) . Moreover, Tn−1 polation: ∗ Tn−1 (

θn π + kπ θn π + kπ ) = Pβq ( ), n n

k = 0, 1, . . . , 2n − 2.

(7.54)

Proof. By using the same reasoning as in the proof of Lemma 7.1, we arrive at equality (7.22) which, in this case, takes the form θn π )Wn (t), n

(7.55)

(n) c0 (n) (cj cos jt + dj sin jt), + 2

(7.56)

∗ (t) = −2 sin n(t − Pβq (t) − Tn−1

where

(n)

Wn (t) =

∞

j=1

(n) cj

=

∞ ν=0

q (2ν+1)n+j sin((2ν + 1)θn π −

βπ ), 2

(7.57)

Section 7

Serdyuk Theorem (n)

dj

=

∞

q (2ν+1)n+j cos((2ν + 1)θn π −

ν=0 (n)

By virtue of (7.53), dj (7.56) and (7.57), we get

537 βπ ). 2

(7.58)

= 0, j = 1, 2, . . . , and, hence, in view of relations

∞

∞

j=1

ν=0

1 j βπ Wn (t) = ( + q cos jt) q (2ν+1)n sin((2ν + 1)θn π − ). 2 2

(7.59)

Note that the function ∞

1 − q2 1 1 j q cos jt = , q ∈ (0, 1), g(t) = + 2 2 1 − 2q cos t + q 2 j=1

is strictly positive. Hence, to prove the theorem (in view of Theorem 2.1, Corollary 2.1, and equalities (7.55) and (7.59)), it suffices to show that the quantity σn,q,β =

∞

q (2ν+1)n sin((2ν + 1)θn π −

ν=0

βπ ) 2

differs from zero for all q ∈ (0, 1), β ∈ R, and n ∈ N. Let Gρ (x) =

∞

ρ2k+1 cos(2k+1)x =

ρ(1 − ρ2 ) cos x , ρ ∈ (0, 1), (7.60) (1 + ρ2 )2 − 4ρ2 cos2 x

ρ2k+1 sin(2k +1)x =

ρ(1 + ρ2 ) sin x , ρ ∈ (0, 1). (7.61) (1 + ρ2 )2 − 4ρ2 cos2 x

j=0

Hρ (x) =

∞ j=0

It follows from (7.61) that the function Hρ (x) is positive on (0, π) and negative on (π, 2π). By virtue of (7.60), Gρ (x) = −ρ(1 − ρ2 ) sin x

(1 + ρ2 )2 + 4ρ2 cos2 x . ((1 + ρ2 )2 − 4ρ2 cos2 x)2

Therefore, the function Gρ (x) decreases on (0, π), increases on (π, 2π), and, in addition, π 3π Gρ (x) > 0, x ∈ [0, ) ∪ ( , 2π], 2 2 π 3π Gρ (x) < 0, x ∈ ( , ). 2 2

Best Approximations in the Spaces C and L

538

Chapter 7

We have ∞

q (2ν+1)n cos((2ν + 1)θn π −

ν=0

βπ βπ βπ ) = Gqn (θn π) cos + Hqn (θn π) sin . 2 2 2

Thus, in view of the indicated properties of the functions Gρ (x) and Hρ (x) for ρ = q n and equality (7.53), we conclude that π θn π ∈ [ , π] for β ∈ [4m, 1 + 4m] ∪ [2 + 4m, 3 + 4m], m ∈ Z, 2

(7.62)

and π θn π ∈ [0, ] for β ∈ [1 + 4m, 2 + 4m] ∪ [3 + 4m, 4(m + 1)], m ∈ Z. (7.63) 2 Hence, by using relations (7.62) and (7.63) and the equality σn,q,β = Hqn (θn π) cos

βπ βπ − Gqn (θn π) sin , 2 2

we find σn,q,β > 0 for β ∈ [4m, 1 + 4m] ∪ [3 + 4m, 4(m + 1)], m ∈ Z, σn,q,β < 0 for β ∈ [1 + 4m, 3 + 4m],

m ∈ Z.

(7.64) (7.65)

This means that σn,q,β = 0 for any n ∈ N, β ∈ R, and q ∈ (0, 1). Lemma 7.2 is proved. Thus, the Poisson kernels Pβq (t) satisfy the condition A∗n for any β ∈ R, q ∈ (0, 1), and any natural n. Therefore, by virtue of equality (7.41) for ψ(k) = q k and Theorem 3.1, we arrive at the following statement: Theorem 7.3. If β ∈ R and q ∈ (0, 1), then, for any n ∈ N, the kernel Pβq (t) satisfies the condition A∗n and q ) = En (Lqβ,1 )1 = En (Cβ,∞

1 En (Pβq )1 = Pβq ∗ sgn sin n(·)C π

∞

=

4 q (2ν+1)n βπ | sin((2ν + 1)θn π − )|, π 2ν + 1 2 ν=0

where θn is the unique root of equation (7.53) on [0, π) .

Section 8

8.

Bernstein Inequalities for Polynomials

539

Bernstein Inequalities for Polynomials

8.1. In the next two sections, we present the theorems in which the differential-difference characteristics of a function f ∈ X are established according to a given sequence En (f )X of its best approximations. In approximation theory, as indicated in Chapter 6, theorems of this sort are called inverse. The first inverse theorems were established by Bernstein at the beginning of the last century. The proofs of these theorems are based on the inequalities for the norms of polynomials and their derivatives. Inequalities of this sort are called Bernstein inequalities. Definition 8.1. Assume that ψ = ψ(k) is an arbitrary sequence and β is a real number. We say that a couple (ψ, β) belongs to a set BX ((ψ, β) ∈ BX) if the following inequality holds for any trigonometric polynomial Tn (·) of degree n: −1 (Tn (·))ψ β X ≤ O(1)|ψ(n)| Tn (·)X,

ψ (Tn (·))ψ β = Dβ Tn (·),

(8.1)

where, O(1) is a quantity uniformly bounded in n and Tn ∈ T2n+1 . In the case where X = C, ψ(k) = k −1 , k = 1, 2, . . . , and β = 1, relation (8.1) turns into the well-known Bernstein inequality (where the quantity O(1) can be replaced by 1 ). For ψ(k) = k −r , r ∈ N, and β = r, this inequality follows from results obtained by Stechkin [1]. Inequality (8.1) is also known in some other cases. 8.2. Let us now indicate several sufficient conditions for the validity of the inclusion (ψ, β) ∈ BX. First, we consider the case where X = Lp , 1 < p < ∞, and establish the following assertion: Proposition 8.1. Assume that the function ψ(·) is such that sup q

q+1 2

|ψn−1 (k + 1) − ψn−1 (k)| ≤ Kλn ,

K = const,

(8.2)

k=2q

where ψn−1 (k) =

−1 ψ (k), 1 ≤ k ≤ n, 0,

k > n,

(8.3)

540

Best Approximations in the Spaces C and L

Chapter 7

λn = max |ψn−1 (k)| = max |ψ(k)|−1 .

(8.4)

and

k

k≤n

Then the following inequality holds for any trigonometric polynomial Tn (·) of degree n : (Tn (·))ψ β Lp ≤ Cp λn Tn (·)Lp ,

1 < p < ∞,

(8.5)

where Cp is a quantity which may depend only on the function ψ(·) and the number p. Proof. It is necessary to show that Dψ β Lp ∩T2n+1 ≤ Cp λn ,

(8.6)

where Dψ β is an operator acting upon a function f (·) such that ∞

a0 ak cos kx + bk sin kx S[f ] = + 2

(8.7)

k=1

according to the formula (Dψ β f )(t) =

∞

ψ −1 (k)(ak cos(kt + βπ/2) + bk (sin kt + βπ/2)).

(8.8)

k=1

˜ = (˜ Assume that multiplicators M = (μk ) and M μk ), where ⎧ ⎨0, μk = and

k = 0, k ≥ n + 1,

⎩ψ −1 (k) cos βπ , 1 ≤ k ≤ n, 2 ⎧ ⎨0,

k = 0, k ≥ n + 1,

μ ˜k =

⎩ψ −1 (k) sin βπ , 1 ≤ k ≤ n, 2 specify operators from Lp in Lp by the formulas (M f )(t) =

n k=1

(8.9)

ψ −1 (k) cos

βπ (ak cos kt + bk sin kt) 2

(8.10)

(8.11)

Section 8

Bernstein Inequalities for Polynomials

and ˜ f )(t) = (M

n

ψ −1 (k) sin

k=1

βπ (ak cos kt + bk sin kt). 2

541

(8.12)

Thus, for f ∈ T2n+1 , in view of (8.8), (8.10), and (8.11), we conclude that ˜ ˜ (Dψ β f )(·) = (M f )(·) − (M f )(·),

(8.13)

where f˜(·) is a function trigonometrically conjugate to f (·), namely, f˜(·) = (U f )(·). Therefore (see Section 6.3), ˜ ˜ Dψ β Lp ∩T2n+1 = M − M U p ≤ M p + M p U p .

(8.14)

To complete the proof of inequality (8.6), it remains to note that, by virtue of relations (8.2) and (8.4), sup |μk | ≤ λn , k

sup q

q+1 2

|μk+1 − μk | ≤ Kλn ,

k=2q

sup |˜ μk | ≤ λn , k

sup q

q+1 2

|˜ μk+1 − μk | ≤ Kλn

k=2q

and apply the Marcinkiewicz and Riesz theorems. Proposition 8.1 yields the following corollary: Corollary 8.1. Assume that ψ(k), k ∈ N, is an arbitrary nonincreasing sequence of nonnegative numbers. Then ψ ∈ BLp , 1 < p < ∞. In particular, if M is the set of convex (downward) sequences vanishing at infinity, then M ⊂ BLp . Indeed, in this case, λn = ψ −1 (n) and, hence, inequalities (8.1) and (8.5) coincide. 8.3. We now consider the cases where X = C or X = L1 . Let Tn (x) be an arbitrary trigonometric polynomial of degree n. Then, for its (ψ, β)-derivative, we can write π n 1 ψ(n) βπ ψ ψ(n)(Tn (x))β = Tn (x + t)( cos(kt − ))dt π ψ(k) 2 −π

1 = π

k=1

π Tn (x + t)τ2n−1 (t)dt, −π

(8.15)

Best Approximations in the Spaces C and L

542 where

τ2n−1 (t) =

2n−1

(n)

λk cos(kt −

k=1

(8.16)

ψ(n)/ψ(k), 1 ≤ k ≤ n,

and (n) λk

βπ ) 2

=

2 − k/n,

Chapter 7

(8.17)

n ≤ k ≤ 2n.

Thus, by applying the Minkowski inequality, we find ψ(n)(Tn (·))ψ β X ≤

1 Tn (·)Xτ2n−1 1 π

( · 1 = · L1 ).

(8.18)

Hence, to prove that the couple (ψ, β) belongs to the set BX , it suffices to check the validity of the inequality τ2n−1 1 ≤ K, n = 1, 2, . . .

(8.19)

(n)

There are numerous conditions for the coefficients λk guaranteeing that estimate (8.19) is satisfied. Some conditions of this sort are studied in Chapter 1. We now use, e.g., relations (1.8.14) and (1.8.15), which enable us to conclude that τ2n−1 1 = O(1)(| sin(βπ/2)|

2n−1

k −1 |λk | (n)

k=1

+

2n−1

(2n − k)−1 |λk | + (n)

k=1

where

2n−1 1 (n) k(2n − k)|Δ2 λk−1 |), (8.20) 2n k=1

(n)

(n)

(n)

(n)

Δ2 λk−1 = λk−1 − 2λk + λk+1 ,

(n)

λ0

=0

and O(1) is a quantity uniformly bounded in n. (n) According to relation (8.17), for k ∈ [n, 2n], we have λk = (2n − k)/n and, thus, 2n−1

k −1 |λk | = (n)

k=1 2n−1 k=n

2n−1 k=1

−1

(2n − k)

2n−1 2 1 ( − )≤2 k −1 = O(1), k n

(n) |λk |

(8.21)

k=n

=

2n−1 k=n

2n − 1 = O(1) n(2n − k)

(8.22)

Section 8

Bernstein Inequalities for Polynomials

and

543

2n−1 1 (n) k(2n − k)Δ2 λk−1 = 0. 2n k=n+1

Therefore, in the analyzed case, relation (8.20) takes the form τ2n−1 1 = O(1)(| sin(βπ/2)|

n−1

k −1 |λk | (n)

k=1

+

n−1

1 (n) + k(2n − 1)|Δ2 λk−1 |) (8.23) 2n n−1

−1

(2n − k)

(n) |λk |

k=1

k=2

and, finally, we arrive at the following statement: Proposition 8.2. Assume that the sequence ψ(k) and the number β are such that ψ(n)(| sin(βπ/2)|

n−1

(k|ψ(k)|)−1 +

k=1

+

n−1

((2n − k)|ψ(k)|)−1

k=1

1 2n

n−1

k(2n − k)|Δ2 (1/ψ(k − 1))|) = O(1), (8.24)

k=2

where O(1) is a quantity uniformly bounded in n. Then (ψ, β) ∈ BX. If the sequence ψ(k) is nonincreasing and positive, then ψ(n)

n−1

((2n − k)|ψ(k)|)−1 = O(1).

(8.25)

k=1

If, in addition, the quantities Δ2 (1/ψ(k − 1)) do not change their signs, then ψ(n) k(2n − k)|Δ2 (1/ψ(k − 1))| = O(1). 2n n−1 k=2

Hence, Proposition 8.2 implies the following assertion: Corollary 8.2. If ψ(k), k ∈ N, is an arbitrary nonincreasing sequence of nonnegative numbers for which either Δ2 (1/ψ(k − 1)) ≥ 0,

(8.26)

Best Approximations in the Spaces C and L

544

Chapter 7

or Δ2 (1/ψ(k − 1)) ≤ 0, k = 1, 2, . . . , n − 1,

(8.27)

and, in addition, | sin(βπ/2)|

n−1

ψ(n)(kψ(k))−1 = O(1),

(8.28)

k=1

then (ψ, β) = BX. 8.4. If ψ ∈ F, then, in view of Corollary 3.13.3 and relation (3.13.18), we have η(t) − t ψ(t) ≤ Ktψ (t) (8.29) ≤ K1 tψ (t), t ≥ 1, t and, thus, ψ(n)

n−1 k=1

1 1 = ψ(n) + O(1) kψ(k) (k + 1)ψ(k) n−2 k=1

≤ ψ(n)

n−1 k=1 n−1

≤ ψ(n)

1 k+1

k+1

dt + O(1) ψ(t)

k

dt + O(1) ≤ Kψ(n) tψ(t)

1

n−1

ψ (t) dt + O(1) ψ 2 (t)

1 n−1

d(

= Kψ(n)

1 ) + O(1) = O(1). ψ(t)

(8.30)

1

This means that relation (8.28) holds for any sequence ψ from F and all β ∈ R1 . At the same time, if ψ ∈ M0 = M0 \ FC , then, as can be shown by analyzing the function ψ(t) = ln−1 (t + e) from M0 as an example, the validity of condition (8.28) can be guaranteed only by the equality sin(βπ/2) = 0, i.e., this condition is satisfied for β = 2k, k = 0, ±1, . . . . As a result, we arrive at the following statement: Corollary 8.3. Assume that X is either C or Lp , 1 ≤ p < ∞. If ψ ∈ F and one of the conditions (8.26) or (8.27) is satisfied, then, for any β ∈ R1 , the couple (ψ, β) belongs to BX.

Section 9

Inverse Theorems

545

At the same time, if ψ ∈ M0 = M0 \ F and one of the conditions (8.26) or (8.27) is satisfied, then (ψ, β) ∈ BX provided β = 2k, k = 0, ±1, . . . . In the case where X = Lp , 1 < p < ∞, the inclusion (ψ, β) ∈ BX holds for any ψ ∈ M and β ∈ R1 .

9.

Inverse Theorems

9.1. The principal result in this section can be formulated in the form of the following theorem: Theorem 9.1. Assume that X is either C or Lp , 1 ≤ p < ∞, f ∈ X, and En (f )X =

inf

Tn−1 ∈T2n−1

f (·) − Tn−1 (·)X.

(9.1)

Then the following assertions are true: (i) if (ψ, β) ∈ BX, f ∈ X, and the series ∞

Ek (f )X|ψ(k)|−1

(9.2)

k=1

is convergent, then the derivative fβψ ∈ X exists and satisfies the inequality ∞

En (fβψ )X ≤ K1

Ek (f )X|ψ(k)|−1 ,

n ∈ N;

(9.3)

k=n

(ii) if (ψ, β) ∈ BX, ψ ∈ M0 , f ∈ X, and the series ∞

Ek (f )X(kψ(k))−1

(9.4)

k=1

is convergent, then the derivative fβψ ∈ X exists and satisfies the inequality En (fβψ )X ≤ K2 (

∞ En (f )X Ek (f )X(kψ(k))−1 ), n ∈ N ; + ψ(n) k=n+1

(9.5)

Best Approximations in the Spaces C and L

546

Chapter 7

(iii) if (ψ, β) ∈ BX, ψ ∈ F, η(ψ; t) − t ≥ K3 > 0, f ∈ X, and the series ∞

Ek (f )X(ψ(k)(η(k) − k))−1

(9.6)

k=1

is convergent, then the derivative fβψ ∈ X exists and satisfies the inequality En (fβψ )X ≤ K4 (

∞ En (f )X + Ek (f )X(ψ(k)(η(k) − k))−1 ), (9.7) ψ(n) k=n+1

where n ∈ N, η(t) = η(ψ; t) = ψ −1 (ψ(t)/2), and Ki , i = 1, 4, are quantities which may depend on the function ψ(·). Proof. First, we prove assertion (i). Assume that {tn (·)}∞ n=1 is a sequence of trigonometric polynomials delivering, for any n ∈ N, the best approximation of the function f in the space X. Then the series ∞ tn (x) + (tk (x) − tk−1 (x)) (9.8) k=n+1

converges to f (x) in the norm of the space X because its partial sums Sm (x) for m > n coincide with the polynomials tm (x). Consider a series (tn (x))ψ β

+

∞

(tk (x) − tk−1 (x))ψ β.

(9.9)

k=n+1

Let us show that this series converges in X to a sum S(x) whose Fourier series has the form ∞ k=1

βπ βπ 1 (ak (f ) cos(kx + ) + bk (f ) sin(kx + )). ψ(k) 2 2

(9.10)

This proves the existence of the derivative fβψ (·). The difference uk (x) = tk (x) − tk−1 (x) is a polynomial of degree k. Thus, since (ψ, β) ∈ BX, in view of inequality (8.1), we obtain (uk (x))ψ β ≤ K

uk (x) K ≤ (tk (x) − f (x) + tk−1 (x) − f (x)) |ψ(k)| |ψ(k)|

≤ 2KEk (f )(ψ(k))−1 .

(9.11)

Section 9

Inverse Theorems

547

Here and in what follows, by K we denote the quantities which may depend only on the function ψ(·). For the sake of simplicity, the subscript X in the quantities · X and Ek (f )X is omitted. Therefore, according to (9.9) and (9.11), ∞

(uk (x))ψ β ≤ K

k=n+1

∞

Ek (f )(ψ(k))−1 .

(9.12)

k=n+1

Under the assumptions of the theorem, this series is convergent. This enables us to conclude that series (9.9) indeed converges in the norm of the space X to a certain function S(x) from this space. (n) (n) Let ak = ak (tn ) and let bk = bk (tn ), k = 0, 1, . . . , be the Fourier (n) coefficients of the polynomials tn (·). Then the corresponding coefficients αk (n) and βk of the polynomials (tn (·))ψ β have the form (n)

αk = (n)

βk

=

1 (n) (n) (ak cos(βπ/2) + bk sin(βπ/2)), ψ(k)

(9.13)

1 (n) (n) (b cos(βπ/2) − ak sin(βπ/2)). ψ(k) k

(9.13 )

Since the equality S(x) = lim (tn (x))ψ β n→∞

holds at least in a sense of convergence in L1 , we get (n)

ak (s) = lim αk n→∞

(n)

and bk (s) = lim βk , n→∞

k = 0, 1, . . . .

(9.14)

k = 0, 1, . . . .

(9.15)

It is clear that (n) lim a n→∞ k

= ak (f ) and

(n) lim b n→∞ k

= bk (f ),

Relations (9.13)–(9.15) imply that ak (s) =

1 (ak (f ) cos(βπ/2) + bk (f ) sin(βπ/2)), ψ(k)

(9.16)

bk (s) =

1 (bk (f ) cos(βπ/2) − ak (f ) sin(βπ/2)). ψ(k)

(9.17)

Best Approximations in the Spaces C and L

548

Chapter 7

Thus, we conclude that the Fourier series of the function S(x) indeed coincides with series (9.10). This means that the function f (x) possesses the (ψ, β)derivative fβψ (·) which belongs to the space X, and satisfies the equality ∞

fβψ (·) = (tn (·))ψ β +

(uk (x))ψ β,

n ∈ N,

(9.18)

k=n+1

in the norm of this space. Inequality (9.3) follows from (9.18) in view of estimate (9.12). 9.2. To prove (ii) and (iii), we follow the same scheme as in case (i). Namely, we assume that {tn (·)}∞ n=1 is the sequence of trigonometric polynomials of the best approximation of the function f in the space X. For any natural n, we set n0 = n, n1 = [η(n)]+1, . . . , nk = [η(nk−1 )]+1, . . . , where [α] is the integer part of a number α. In this case, the series tn0 (x) +

∞

(tnk (x) − tnk−1 (x))

(9.19)

k=1

converges to f (x) in the norm of the space X. We now consider the following analog of series (9.9): (tn0 (x))ψ β

+

∞

(tnk (x) − tnk−1 (x))ψ β

(9.20)

k=1

and prove that this series converges in the space X to the sum S(x) whose Fourier series has the form (9.10). In view of relation (8.1), we arrive at an analog of estimate (9.11) for the difference uk (x) = tnk − tnk−1 (x), namely, −1 (uk (x))ψ β ≤ KEnk−1 +1 (f )(ψ(nk )) .

(9.21)

This yields ∞ k=1

(uk (x))ψ β

−1

≤ K(En+1 (f )(ψ(n))

+

∞

Enk +1 (f )(ψ(nk ))−1 ). (9.22)

k=1

Note that if ψ ∈ M0 , then there exists a constant K such that τ ≤ C, η¯(τ ) = η¯(ψ; τ ) = ψ −1 (2ψ(τ )), τ ≥ η(1) τ − η¯(τ )

(9.23)

Section 9

Inverse Theorems

and

τ − η¯(τ ) ≤ C, η(t) − t

τ ∈ [t, η(t)],

549

τ ≥ η(1).

(9.24)

Indeed, by setting z = η¯(τ ), in view of the definition of the set M0 (see Section 3.12), we obtain τ η(z) z = =1+ = 1 + μ(ψ; z) ≤ K. τ − η¯(τ ) η(z) − z η(z) − z

(9.25)

The proof of relation (9.24) is also quite simple: τ η(t) τ τ − η¯(τ ) ≤ ≤K ≤ K. η(t) − t η(t)(η(t) − t) η(τ ) By virtue of estimates (9.23) and (9.24) and the facts that the inequality ψ(τ ) ≥ ψ(η(τ )) = ψ(τ )/2 holds on any interval [t, η(t)] and, without loss of generality, we can assume that η(t) − t > 1, we find n k −1 1 Eν+1 (f ) Enk +1 (f ) ≤K ψ(nk ) ψ(ν) η(nk−1 ) − nk−1 ν=n k−1

≤K

n k −1 ν=nk−1

≤

n k −1 ν=nk−1

Eν+1 (f ) ν ν − η¯(ν) νψ(ν) ν − η¯(ν) η(nk−1 ) − nk−1

Eν+1 (f ) . νψ(ν)

(9.26)

Substituting this estimate in (9.22), we get ∞ k=1

(uk (x))ψ β

−1

≤ K(En+1 (f )(ψ(n))

+

∞

Ek (f )(kψ(k))−1 .

(9.27)

k=n+1

According to the conditions of case (ii), the series on the right-hand side in (9.27) converges in the norm of the space X. Hence, series (9.20) converges in the space X to a certain function s ∈ X. To complete the proof of case (ii), it remains to show that S[s] = S[fβψ ]. This can be done by repeating the reasoning used to establish equality (9.18)) and, hence, inequality (9.5) (by combining an analog of equality (9.18) with relation (9.27).

Best Approximations in the Spaces C and L

550

Chapter 7

To prove (iii), we establish the following analog of estimate (9.27): ∞

(uk (x))ψ β

k=1 −1

≤ K(En+1 (f )(ψ(n))

∞

+

Ek (ϕ)(ψ(k)(η(k) − k))−1 . (9.28)

k=n+1

We start from relation (9.22), according to which, to prove inequality (9.28), it suffices to show that ∞ ∞ Enk +1 (f )(ψ(nk ))−1 ≤ K Ek (f )(ψ(k)(η(k) − k))−1 . (9.29) k=1

k=n+1

To deduce (9.29), we first assume that η(nk−1 ) − nk−1 ≥ 1 ∀k ∈ N.

(9.30)

In this case, 1≤

η(nk−1 ) − [η(nk−1 )] η(nk−1 ) − nk−1 =1+ 0 and, therefore, Enk +1 (f )(ψ(nk ))−1 ≤ K6 (ψ(nk−1 ))−1 (η(nk−1 ) − nk−1 )−1 ]

[η(nk−1 )]

≤ K7

ν=nk−1

Eν+1 (f )(ψ(ν)(η(ν) − ν))−1 .

(9.33 )

Section 9

Inverse Theorems

551

Thus, by virtue of (9.33) and (9.33 ), we find ∞

−1

Enk +1 (ψ(nk ))

≤K

∞ [η(n k−1 )]

Eν+1 (f )(ψ(ν)(η(ν) − ν))−1

k=1 ν=nk−1

k=1

=K

∞

Ek+1 (f )(ψ(k)(η(k) − k))−1 ,

k=n

which implies inequality (9.29) and, hence, inequality (9.28). The proof of case (iii) is now completed by repeating the reasoning used to complete the proof in case (ii). Theorem 9.1 is proved.

8. INTERPOLATION

1.

Interpolation Trigonometric Polynomials

1.1. Assume that, in the interval [0, 2π), we have a collection of points (nodes of interpolation): I = {0 ≤ x1 < x2 < . . . < x2n+1 < 2π}

(1.1)

and an arbitrary set of real numbers Y = {y1 , y2 , . . . , y2n+1 }.

(1.2)

A trigonometric polynomial Tn (x) = Tn (Y, I, x) taking values yi , i = 1, 2n + 1, at the points xi i.e., Tn (xi ) = yi ,

i = 1, 2n + 1,

(1.3)

is called an interpolation trigonometric polynomial for a given set Y over the system of nodes I. The interpolation polynomial Tn (Y, I, x) exists and is unique for any collection Y and system of nodes I. Moreover, it can be represented in the explicit form as follows: Tn (Y, I, x) =

2n+1 k=1

yk

x−xk−1 2 xk −xk−1 2

1 sin x−x 2 . . . sin

1 sin xk −x . . . sin 2

sin sin

x−xk+1 . . . sin x−x22n+1 2 xk −xk+1 . . . sin xk −x22n+1 2

. (1.4)

It is easy to see that the numerator of each term on the right-hand side in (1.4) is a trigonometric polynomial of degree n and equalities (1.3) are satisfied. The uniqueness of the polynomial Tn (Y, I, x) follows from Corollary 6.1.2. 553

554

Interpolation

Chapter 8

Let tn,j (I, x) be a polynomial of degree n equal to 1 for x = xj and equal to 0 at all other nodes. The polynomials tn,j (I, x), j = 1, 2n − 1, are called the fundamental polynomials corresponding to the system of nodes I. In view of (1.4), we get tn,j (I, x) =

x−xj−1 2 xj −xj−1 2

1 sin x−x 2 . . . sin

sin

xj −x1 2

. . . sin

sin sin

x−xj+1 . . . sin x−x22n+1 2 xj −xj+1 x −x . . . sin j 22n+1 2

(1.5)

and, for any set Y and any system of nodes I, Tn (Y, I, x) =

2n+1

yj tn,j (I, x).

(1.6)

j=1

In what follows, we study the procedure of interpolation only in the case of equidistant nodes: (n)

(n)

xj = xj,ξ = xj,ξ(n) = ξ +

2πj , j = 0, 1, . . . , 2n, 2n + 1

(1.7)

where ξ = ξ (n) is a number which may depend on n ∈ N. A polynomial which coincides with a given function f (·) at points (1.7) is called an interpolation trigonometric polynomial of degree n for the function f (·) and the system of nodes (1.7). This polynomial is denoted by In (f ; ξ; x). 1.2. The Dirichlet kernel 1 sin(n + 1/2)t Dn (t) = + cos kt = 2 2 sin t/2 n

(1.8)

k=1

is a trigonometric polynomial of degree n and 0, j = 1, 2, . . . , 2n, 2πj Dn ( )= 2n + 1 n + 1/2, j = 0. Therefore, the polynomial tn,j (ξ, x) = (n)

2 Dn (x − xj ) 2n + 1

is equal to 1 for x = xj = xj,ξ and to 0 at all other points xk , i.e., the polynomials tn,j (ξ, x), j = 1, 2n − 1, are fundamental for system (1.7). Consequently,

Section 1

Interpolation Trigonometric Polynomials

555

by virtue of (1.6), we can write 2 f (xj )Dn (x − xj ). 2n + 1 2n

In (f, ξ, x) =

(1.9)

j=0

We now represent this polynomial in the canonical form: n n 1 (n) (n) (n) iνx (aν cos νx + bν sin νx) = c(n) . (1.10) In (f, ξ, x) = a0 + ν e 2 ν=−n ν=1

For this purpose, the kernel Dn (u) in relation (1.9) is rewritten in the form (1.8). By comparing similar terms in equalities (1.9) and (1.10), we conclude that 2 f (xj ) cos νxj , 2n + 1

ν = 0, 1, . . . n,

(1.11)

2 f (xj ) sin νxj , 2n + 1

ν = 1, 2, . . . n.

(1.11 )

ν = −n, . . . , 0, . . . , n.

(1.12)

2n

aν(n) =

j=0 2n

bν(n) =

j=0

Similarly, 1 f (xj )e−iνxj , 2n + 1 2n

cν =

j=0

(n)

(n)

The numbers aν and bν are called the Fourier–Lagrange coefficients of (n) the function f corresponding to the system of nodes (1.7) and the numbers cν are called the complex Fourier–Lagrange coefficients of the function f. The following assertion establishes the relationship between the complex Fourier coeffi(n) cients fˆk of the function f and its Fourier–Lagrange coefficients cν . (n) Proposition 1.1. Assume that fˆν and cν are, respectively, the complex Fourier coefficients and Fourier–Lagrange coefficients of the function f for system (1.7). Then the following equalities are true:

cν(n)

=

∞

fˆν+m(2n+1) ei(2n+1)mξ ,

ν = −n, . . . , 0, . . . , n,

(1.13)

m=−∞

provided that the Fourier series S[f ] of the function f converges at the points (n) xj = xj,ξ to the values f (xj ) and the sum of the series on the right-hand side in (1.13) is equal to the limit of symmetric expanding partial sums.

556

Interpolation

Chapter 8

Proof. In view of relation (1.2), we have (2n +

1)cν(n)

2n

=

f (xj )e−iνxj

j=0 2n ∞

=

fˆk ei(k−ν)xj =

j=0 k=−∞

Since xj = ξ + 2n

i(k−ν)xk

e

=

j=0

2πj 2n+1

2n

fˆk

k=−∞

2n

ei(k−ν)xj .

j=0

and 2πj

ei(k−ν)(ξ+ 2n+1 ) =

j=0

0 =

∞

ei(k−ν)ξ (1 − ei2πj(k−ν) ) 1 − ei

2πj(k−ν) 2n+1

for (k − ν) = (2n + 1)m, m ∈ Z,

(2n + 1)ei(k−ν)ξ for (k − ν) = (2n + 1)m, m ∈ Z,

we get c(n) ν =

2n ∞ ∞ 1 fˆk fˆν+m(2n+1) ei(2n+1)mξ . ei(k−ν)xj = 2n + 1 m=−∞ j=0

k=−∞

Proposition 1.1 is proved. (n) (n) (n) Since cν = aν − ibν and fˆν = aν − ibν , where aν = aν (f ) and bν = bν (f ), it follows from relation (1.13) that

a(n) ν = aν +

∞

(am(2n+1)+ν + am(2n+1)−ν ) cos(2n + 1)mξ

m=1

+ (bm(2n+1)+ν + bm(2n+1)−ν ) sin(2n + 1)mξ,

(1.14)

ν = 0, 1, . . . , n, and bν(n) = bν +

∞

(bm(2n+1)+ν − bm(2n+1)−ν ) cos(2n + 1)mξ

m=1

+ (−am(2n+1)+ν + am(2n+1)−ν ) sin(2n + 1)nξ, ν = 1, 2 . . . , n.

(1.15)

Section 2

Lebesgue Constants and Nikol’skii Theorems

557

These equalities establish the relationship between Fourier–Lagrange coeffi(n) (n) cients aν and bν of the function f and its Fourier coefficients ak and bk . If ξ = 0, then we set (n) df

(n)

xj = xj,0 = df Sñ (f ; x) = In (f, 0, x) =

2πj , 2n + 1

j = 0, 1, . . . , 2n,

2 (n) f (xj )Dn (x − xj ). 2n + 1 2n

(1.16)

j=0

In this case, relations (1.14) and (1.15) take the form aν(n) = aν +

∞

(am(2n+1)+ν + am(2n+1)−ν ),

ν = 0, 1, . . . , n,

(1.17)

(bm(2n+1)+ν − bm(2n+1)−ν ),

ν = 1, 2, . . . , n.

(1.18)

m=1

b(n) ν

= bν +

∞ m=1

These equalities are used in what follows.

2.

Lebesgue Constants and Nikol’skii Theorems

∗ 2.1. Let f ∈ C and let Tn−1 be the polynomial of its best approximation in the space C. The operator generating the sequence Sñ (f ; x) is linear and, hence, ∗ ∗ (x)| + |Sñ (f − Tn−1 ; x)|. |f (x) − Sñ (f ; x)| ≤ |f (x) − Tn−1

(2.1)

Since ∗ ; x)| = |Sñ (f − Tn−1

2 (n) (n) (n) ∗ (f (xj ) − Tn−1 (xj ))Dn (x − xj )| | 2n + 1 2n

j=0

∗ ≤ f − Tn−1 C

2 (n) |Dn (x − xj )| 2n + 1 2n

j=0

¯ n (x)En−1 (f )C , =L where

2 (n) |Dn (x − xj )|, 2n + 1

(2.2)

2n

¯ n (x) = L

j=0

(2.3)

558

Interpolation

Chapter 8

we always have ¯ n (x))En (f )C . |f (x) − Sñ (f ; x)| ≤ (1 + L

(2.4)

This estimate can be regarded as an interpolation analog of the classical ¯ n (x) is called a Lebesgue function, in Lebesgue inequality. Hence, the function L a sense that it plays the same role as the well-known Lebesgue constants because ¯ n (x) = sup |Sñ (f ; x)|C . L

(2.5)

|f |≤1

The following statement describes the asymptotic behavior of the function ¯ n (x): L Theorem 2.1. The equality ¯ n (x) = 2 | sin 2n + 1 x| ln n + O(1), L π 2

(2.6)

where O(1) is a quantity uniformly bounded in x and n, holds as n → ∞ at all points x ∈ R1 . ¯ n (x) is even. Moreover, one can easily Proof. It is clear that the function L see that this function is periodic with period 2π/(2n + 1). Hence, it suffices to establish relation (2.6) only for x ∈ (0, π/(2n + 1)]. By setting h = π/(2n + 1), in view of (2.3), we find n 1 ¯ Ln (x) = |Dn (x − 2jh)| + O(1). n j=−n

At the same time, Dn (x − 2jh) = sin

2n + 1 1 1 1 x( + − ) 2 x − 2jh 2 sin(x − 2jh)/2 x − 2jh

and the function u−1 − (cosec u/2)/2 is bounded for u ∈ (−π, π). Therefore, n −1 2n + 1 1 1 1 ¯ Ln (x) = | sin x| ( + ) + O(1) 2 n |x − 2jh| |x − 2jh| j=1

2n + 1 1 −1 2 2n + 1 j + O(1) = | sin x| x| ln n + O(1). 2 nh π 2 n

= | sin

j=−n

j=1

Section 2

Lebesgue Constants and Nikol’skii Theorems

559

2.2. Comparing equality (2.6) with a similar formula for Lebesgue constants 1 Ln = π

2π |Dn (t)|dt =

4 ln n + O(1), π2

(2.7)

0

we conclude that ¯ n (x) = π | sin 2n + 1 x|Ln + O(1). L (2.8) 2 2 Combining relations (2.4) and (2.6), we arrive at the following corollary: Corollary 2.1. For any f ∈ C and x ∈ R1 , the following inequality is true: 2 2n + 1 |f (x) − Sñ (f ; x)| ≤ ( | sin x| ln n + O(1))En (f )C , π 2 where O(1) is a quantity uniformly bounded in x and n.

(2.9)

2.3. Despite its simple form and generality, estimate (2.9) is often not only exact in order but also asymptotically exact. Thus, in particular, this is true in the r for all r ∈ N. In this case, according to Corollary 6.5.2, we have classes W∞ Kr , nr where Kr are the Favard constants (see (6.5.32)). r , r ∈ N, then In view of relations (2.9) and (2.10), if f ∈ W∞ r En (W∞ )=

(2.10)

2Kr ln n 2n + 1 | sin (2.11) x| + O(1)n−r . |f (x) − Sñ−1 (f ; x)| ≤ π nr 2 In [7], Nikol’skii proved that, for any x, one can indicate a function ϕn,r (t) r for which the expression in the class W∞ |ϕn,r (x) − Sñ−1 (ϕn,r ; x)| coincides with the right-hand side of (2.11). Thus, the following assertion is proved: Theorem 2.2. If r ∈ N, then the following equality holds as n → ∞ : ˜ n (W r ; x) = sup |f (x) − Sñ−1 (f ; x)| E ∞ r f ∈W∞

2Kr ln n 2n + 1 | sin x| + O(1)n−r , r π n 2 where O(1) is a quantity uniformly bounded in x and n. =

(2.12)

560

Interpolation

Chapter 8

This enables us to conclude that estimate (2.9) gives asymptotically exact rer for all r ∈ N. However, as the smoothness of functions sults in the classes W∞ increases further, in particular, for the classes of infinitely differentiable functions, the estimates obtained by using inequality (2.9) are no longer asymptotically exact. Moreover, they are not exact even in order. Hence, to establish the asymptotic equalities for deviations ρñ (f ; x) = f (x) − Sñ−1 (f ; x) in the classes of functions with high smoothness, it is necessary to perform additional investigations. This is done in the subsequent sections.

3.

Approximation by Interpolation Polynomials in the Classes of Infinitely Differentiable Functions 3.1. Consider the problem of asymptotic equalities for the quantities ψ ˜ n (C ψ ; x) = sup{|˜ ρn (f ; x)| = |f (x) − Sñ−1 (f ; x)| : f ∈ Cβ,∞ } E β,∞

(3.1)

with ψ ∈ M+ ∞. In this case, as already indicated (see Section 5.14), the functions ψ(·) vanish faster than any power function, namely, lim k r ψ(k) = 0 ∀r ∈ N.

k→∞

This means that the Fourier series of the function f from Cβψ can be differentiated infinitely many times and, as a result, we obtain uniformly convergent series, i.e., the classes Cβψ consist of infinitely differentiable functions. The basic idea used in the solution of this problem is connected with its reduction to a similar problem for the quantities ψ ψ En (Cβ,∞ ) = sup{|ρ(f ; x)| = |f (x) − Sn−1 (f ; x)| : f ∈ Cβ,∞ }.

This idea is realized in the following assertion: 1 Theorem 3.1. Assume that ψ ∈ M+ ∞ and β ∈ R. Then, for any x ∈ R , the following equality is true as n → ∞ :

˜ n (C ψ ; x) = 2| sin 2n − 1 x|En (C ψ ) E β,∞ β,∞ 2 + O(1)(ψ(n) + ψ(3n) ln+ (η(n) − n))

Section 3

Approximation in Classes of Infinitely Differentiable Functions

561

8 2n − 1 | sin x|ψ(n) ln+ (η(n) − n) π2 2

=

+ O(1)(ψ(n) + ψ(3n) ln+ (η(n) − n)),

(3.2)

where η(n) = η(ψ; n) = ψ −1 (ψ(n)/2) and O(1) are quantities uniformly bounded in x, n, and β. Proof. In view of equalities (1.16), (1.11), and (1.11 ), the interpolation polynomial Sñ (f ; x) admits a representation (n) a (n) ak cos kx + bk sin kx, Sñ (f ; x) = 0 + 2 n

(n)

(3.3)

k=1

where (n) ak

2 (n) (n) = f (xi ) cos kxi 2n + 1 2n

(3.4)

i=0

and (n)

2 (n) (n) f (xi ) sin kxi , 2n + 1 2n

bk =

k = 1, n.

(3.5)

i=0

If f ∈ Lψ β , then its Fourier coefficients ak = ak (f ) and bk = bk (f ) have the form π 1 πβ ψ(k) cos(kt + ak = )ϕ(t)dt (3.6a) π 2 −π

and 1 bk = π

π ψ(k) sin(kt + −π

πβ )ϕ(t)dt, 2

(3.6b)

where ϕ(·) = fβψ (·). Combining equalities (3.3)–(3.6), (1.14), and (1.15), we get 1 f (x) − Sñ (f ; x) = π

π ϕ(t) −π

∞

(2k+1)n+k

ψ(ν)(cos(ν(t − x) +

k=1 ν=(2k−1)n+k

− cos(νt + (k(2n + 1) − ν)x +

πβ ))dt 2

πβ ) 2

562

Interpolation 2 = π

π ϕ(t)

sin

k=1

−π

×

∞

2n

Chapter 8

k(2n + 1)x 2

ψ((2k − 1)n + k + i) sin(((2k − 1)n + k + i)t

i=0

− ((k − 1)n +

k πβ + i)x + )dt 2 2

= J (1) (x) + J (2) (x),

(3.7)

where (1)

J (1) (x) = Jψ,β,n (x) 2 2n + 1 = sin x π 2 df

π

−π

ϕ(t)

2n

ψ(n + 1 + i)

i=0

× sin(i(t − x) + (n + 1)t −

x πβ + )dt 2 2

and (2)

J (2) (x) = Jψ,β,n (x) 2 = π df

π ϕ(t)

∞ k=2

−π

k(2n + 1)x ψ((2k − 1)n + k + i) 2 2n

sin

i=0

× sin(((2k − 1)n + k + i)t − ((k − 1)n +

πβ k + i)x + )dt. 2 2

Further, let (3)

J (3) (x) = Jψ,β,n (x) df

=

2 2n + 1 sin x π 2 π ∞ 1 πβ ψ(ν) sin(ν(t − x) + (n + )x + × ϕ(t) )dt. 2 2 −π

ν=3n+2

Section 3

Approximation in Classes of Infinitely Differentiable Functions

563

Since J (1) (x) + J (3) (x) 2 2n + 1 = sin x π 2

π ϕ(t)

∞ ν=n+1

−π

πβ 1 )dt, ψ(ν) sin(ν(t − x) + (n + )x + 2 2

in view of relation (3.7), we obtain f (x) − Sñ (f ; x) 2 2n + 1 = sin x π 2

π

∞

ϕ(t + x)

ψ(ν) cos(νt + γn+1,x,β )dt

ν=n+1

−π

+ J (2) (x) − J (3) (x), (3.9) where

1 π(β − 1) df γn+1,x,β = (n + )x + . 2 2

3.2. We now establish the upper bounds of the quantities |J (2) (x)| and ψ |J (3) (x)| in the set Cβ,∞ . Note that, in this case, we have |ϕ(·)| ≤ 1 (almost everywhere) and, therefore, |J

(2)

2 (x)| ≤ | π

2π ∞

k(2n + 1)x ψ((2k − 1)n + k + i) 2 2n

sin

i=0

0 k=2

× sin(((2k − 1)n + k + i)t +

k(2n + 1)x πβ + |dt 2 2

∞ 2n 2 | ψ((2k − 1)n + k + i) ≤ π 2π

k=2 0

i=0

× sin(((2k − 1)n + k + i)t +

|J

(3)

2 (x)| ≤ π

π | −π

∞ ν=3n+2

k(2n + 1)x πβ + )|dt, (3.10) 2 2

ψ(ν) cos(νt + γn+1,x,β )|dt.

(3.11)

564

Interpolation

Chapter 8

Assume that Ψn,β k (t) =

2n

ψ((2k − 1)n + k + i)

i=0

× sin(((2k − 1)n + k + i)t +

k(2n + 1)x πβ + ), (3.12) 2 2

−1 e(1) n = {t ∈ [−π, π] : |t| ≤ min{π, (η(3n + 2) − (3n + 2)) }},

and

df

(1) e(2) n =[−π, π] \ en .

It is clear that ∞ ∞ 2n n,β |Ψk (t)|dt ψ((2k − 1)n + k + i)dt k=2 (1) i=0 en

k=2 (1) en

−1

= 2 min{π, (η(3n + 2) − (3n + 2))

}

∞

ψ(k)

k=3n+2 −1

≤ 2 min{π, (η(3n + 2) − (3n + 2))

∞ }(ψ(3n + 2) +

ψ(v)dv). (3.13)

3n+2

According to Proposition 6.8.2, if ψ ∈ M+ ∞ , then relation (6.18.13) is true, which means that, for any m ∈ N, ∞ ψ(v)dv ≤ Kψ(m)(η(m) − m).

(3.14)

m

Hence, −1

min{π, (η(3n + 2) − (3n + 2))

∞ }(ψ(3n + 2) +

ψ(v)dv)

3n+2

≤ Kψ(3n + 2)

(3.15)

and, consequently, ∞ π ∞ 2 2 n,β |Ψk (t)|dt = |Ψn,β k (t)|dt + O(1)ψ(3n + 2). π π k=2−π

k=2 (2) en

(3.16)

Section 3

Approximation in Classes of Infinitely Differentiable Functions

565

We now apply the Abel transformation to the expression for Ψn,β k (t). This gives 2n

αi βi =

i=0

2n−1

(αi − αi+1 )Bi + α2n B2n ,

where Bi =

i

βm .

(3.17)

m=0

i=0

In relation (3.17), we set αi = ψ((2k − 1)n + k + i), βi = sin(it + ξ), 2n + 1 ξ = ((2k − 1)n + k)t + kx + βπ/2, 2 Δψ((2k−1)n+k+i) = ψ((2k−1)n+k+i)−ψ((2k−1)n+k+i+1), i = 0, 2n, and, after elementary transformations, obtain 2n

ψ((2k − 1)n + k + i) sin(it + ξ)

i=0

=

2n−1

Δψ((2k − 1)n + k + i)Bi,ξ (t) + ψ((2k + 1)n + k)B2n,ξ (t), (3.18)

i=0

where Bi,ξ

i

it sin i+1 2 t sin( 2 + ξ) = sin(mt + ξ) = , sin 2t m=0 df

t = 2πν, ν ∈ Z.

In view of (3.17), for any i ∈ N, we get it t it it |Bi,ξ (t)| = |(sin cot + cos ) sin( + ξ)| 2 2 2 2 2 t ≤ 1 + | cot | ≤ 1 + , 0 < |t| ≤ π. 2 |t|

(3.19)

(3.20)

By virtue of relations (3.10), (3.16), and (3.18)–(3.20), we conclude that ∞ π 2 (2) |Ψn,β |J (x)| ≤ k (t)|dt π k=2−π

∞ 2n−1 2 ≤ ( Δψ((2k − 1)n + k + i) |Bi,ξ (t)|dt π k=2 i=0

(2)

en

|B2n,ξ (t)|dt) + O(1)ψ(3n + 2)

+ ψ((2k + 1)n + k) (2)

en

566

Interpolation

Chapter 8

∞ 2n−1 2 2 ( Δψ((2k − 1)n + k + i) (1 + )dt ≤ π |t| k=2 i=0

(2)

en

(1 +

+ ψ((2k + 1)n + k) (2)

2 )dt) + O(1)ψ(3n + 2) |t|

en

= O(1)(1 + ln+ (η(3n + 2) − (3n + 2))) ∞ 2n−1 ( Δψ((2k − 1)n + k + i) × k=2 i=0

+ ψ((2k + 1)n + k) + ψ(3n + 2)) = O(1)(1 + ln+ (η(3n + 2) − (3n + 2)) ×

∞

ψ((2k − 1)n + k) + ψ(3n + 2)).

(3.21)

k=2

The quantity μ(ψ; t) is monotonically increasing and relation (3.14) is true. Therefore, ∞

ψ((2k − 1)n + k)

k=2

∞ ψ((2n + 1)t − n)dt

≤ ψ(3n + 2) + 2

1 = ψ(3n + 2) + 2n + 1

∞ ψ(v)dv 3n+2

Kψ(3n + 2)(η(3n + 2) − (3n + 2)) 2n + 1 O(1) = ψ(3n + 2)(1 + ) = ψ(3n + 2)(1 + o(1)). μ(3n + 2) ≤ ψ(3n + 2) +

(3.22)

Relations (3.21) and (3.22) imply that |J (2) (x)| = O(1)ψ(3n + 2)(1 + ln+ (η(3n + 2) − (3n + 2))).

(3.23)

We now prove that a similar relation holds for the quantity J (3) (x). In view of relation (3.14), we find

Section 3

Approximation in Classes of Infinitely Differentiable Functions

∞

|

567

ψ(ν) cos(νt + γn+1,x,β )|dt

ν=3n+2

(1)

en

∞

≤ (1)

en

ψ(ν)dt

ν=3n+2

−1

≤ 2 min{π, (η(3n + 2) − (3n + 2))

∞ }(ψ(3n + 2) +

ψ(v)dv)

3n+2

≤ 2(π + K)ψ(3n + 2).

(3.24)

Hence, 2 π

π

∞

| −π

ψ(ν) cos(νt + γ)|dt

ν=3n+2

2 = π

∞

|

ψ(ν) cos(νt + γ)|dt + O(1)ψ(3n + 2), (3.25)

ν=3n+2

(2)

en

where

1 π(β − 1) γ = γn+1,x,β = (n + )x + . 2 2 By applying the Abel transformation, we find

∞

ψ(k) cos(kt + γ) = −ψ(3n + 2)Aγ,3n+1 (t) +

k=3n+2

∞

Δψ(k)Aγ,k (t),

k=3n+2

(3.26) where Aγ,k (t)=

sin((k + 12 )t + γ) , 2 sin 2t

t = 2νπ, ν ∈ Z.

Since |Aγ,k (t)| ≤

π 1 t ≤ 2|t| , 2| sin 2 |

0 < |t| < π, k = 1, 2, . . . ,

we can write π dt |Aγ,k (t)|dt ≤ = O(1)(1 + ln+ (η(3n + 2) − (3n + 2))). (3.27) 2 |t| (2)

en

(2)

en

568

Interpolation

Chapter 8

Relations (3.26) and (3.27) yield the following estimate: | (2)

en

∞

ψ(k) cos(kt + γ)|dt

k=3n+2

≤ ψ(3n + 2)

|Aγ,3n+1 (t)|dt +

∞

(2)

en

|Aγ,k (t)|dt

Δψ(k)

k=3n+2

(2)

en

= O(1)ψ(3n + 2)(1 + ln+ (η(3n + 2) − (3n + 2))).

(3.28)

Combining this estimate with estimates (3.10) and (3.25), we get |J (3) (x)| = O(1)ψ(3n + 2)(1 + ln+ (η(3n + 2) − (3n + 2))).

(3.29)

ψ 3.3. The set Cβ,∞ is invariant under the shifts of the argument and, hence, in view of equalities (3.9), (3.23), and (3.29), we find ψ Eñ+1 (Cβ,∞ ; x) = sup |f (x) − Sñ (f ; x)| ψ f ∈Cβ,∞

2n + 1 2 x| sup | = | sin π 2 |ϕ|≤1

π ϕ(t + x)

−π

×

∞

ψ(k) cos(kt + γn+1,x,β )dt| + rn+1 , (3.30)

k=n+1

where rn+1 = O(1)ψ(3n + 2)(1 + ln+ (η(3n + 2) − (3n + 2))) and O(1) is a quantity uniformly bounded in x, n, and β. By Fn (t) we denote the series on the right-hand side of relation (3.30) and set ψ1 (k) = ψ(k) cos γ

and ψ2 (k) = −ψ(k) sin γ,

Then Fn (t) =

∞ k=n+1

where

(ψ1 (k) cos kt + ψ2 (k) sin kt).

γ = γn+1,x,β .

(3.31)

Section 3

Approximation in Classes of Infinitely Differentiable Functions

569

Note that, in the analyzed case, ψ ∈ M+ ∞ , we have ±ψ1 , ±ψ2 ∈ M . Therefore (see relation (5.17.9)),

π In = sup | |ϕ|≤1

ϕ(x + t)Fn (t)dt| = Fn (·)1 .

(3.32)

−π

According to Subsection 5.17.2, we find Fn (·)1 =

2n−1 ∞ 4 ψ(k) |ψ2 (k)| +2 + O(1)ψ(n). π k+1−n k k=n

(3.33)

k=2n

Since M+ ∞ ⊂ F, we get ∞ ψ(k) = O(1)ψ(n). k

k=2n

Consequently, In = Fn (·)1 =

∞ 4 ψ(k) + O(1)ψ(·). π k+1−n

(3.34)

k=n

On the other hand, in view of relation (5.17.9), ψ )= qn (Cβ,∞

1 Fn (·). π

(3.35)

Combining relations (3.30)–(3.35), we get ˜ n (C ψ ; x) = 2| sin 2n − 1 x|En (C ψ ) + Rn , E β,∞ β,∞ 2

(3.36)

where Rn = O(1)(ψ(n) + ψ(3n − 1) ln+ (η(3n − 1) − (3n − 1)).

(3.37)

3.4. In the analyzed case, the quantity Rn can be represented in the form Rn = O(1)(ψ(n) + ψ(3n) ln+ (η(n) − n)).

(3.38)

To prove (3.38), we first note that, for any ψ ∈ M and m ∈ N, ψ(m) ln+ (η(m) − m) ≤ 2ψ(m + 1) ln+ (η(m) − m).

(3.39)

570

Interpolation

Chapter 8

Indeed, if η(m) − m ≤ 1, then ln+ (η(m) − m) = 0 and, hence, relation (3.39) is evident. At the same time, if η(m) − m > 1, then, in view of the fact that the function ψ(·) is decreasing, we get ψ(m) = 2ψ(η(m)) ≤ 2ψ(m + 1), which yields inequality (3.39). In this inequality, we set m = 3n − 1. As a result, it follows from relation (3.37) that Rn = O(1)(ψ(n) + ψ(3n) ln+ (η(3n − 1) − (3n − 1))).

(3.40)

To prove (3.38), it remains to show that η(3n − 1) − (3n − 1) ≤ O(3)(η(n) − n).

(3.41)

This inequality is a consequence of the following assertion: Lemma 3.1. Assume that ψ ∈ M+ ∞ , a > 1, and 0 ≤ γ ≤ a − 1. Then η(at − γ) − (at − γ) ≤ a(η(t) − t),

t ≥ 1.

(3.42)

Proof. We have η(at−γ)−(at−γ) = (η(t)−t)(1+

η(at − γ) − (at − γ) − (η(t) − t) ). (3.43) η(t) − t

As always, we set η (t) = η (t + 0). This gives η(at − γ) − (at − γ) − (η(t) − t) (η (θt) − 1)((a − 1)t − γ) ≤ η(t) − t η(t) − t ≤ (a − 1)(η (θt) − 1)μ(t),

(3.44)

where 1 ≤ θ ≤ a − γ/t. In view of the equality η(t)−t = t/μ(t), t ≥ 1, and the fact that the quantity 1/μ(t) monotonically decreases to zero, we obtain η (t) − t ≤ 1/μ(t). This inequality enables us to conclude that (a − 1)(η (θt) − 1)μ(t) ≤ (a − 1)

μ(t) ≤ a − 1. μ(θt)

Combining relations (3.43)–(3.45), we arrive at inequality (3.42).

(3.45)

Section 3

Approximation in Classes of Infinitely Differentiable Functions

571

By setting a = 3, γ = 1, and t = n in (3.42), we arrive at estimate (3.41) and, hence, at relation (3.38). Substituting the quantity Rn from relation (3.38) in (3.36), we establish the intermediate equality in (3.2). To deduce the last equality in (3.2), it suffices to apply relation (5.10.14), i.e., ψ )= En (Cβ,∞

4 ψ(n) ln+ (η(n) − n) + O(1)ψ(n). π2

(3.46)

3.5. Theorem 3.1 is proved. We now make several remarks to this theorem. First, we note that relation (3.2) is an asymptotic equality and, thus, gives the solution of the corresponding Kolmogorov–Nikol’skii problem whenever lim (η(n) − n) = ∞

(3.47)

n→∞

because, for ψ ∈ M+ ∞ and any a > 1, we have lim

t→∞

ψ(at) = 0. ψ(t)

(3.48)

Let us prove this relation. If ψ ∈ M, then (see, e.g., (3.12.24)) ψ(t) ≤ 2|ψ (t)|(η(t) − t),

t ≥ 1.

(3.49)

Hence, for any r ≥ 0, we get (tr (ψ(t)) = rtr−1 ψ(t) − tr |ψ (t)| ≤ tr |ψ (t)|(2r

η(t) − t − 1). t

(3.50)

For ψ ∈ M+ ∞ , the quantity (η(t) − t)/t approaches zero as t → ∞, and, therefore, one can indicate a number t0 = t0 (r, ψ) such that the function ϕ(t) = tr ψ(t) does not increase for t > t0 , . Thus, the function ψ(t) admits a representation ψ(t) = ϕ(t)t−r , (3.51) where ϕ(t) is a function nonincreasing for t > t0 . Let ε be an arbitrarily small positive number and let r = loga 1ε . It is necessary to find a number t0 = t0 (ε, ψ) such that relation (3.51) is true for the indicated value of r . Thus, ψ(at) ϕ(at) 1 r = ( ) ≤ a−r = ε, ψ(t) ϕ(t) a which yields (3.48).

572

Interpolation

Chapter 8

r

3.6. The function ψ(t) = e−αt , α > 0, belongs to M+ ∞ for any r > 0. If, in addition, 0 < r < 1, then the indicated function satisfies relation (3.47). This means that Theorem 3.1 implies the following assertion: r

Corollary 3.1. Assume that ψ(t) = e−αt , α > 0, 0 < r < 1. Then the following relation is true as n → ∞ at any point x ∈ R1 : ˜ n (C ψ ) = 8 | sin 2n − 1 x|(1 − r)e−αnr ln n + O(1)e−αnr , E β,∞ π2 2

(3.52)

where O(1) is a quantity uniformly bounded in x and n.

4.

Approximation by Interpolation Polynomials on the Classes of Analytic Functions

4.1. In this section, we continue our analysis of the quantities ρñ (f ; x) in the sets Cβψ . In this case, the functions ψ(·) satisfy the following condition: lim

k→∞

ψ(k + 1) = q, ψ(k)

q ∈ [0, 1),

(4.1)

i.e., ψ ∈ Dq . Recall that, in this case, the sets Cβψ consist of 2π-periodic functions admitting regular extensions into the strip |Imz| ≤ | ln q|. The principal results obtained in this section are formulated in the following theorems: Theorem 4.1. Let ψ ∈ Dq , 0 < q < 1, let ψ(k) > 0, and let β ∈ R1 . In this case, if f ∈ Cβψ C, then |˜ ρ(f ; x)| ≤ ψ(n)| sin +

q 2n − 1 16 x|( 2 K(q) + O(1)( 2 π n(1 − q)

εn ))En (fβψ )C (1 − q)2

(4.2)

for any x ∈ R1 . Furthermore, for any x ∈ R, n ∈ N, and f ∈ Cβψ C, one can find a function F (t) = F (f ; n; x; t) such that En (Fβψ )C = En (fβψ )C and the following equality is true: |˜ ρn (F ; x)| = ψ(n)| sin

2n − 1 16 εn + 1/n )En (Fβψ )C . (4.3) x|( 2 K(q) + O(1) 2 2 π (1 − q)

Section 4

Approximation on the Classes of Analytic Functions

573

In relations (4.2) and (4.3), π/2 du , K(q) = 1 − q 2 sin2 u

ψ(k + 1) εn = sup | − q|, ψ(k) k≥n

0

and O(1) are quantities uniformly bounded in x, n, q, β, and f ∈ Cβψ C. Theorem 4.1 implies, in particular, that inequality (4.2) is asymptotically exact for all x ∈ R in the entire space Cβψ C. This inequality remains asymptotically exact in certain important subsets of Cβψ C. Thus, the following assertion is true: Theorem 4.2. Let ψ ∈ Dq , 0 < q < 1, let ψ(k) > 0, let β ∈ R, and let ω(t) be an arbitrary modulus of continuity. Then the following equalities hold as n → ∞ for any x ∈ R : ˜ n (C ψ ; x) E β,∞ = | sin

q 2n − 1 16 εn )), (4.4) x|ψ(n)( 2 K(q) + O(1)( + 2 π n(1 − q) (1 − q)2

˜ n (C ψ Hω ; x) E β = | sin where

ω(1/n)(εn + 1/n) 2n − 1 8 ), (4.5) x|ψ(n)( 2 en (ω)K(q) + O(1) 2 π (1 − q)2 π/2 2t en (ω) = θω ω( ) sin tdt, n 0

θω ∈ [1/2, 1] (with θω = 1 if ω(t) is a convex modulus of continuity), K(q) and εn are the same quantities as in Theorem 4.1, and O(1) are quantities uniformly bounded in x, n, q, and β. Note that the asymptotic equalities (4.4) and (4.5) for the quantities ˜ n (C ψ ; x) and E ˜ n (C ψ Hω ; x) are interpolation analogs of the asymptotic E β,∞ β ψ ) and equalities obtained in Section 5.20 for the upper bounds En (Cβ,∞ ψ En (Cβψ Hω ) of approximations by Fourier sums on the classes Cβ,∞ and Cβψ Hω in the space C . Moreover, the following equalities are true:

574

Interpolation

Chapter 8

˜ n (C ψ ; x) = 2| sin 2n − 1 x|(En (C ψ )C E β,∞ β,∞ 2 + O(1)ψ(n)(

q εn )), + (n(1 − q) (1 − q)2

˜ n (C ψ Hω ; x) = 2| sin 2n − 1 x|(En (C ψ Hω )C E β β 2 + O(1)

ψ(n)ω(1/n)(εn + 1/n) ), (1 − q)2

where the quantities εn and O(1) have the same meaning as in Theorem 4.2. As already indicated, the Poisson kernels Pβq (t) =

∞ k=1

q k cos(kt −

βπ ), 2

q ∈ (0, 1), β ∈ R

can be regarded as an important example of kernels whose coefficients ψ(k) satisfy the condition ψ ∈ Dq , 0 < q < 1. In the case where ψ(k) = q k , the classes Cβψ N are denoted by Cβq N and the corresponding (ψ, β)-derivatives and (ψ, β)integrals of the function f are denoted by fβq (·) and Jβq (f ; ·), respectively. In view of equality (5.16.3), Theorem 4.2 yields the following corollary: Corollary 4.1. Let 0 < q < 1, let β ∈ R, and let ω(t) be an arbitrary modulus of continuity. Then, for any x ∈ R, the following asymptotic equalities hold as n → ∞ : q ˜ n (C q ; x) = | sin 2n − 1 x|q n ( 16 K(q) + O(1) E ), β,∞ 2 2 π n(1 − q) ˜ n (C q Hω ; x) = | sin 2n − 1 x|q n ( 8 en (ω)K(q) + O(1) ω(1/n) ), E β 2 π2 n(1 − q)2 where the quantities en (ω) and K(q) are the same as in Theorem 4.2 and O(1) are quantities uniformly bounded in x, n, q, and β. The conditions of Theorem 4.2 are also satisfied by the coefficients ψ(k) of the biharmonic Poisson kernel ∞

1 1 − q2 βπ Bq,β (t) = + (1 + k)q k cos(kt − ), 2 2 2 k=1

0 < q < 1, β ∈ R,

Section 4

Approximation on the Classes of Analytic Functions

575

and the coefficients of the Neumann kernel Nq,β (t) =

∞ k q k=1

k

cos(kt −

βπ ), 2

0 < q < 1, β ∈ R.

It is easy to see that the coefficients ψ(k) of the kernels Bq,β (t) and Nq,β (t) satisfy the relations εn = sup | k≥n

ψ(k + 1) ψ(n + 1) q − q| = | − q| ≤ , n ∈ N. ψ(k) ψ(n) n

Hence, Theorem 4.2 and Corollaries 5.20.3 and 5.20.4, imply the following statement: Corollary 4.2. Let ψ(k) = (1 +

1 − q2 k)q k , 2

0 < q < 1, k ∈ N,

let β ∈ R, and let ω(t) be an arbitrary modulus of continuity. Then the following equalities hold for any x ∈ R as n → ∞ : ˜ n (C ψ ; x) E β,∞ = | sin

16 q 2n − 1 1 − q2 ) (4.6) x|(1 + n)q n ( 2 K(q) + O(1) 2 2 π n(1 − q)2

and ˜ n (C ψ Hω ; x) E β = | sin

8 ω(1/n) 2n − 1 1 − q2 ). (4.7) x|(1 + n)q n ( 2 en (ω)K(q) + O(1) 2 2 π n(1 − q)2

At the same time, if ψ(k) = q k /k, 0 < q < 1, k ∈ N, and β ∈ R, then n q ˜ n (C ψ ; x) = | sin 2n − 1 x| q ( 16 K(q) + O(1) ), E β,∞ 2 2 n π n(1 − q)2

(4.8)

n ˜ n (C ψ Hω ; x) = | sin 2n − 1 x| q ( 8 en (ω)K(q) + O(1) ω(1/n) ) E β 2 n π2 n(1 − q)2

(4.9)

576

Interpolation

Chapter 8

for any x ∈ R as n → ∞. In equalities (4.6)–(4.9), the quantities en (ω) and K(q) have the same meaning as in Theorem 4.2 and O(1) are quantities uniformly bounded in x, n, q, and β. The following theorem can be regarded as a complement to Theorem 4.2 for q = 0: Theorem 4.3. Let ψ ∈ D0 , ψ(k) > 0, let β ∈ R, and let ω(t) be an arbitrary modulus of continuity. Then, for any x ∈ R, the following asymptotic equalities hold as n → ∞ : ˜ n (C ψ ; x) = | sin 2n − 1 x|( 8 ψ(n) E β,∞ 2 π ∞ ψ(n + 1) 1 ψ(k))), (4.10) , }+ + O(1)(ψ(n + 1) min{ ψ(n) n k=n+2

˜ n (C ψ Hω ; x) E β ∞ 1 2n − 1 4 ψ(k)), (4.11) = | sin x|( ψ(n)en (ω) + O(1)ω( ) 2 π n k=n+1

where

π/2 2t en (ω) = θω ω( ) sin tdt, n 0

θω ∈ [2/3, 1] (with θω = 1 if ω(t) is a convex function), and O(1) are quantities uniformly bounded in all analyzed parameters. Note that the conditions ψ ∈ D0 and ψ(k) > 0 guarantee the validity of the formula ∞ ψ(m) = o(1) ψ(k). k=m+1

4.2. To prove Theorems 4.1–4.3, we need the following lemma on the integral representation of the quantities ρñ (f ; x) : Lemma 4.1. Let ψ(k) > 0, let

∞

ψ(k) < ∞, and let β ∈ R. Then the

k=1

following equality holds for any function f ∈ Cβψ at any point x ∈ R1 :

Section 4

Approximation on the Classes of Analytic Functions

577

ρñ (f ; x) 2 2n − 1 = sin x π 2

π δn (t + x)(

∞

ψ(ν) cos(νt + γn ) + rn (t))dt, (4.12)

ν=n

−π

where δn (τ ) = fβψ (τ ) − tn−1 (τ ), tn−1 (·) is an arbitrary trigonometric polynomial from T2n−1 and the quantities rn (t) and γn are specified by the equalities rn (t) = rn (ψ; β; x; t) =

∞

∞

k=1 ν=(2k+1)n−k

1 ψ(ν) sin(νt + (k + )(2n − 1)x + πβ/2), 2

γn = γn (β; x) = ((2n − 1)x + π(β − 1))/2.

(4.13) (4.14)

Proof. In view of relations (3.3)–(3.6), by setting (n)

σk = σk

= (2k − 1)n + k,

(4.15)

(n)

αk = αk (β; x) = (2n + 1)kx + βπ/2, k = 1, 2, . . . , for any function f ∈ 1 ρñ (f ; x) = π 1 = π

Cβψ ,

we conclude that

π ϕ(t + x)

∞ σk+1 −1

ψ(ν)[cos(νt +

k=1 ν=σk

−π

π ϕ(t + x) ∞

πβ ) − cos(νt + αk )]dt 2

∞ ∞ πβ { ψ(ν)[cos(νt + ) − cos(νt + αk )] 2 ν=σ k=1

−π

−

(4.16)

k

ψ(ν)[cos(νt +

ν=σk+1

πβ ) − cos(νt + αk )]}dt 2

(4.17)

or 1 ρñ (f ; x) = π

π ϕ(t + x)(

∞

ψ(ν)[cos(νt +

ν=σ1

−π

+

∞

{

∞

ψ(ν)[cos(νt +

k=1 ν=σk+1

−

∞ ν=σk+1

πβ ) − cos(νt + α1 )] 2

ψ(ν)[cos(νt +

πβ ) − cos(νt + αk+1 )] 2

πβ ) − cos(νt + αk )]})dt, 2

578

Interpolation

Chapter 8

whence it follows that π ∞ ∞ 1 ϕ(t + x) ψ(ν) ρñ (f ; x) = π ν=σ k=0

−π

k+1

× [cos(νt + αk ) − cos(νt + αk+1 )]dt 2 2n + 1 = sin x π 2

π ϕ(t + x)

∞ ∞

ψ(ν)

k=0 ν=σk+1

−π

1 πβ × sin(νt + (k + )(2n + 1)x + )dt. (4.18) 2 2 If we now replace n − 1 by n in (4.18) and take into account the fact that (n−1) σk+1 = (2k + 1)n − k, then we get 2 2n − 1 ρñ (f ; x) = sin x π 2

π

fβψ (t + x)

−π

×(

∞

ψ(ν) cos(νt + γn ) + rn (t))dt, (4.19)

ν=n

where rn (t) and γn are specified by equalities (4.13) and (4.14). The functions ∞ ψ(ν) cos(νt + γn ) and rn (t) are orthogonal to any trigonometric polynomial ν=n

tn−1 ∈ T2n−1 and, therefore, fβψ (u) in relation (4.19) can be replaced by δn (u). As a result, relation (4.19) yields (4.12) for any function f ∈ Cβψ . 4.3. Proof of Theorem 4.1. Let ψ ∈ Dq , 0 < q < 1. Then, by virtue of (4.13), we have ∞ ∞ rn (t) ≤ ψ(ν) (4.20) k=1 ν=(2k+1)n−k

and ∞ ν=(2k+1)n−k

$

(2n−1)k−1

ψ(ν) = ψ(n)

i=0

× (1 +

ψ(n + i + 1) ψ(n + i) ∞ m−1 $ ψ((2k + 1)n − k + j + 1)

m=1 j=0

ψ((2k + 1)n − k + j)

)

Section 4

Approximation on the Classes of Analytic Functions $

579

(2n−1)k−1

≤ ψ(n)

(q + εn+i )

i=0

× (1 +

∞ m−1 $

(q + ε(2k+1)n−k+j ))

m=1 j=0

≤ ψ(n)

(q + εn )(2n−1)k . 1 − q − ε3n−1

Therefore, ∞ ψ(n) (q + εn )(2n−1)k |rn (t)| ≤ 1 − q − ε3n−1 k=1

=

ψ(n)(q + εn )2n−1 qψ(n) = o(1) . (1 − q − ε3n−1 )(1 − (q + εn )2n−1 ) (1 − q)n

(4.21)

By virtue of Lemma 5.20.1, we have ∞

ψ(ν) cos(νt + γn ) = ψ(n)(q −n

ν=n

∞

q k cos(kt + γn ) + r¯n (t)),

(4.22)

k=n

where ∞ $ i−1 ψ(n + l + 1) r¯n (t) = r¯n (ψ, γn , t) = ( − q i ) cos((n + i)t + γn ). ψ(n + l) df

i=1 l=0

In this case, the following inequality holds for the quantity r¯n (t) beginning with a certain number n0 : |¯ rn (t)| ≤

εn , (1 − q − εn )(1 − q)

εn = sup | k≥n

ψ(k + 1) − q|. ψ(k)

(4.23)

Combining relations (4.12) with (4.22) and (4.21) with (4.23), we obtain 2 2n − 1 xψ(n) ρñ (f ; x) = sin π 2

π

−π

+ O(1)(

δ(t + x)(q −n

∞

q ν cos(νt + γn )

ν=n

q εn + ))dt. 2 (1 − q) n(1 − q)

(4.24)

580

Interpolation

Chapter 8

In relation (4.24), we choose tn−1 (·) in the form of the polynomial of the best approximation of the function fβψ (·) in the space C. This gives 2 2n − 1 |˜ ρn (f ; x)| ≤ | sin x|ψ(n)(q −n π 2

π ∞ | q ν cos(νt + γn )|dt

−π

+ O(1)(

ν=n

q εn + ))En (fβψ )C . 2 (1 − q) n(1 − q)

(4.25)

In Section 5.20, it is shown that, for any 0 < q < 1 and α ∈ R, π ∞ 8 q | q ν cos(νt + α)|dt = q n ( K(q) + O(1) ), π n(1 − q) ν=n

(4.26)

−π

where O(1) is a quantity uniformly bounded in n, q, and α. Substituting this equality with α = γn in relation (4.25), we arrive at relation (4.2). Let us now prove the second part of Theorem 4.1. By virtue of relations (4.19) and (4.21)–(4.23) and the fact of orthogonality of the function rn (t) to any polynomial tn−1 ∈ T2n−1 , for any f ∈ Cβψ , we obtain 2 2n − 1 xψ(n)(q −n ρñ (f ; x) = sin π 2

π

∞

q ν cos(νt + γn )dt

ν=n

−π

+ O(1)(

fβψ (t + x)

q εn + )En (fβψ )C ). (1 − q)2 n(1 − q)

(4.27)

Note that, for any x ∈ R, we can write 1 π

π

fβψ (t + x)

∞

q ν cos(νt + γn )dt = ρn (gx ; x),

ν=n

−π

where df

ρn (f ) = ρn (f ; x) = f (x) − Sn−1 (f ; x), andSn (f ) = Sn (f ; x) are partial Fourier sums of order n for a function f from the space L and 1 gx (·) = π df

π −π

fβψ (t

+ ·)

∞ ν=1

ψ(ν) cos(νt + γn )dt.

Section 4

Approximation on the Classes of Analytic Functions

581

Furthermore, according to Theorem 5.19.2, for the function gx (·), one can find (for any n ∈ N ) a function ϕ(t) ¯ = ϕ(n; ¯ x; t) such that En (ϕ) ¯C = En (fβψ )C and the following equality is true: ρn (G)C = (

8q n O(1)q n K(q) + )En (fβψ )C , π2 (1 − q)2 n

(4.28)

where df

q G(τ ) = J2γ (ϕ; ¯ τ) n /π

1 = π

π ϕ(τ ¯ + t)

∞

q ν cos(νt + γn )dt,

ν=1

−π

and O(1) is a quantity uniformly bounded in n, q, and γn . Assume that a point x0 is such that |ρn (G; x0 )| = ρn (G)C .

(4.29)

df

¯ − x + x0 )) is the required function. Indeed, since Fβψ (t) = Then F (t) = Jβψ (ϕ(t ϕ(t ¯ − x + x0 ), we have En (Fβψ )C = En (fβψ )C and, according to relations (4.27)–(4.29), for given x and n, we find 2 2n − 1 |˜ ρn (F ; x)| = | sin x|ψ(n)(q −n | π 2

π

−π

+ O(1)(

ϕ(t ¯ + x0 )

∞

q ν cos(νt + γn )dt|

ν=n

εn q + )En (Fβψ )C ) (1 − q)2 n(1 − q)

= 2| sin

εn + 1/n 2n − 1 En (fβψ )C ) x|ψ(n)(q −n ρn (G)C + O(1) 2 (1 − q)2

= 2| sin

εn + 1/n 2n − 1 8 )En (fβψ )C . x|ψ(n)( 2 K(q) + O(1) 2 π (1 − q)2

Theorem 4.1 is proved. Proof of Theorem 4.2. We consider the upper bounds of the absolute values ψ and of both sides of equality (4.24) for given x and tn−1 ≡ 0 in the class Cβ,∞ 0 take into account the invariance of the set SM under shifts of the argument. As a result, we obtain

582

Interpolation

Chapter 8

˜ n (C ψ ; x) = sup |˜ ρn (f ; x)| E β,∞ ψ f ∈Cβ,∞

2 2n − 1 = | sin x|ψ(n)(q −n sup π 2 ϕ∈S 0

π ϕ(t)

M −π

+ O(1)(

∞

q ν cos(νt + γn )dt

ν=n

q εn )). + n(1 − q) (1 − q)2

(4.30)

Note that π sup |

0 ϕ∈SM

ϕ(t)

∞

q ν cos(νt + γn )dt|

ν=n

−π

π ∞ | q ν cos(νt + γn )|dt + O(1) = −π

ν=n

q n(1 − q)

(see, e.g., Subsection 5.16.4). Substituting this equality in formula (4.30), by virtue of relation (4.26) (with α = γn ), we get ˜ n (C ψ ; x) = 2 | sin 2n − 1 x|ψ(n)(q −n E β,∞ π 2

π ∞ | q ν cos(νt + γn )|dt

−π

ν=n

q εn )) + n(1 − q) (1 − q)2 q 2n − 1 16 εn )). = | sin x|ψ(n)( 2 K(q) + O(1)( + 2 π n(1 − q) (1 − q)2 + O(1)(

Relation (4.4) is proved. Similarly, by analyzing the upper bounds of the absolute values of both sides of equality (4.12) in the class Cβψ Hω for any fixed x and taking into account the invariance of the set Hω under shifts of the argument, equality (4.22), and estimate (4.23), we obtain ˜ n (C ψ Hω ; x) E β

2 2n − 1 = | sin x| sup | π 2 ϕ∈Hω

π

ϕ(t)ψ(n)q −n

−π

×

∞ k=n

q k cos(kt + γn )dt + Rn (ϕ)|, (4.31)

Section 4

Approximation on the Classes of Analytic Functions

583

where df

π

Rn (ϕ) = Rn (ϕ; x) =

δ ∗ (t)(ψ(n)¯ rn (t) + rn (t))dt, δn∗ (τ ) = ϕ(τ ) − t∗n−1 (τ ),

−π

and t∗n−1 (·) is the polynomial of the best approximation of the function ϕ in the space C. Therefore, in view of estimates (4.21) and (4.23), we get |Rn (ϕ)| ≤ 2πδn∗ (·)C ψ(n)¯ rn (·) + rn (·)C q εn = O(1)ψ(n)( )En (ϕ)C . + (1 − q)n (1 − q)2

(4.32)

By using the Jackson inequality in the space C : 1 En (ϕ) ≤ Kω(ϕ; ), n

ϕ ∈ C, n ∈ N,

(4.33)

where K is an absolute constant, and estimate (4.32), we find sup |Rn (ϕ)| = O(1)ψ(n)ω(1/n)(

ϕ∈Hω

q εn ). + (1 − q)n (1 − q)2

(4.34)

It follows from relations (4.31) and (4.34) that ˜ n (C ψ Hω ; x) = 2 | sin 2n − 1 x|ψ(n) E β π 2 × (q

−n

π sup |

ϕ∈Hω

ϕ(t)

−π

+ O(1)ω(1/n)(

∞

q k cos(kt + γn )dt|

k=n

q εn )). + (1 − q)n (1 − q)2

(4.35)

According to Theorem 5.18.1, for any q ∈ (0, 1), β ∈ R, and any modulus of continuity ω(t), we have π sup |

ϕ∈Hω

−π

ϕ(t)

∞ k=n

q k cos(kt +

βπ )dt| 2 =

4 n O(1)q n ω(1/n) q K(q)en (ω) + , (4.36) π (1 − q)2 n

584

Interpolation

where

Chapter 8

π/2 ω(2t/n) sin tdt, en (ω) = θω 0

θω ∈ [1/2, 1] (with θω = 1 if ω(t) is a convex modulus of continuity), and O(1) is a quantity uniformly bounded in n, q, and β. In equality (4.36), we set γn instead of βπ/2 (the possibility of this substitution follows from the uniform boundedness of the quantity O(1) in (4.36) with respect to the parameters n and β ). Comparing the equality obtained as a result with representation (4.35), we arrive at equality (4.5). Theorem 4.2 is proved. Proof of Theorem 4.3. Let ψ ∈ D0 , let ψ(k) > 0, and let β ∈ R. We consider the upper bounds of the absolute values of both sides of equality (4.12) ψ and Cβψ Hω and take into account estimate (4.20) and the in the classes Cβ,∞ 0 and H under shifts of the argument. As a fact of invariance of the sets SM ω result, we obtain ˜ n (C ψ ; x) = sup |˜ ρn (f ; x)| E β,∞ ψ f ∈Cβ,∞

2 2n − 1 = | sin x|( sup | π 2 ϕ∈S 0 M

+ O(1)

∞

π ϕ(t)

−π

∞

∞

ψ(k) cos(kt + γn )dt|

k=n

ψ(ν)),

(4.37)

k=1 ν=(2k+1)n−k

˜ n (C ψ Hω ; x) = 2 | sin 2n − 1 x| E β π 2 π ∞ ψ(k) cos(kt + γn )dt + Rn∗ (ϕ)|, (4.38) × sup | ϕ(t) ϕ∈Hω

k=n

−π

where Rn∗ (ϕ)

=

Rn∗ (ϕ; x)

π = −π

δn∗ (t)rn (t)dt, δn∗ (τ ) = ϕ(τ ) − t∗n−1 (τ ),

Section 4

Approximation on the Classes of Analytic Functions

585

t∗n−1 (·) is the polynomial of the best approximation of the function ϕ in the space C, and O(1) is a quantity uniformly bounded in all analyzed parameters. According to Theorems 5.17.2 and 5.17.3, we have 1 sup | ϕ∈S 0 π M

=

π ϕ(t)

∞

ψ(k) cos (kt + αk )dt|

k=n

−π

∞ 4 ψ(n + 1) 1 ψ(k)), (4.39) ψ(n) + O(1)(ψ(n + 1) min{ ; }+ π ψ(n) n k=n+2

where {αk } is an arbitrary sequence of real numbers and the quantity O(1) has the same meaning as in equality (4.37). By setting αk = γn , k = n, n + 1, . . . , in relation (4.39), comparing the equality obtained as a result with representation (4.37), and taking into account the fact that, for any ψ ∈ D0 and sufficiently large n, ∞

∞

ψ(ν) ≤

k=1 ν=(2k+1)n−k

∞ 1 ψ((2k + 1)n − k), 1 − ε3n−1

(4.40)

k=1

where εn = sup| ψ(k+1) ψ(k) |, we arrive at formula (4.10). k≥n

By using estimates (4.20) and (4.33) and inequality (4.40), we get |Rn∗ (ϕ)|

≤

2πδn∗ C rn C

∞

= O(1)

ψ(k)En (ϕ)C

k=3n−1

= O(1)ω(1/n)

∞

ψ(k).

(4.41)

k=3n−1

By virtue of equality (5.15.4 ), we conclude that 1 sup | ϕ∈Hω π

π ϕ(t)

−π

∞

ψ(k) cos(kt + βk )dt|

k=n

2θω = ψ(n) π

π/2 ∞ ω(2t/n) sin tdt + O(1)ω(1/n) ψ(k), (4.42) 0

k=n+1

586

Interpolation

Chapter 8

where βk is an arbitrary sequence of real numbers, θω ∈ [2/3, 1] with θω = 1 if ω(t) is a convex function, and O(1) is a quantity uniformly bounded in all analyzed parameters. By setting βk = γk , k = n, n + 1, . . . , in relation (4.42) and comparing the equality obtained as a result with representation (4.38) and estimate (4.41), we get equality (4.11). Theorem 4.3 is proved.

5.

Summable Analog of the Favard Method

5.1. As the principal advantage of interpolation polynomials, one can mention the fact that they are completely determined by finitely many values of the approximated function, which is strongly desirable for their practical realization. Moreover, the assertions presented in the previous sections show that the approximating properties of Fourier sums are not much better than the corresponding properties of interpolation polynomials. However (as in the case of Fourier sums), the sequence of polynomials cannot be convergent in the entire class C of continuous 2π-periodic functions, Sñ (f ; x), n ∈ N, i.e., there exist functions f ∈ C for which this sequence is divergent at some points. Hence, it is of interest to construct a sequence of polynomials Un (f ; x) uniformly convergent in the entire space C and determined by finitely many values of the function f (·). At present, there are several methods used to construct sequences of this sort. One of these methods is analyzed in what follows. 5.2. Assume that f ∈ C and n ∈ N. We set xk = and

kπ , k = 0, ±1, . . . , n

π , 2n

xk+1 n−1 1 f (xk + αk ) Fn (t − x)dt, θñ (f ; x) = π k=−n

where

αn =

xk

1 kπ kπ Fn (u) = + cot cos ut 2 2n 2n n−1 k=1

(5.1)

Section 5

Summable Analog of the Favard Method

587

˜ ; x) is a summable analog of is the Favard kernel of order n − 1 and, hence, θ(f the integral representation of the Favard sums θ(f ; x) for the function f (x) : π 1 θn (f ; x) = f (t)Fn (t − x)dt. π −π

It is known that the sums θn (f ; x) possess a series of remarkable properties. One of these properties has already been mentioned in Chapter 6: The Favard sums realize the upper bound of the best approximations in the class W 1 . As always, we assume that Dn (t) is the Dirichlet kernel of order n and 1 Rn (t) = (Dn (t + αn ) + Dn (t − αn )), αn = π/2n, 2 is the Rogosinski kernel. After elementary calculations, we obtain xk+1

Fn (t − x)dt =

π Rn (xk + αn − x), n

xk = kπ/n.

xk

Hence, in view of relation (5.1), we find n−1 1 ˜ θn (f ; x) = f (xk + αn )Rn (xk + αn − x). n k=−n

5.3. In this subsection, we study the behavior of the quantity En (x; ω) = sup |f (x) − θñ (f ; x)|, f ∈Hω

where Hω denotes the class of 2π-periodic continuous functions f (·) satisfying the condition |f (x) − f (x )| ≤ ω(|x − x |), where, in turn, ω(t) is an arbitrary fixed modulus of continuity. The principal results are formulated in the following assertions: Theorem 5.1. The quantity En (x; ω) is an even (π/n)-periodic function satisfying, for all n ∈ N, the inequality π 1 1 π 1 π π En (x; ω) ≤ (ω(x + ) − ω( )) Fn (t + x)dt + (ω( − x) π 2n 2 n π 2n 0

1 π + ω( )) 2 n

π Fn (t − x)dt, 0

x ∈ [0, π/n].

(5.2)

588

Interpolation

Chapter 8

Theorem 5.1 yields the following corollary: Corollary 5.1. The inequality 1 π π 1 1 En (x; ω) < ω( ) + ω( )( + ), 2 n 2n 2 π

(5.3)

holds for all n ∈ N . In particular, this inequality implies that π En (x; ω) < Kω( ), K < 1.32. n

(5.4)

In connection with estimate (5.4), we note that, in the family of all known linear methods Un (f ) of approximation of periodic functions, the least constant K in inequalities of the form π |f (x) − Un (f ; x)| ≤ Kω(f ; ) (5.5) n is attained just for the polynomials θñ (f ; x). In other words, at present, we do not know any other linear method Un (f ) guaranteeing the validity of inequality (5.5) with a constant K smaller than the constant used in estimate (5.4). Proof of Theorem 5.1. First, we show that the function En (x, ω) is π/nperiodic. We set π f1 (t) = f (t − ), t ∈ R1 . n In view of the periodicity of the functions f (·) and Fn (·), we can write xk+1 n−1 1 π π θñ (f ; x) = f1 (xk+1 + αn ) Fn (t + − (x + ))dt π n n k=−n

1 = π

n

xk xk+1

Fn (t − (x +

f1 (xk + αn )

k=−n+1

π ))dt n

xk

xk+1 n−1 π π 1 f1 (xk + αn ) Fn (t − (x + ))dt = θñ (f1 ; x + ) = π n n k=−n

xk

because xn+1

π Fn (t − (x + ))dt = f1 (−π + αn ) n

f1 (x1 + αn ) π

π −π+ n

Fn (t − (x + −π

π ))dt. n

Section 5

Summable Analog of the Favard Method

589

Therefore, for any x, π π f (x) − θñ (f ; x) = f1 (x + ) − θñ (f ; x + ). n n Thus, in view of the fact that the class Hω is invariant under shifts of the argument (i.e., if f ∈ Hω , then the function f1 (x) = f (x + h) also belongs to Hω for any h ), we conclude that the quantity En (x; ω) is π/n-periodic. The proof of the fact that this quantity is even is also quite simple. It is clear that, for this purpose, it suffices to show that, for any function f ∈ Hω and any x, f (x) − θñ (f ; x) = f1 (−x) − θñ (f1 ; −x),

(5.6)

where f1 (t) ≡ f (−t). Indeed, we have −x k

n−1 1 f (−(xk + αn )) θñ (f ; x) = π k=0

Fn (t − x)dt

−x(k+1)

xk+1 n−1 1 f (xk + αn ) Fn (t − x)dt + π k=0

n−1 1 (f1 (xk + αn ) = π k=0

xk xk+1

Fn (t + x)dt xk xk+1

Fn (t − x)dt)

+ f1 (−(xk + αn )) xk

= θñ (f1 ; −x). This yields (5.6) and, hence, the quantity En (x; ω) is even. Since the function En (x; ω) is even and π/n-periodic, it suffices to restrict ourselves to the analysis of the case where, e.g., x ∈ [0, π/2n]. In what follows, we assume that this condition is satisfied. We also note that θñ (f ≡ C = const; x) = C and, therefore, En (x; ω) = sup |θñ (f ; x)|, f ∈Hω,x

where Hω,x is a subset of functions f ∈ Hω such that f (x) = 0.

(5.7)

590

Interpolation

Chapter 8

We now estimate the quantities |θñ (f ; x)| from above. For this purpose, we use the Abel transformation in the form n−1

ak bk =

k=0

n−1

(ak − ak−1 )

k=1

n−1

bi + a0

i=k

n−1

bk

k=0

and represent θñ (f ; x) as follows: xi+1 n−1 n−1 1 θñ (f ; x) = (f (−xk − αn ) − f (−xk + αn )) Fn (x + t)dt π k=1

i=k xi

1 + f (−αn ) π

n−1

xk+1

Fn (t + x)dt

k=0 x k

xi+1 n−1 n−1 1 (f (xk + αn ) − f (xk − αn )) Fn (t − x)dt + π k=1

i=k xi

xk+1 xk+1

1 + f (αn ) π

Fn (t − x)dt.

k=0 x k

Hence, n−1 1 ˜ (f (−(xk + αn )) − f (−(xk − αn )) Fn (t + x)dt θn (f ; x) = π π

k=1

xk

1 + f (−αn ) π

π

1 Fn (t + x)dt + f (αn ) π

0

π Fn (t − x)dt 0

n−1 1 + (f (xk + αn ) − f (xk − αn )) Fn (t − x)dt. (5.8) π π

k=1

xk

This enables us to conclude that, for any function f ∈ Hω,x ,

Section 5

Summable Analog of the Favard Method

1 π | |θñ (f ; x)| ≤ ω( ) π n n−1

591

π Fn (t + x)dt|

k=1 x k

1 + ω(x + αn )| π

π

1 π Fn (t + x)dt| + ω( ) | π n n−1

π

Fn (t − x)dt|

k=1 x k

0

1 + ω(αn − x)| π

π

df ¯ Fn (t − x)dt| = Ω n (x).

(5.9)

0

5.4. The kernel Rn (t), admits the following representation: π cos nt sin 2n 1 kt cos + cos kt = π . 2 2n 2(cos t − cos 2n ) n−1

Rn (t) =

(5.10)

k=1

Therefore, xk+1

Fn (t)dt =

π 2k + 1 Rn ( π) = 0, n 2n

k = 1, 2, . . . , n − 1.

xk

This means that the function π n−1 π−t π kπ Φn (t) = Fn (τ )dτ = − cot sin kt, t ∈ [0, 2π], 2 2n 2n k=1

t

turns to zero at the points kπ/n, i.e., Φn (

kπ ) = 0, n

k = 1, 2, . . . , 2n − 1.

Moreover,

π π , Φn (t) > 0 for t ∈ [0, ). 2 n Let us now examine the sign of the difference Φn (0) =

rn (t) = Φn (t) − Φn (t +

2π ). n

It follows from relation (5.11) that 2π 1 kπ π cos2 rn (t) = ( + cos(t + )). n 2 2n n n−1 k=1

(5.11)

592

Interpolation

Chapter 8

At the same time, according to (5.10), we find 1 π π kπ 1 cos2 (Rn (t + ) + Rn (t − )) = + cos kt 2 2n 2n 2 2n n−1

df

Rn(2) (t) =

k=1

and, therefore, rn (t) =

2π (2) π 3π π π R (t + ) = (Rn (t + ) + Rn (t + )). n n n 2n 2n

(5.12)

Combining equalities (5.12) and (5.10), we find rn (t) =

π sin nt cos(t/2 + π/2n) π sin2 . 2n 2n sin(t/2 + π/n) sin(t/2 + π/2n) sin t/2

This enables us to conclude that sgn rn (t) = sgn

sin nt cos(t/2 + π/2n) . sin(t/2 + π/2) sin(t/2 + π/2n)

(5.13)

for t ∈ [0, 2π]. Now let t ∈ [xk , xk+1 ], xk = kπ/n, k = 0, 1, . . . , 2n−1. Hence, by virtue of equality (5.13), we get ⎧ sgn sin nt = (−1)k , 0 ≤ k < n − 1, ⎪ ⎪ ⎪ ⎪ ⎪ ⎨−sgn sin nt = (−1)k+1 , n − 1 ≤ k ≤ 2n − 2, sgn rn (t) = (5.14) ⎪ sgn sin nt = 1, k = 2n − 2, ⎪ ⎪ ⎪ ⎪ ⎩ −sgn sin nt = 1, k = 2n − 1. Relation (5.14), to a significant extent, characterizes the behavior of the function Φn (t). Thus, in particular, it follows from this relation that, for any x ∈ (0, π/n) and odd n, Φn (xk − x) > Φn (xk+2 − x) > Φn (xk+4 − x) > . . . > Φn (π − x), k = 1, 3, . . . , n − 2, and Φn (xk − x) < Φn (xk+2 − x) < Φn (xk+4 − x) π π < . . . < Φn (π − − x) < Φn (π + − x), n n k = 2, 4, . . . , n − 1.

(5.15)

Section 5

Summable Analog of the Favard Method

593

At the same time, for even n, Φn (xk − x) > Φn (xk+2 − x) > Φn (xk+4 − x) π π > . . . > Φn (π − − x) > Φn (π + − x), n n k = 1, 3, . . . , n − 1, and Φn (xk − x) < Φn (xk+2 − x) < Φn (xk+4 − x) . . . Φn (π − x),

(5.15 )

k = 2, 4, . . . , n − 2. Assume that (1) Ik (x)

π Fn (t + x)dt = Φn (xk + x) − Φn (π + x)

= xk

and (2) Ik (x)

π Fn (t − x)dt = Φn (xk − x) − Φn (π − x).

= xk

Thus, it follows from inequalities (5.15)–(5.15 ) that (1)

sgn Ik (x) = sgn Φn (xk + x) = (−1)k , k = 0, 1, . . . , n − 1, (2)

sgn Ik (x) = sgn Φn (xk − x) = (−1)k−1 , k = 1, 2, . . . , n − 1,

(5.16) (5.16 )

and (2)

sgn I0 (x) = 1.

(5.17)

In view of relations (5.16), (5.17), and (5.9), we find ¯ n (x) = 1 ω( π ) Ω (−1)k π n n−1 k=1

π

1 Fn (t + x)dt + ω(x + αn ) π

xk

1 π (−1)k−1 + ω( ) π n n−1 k=1

π Fn (t + x)dt 0

π Fn (t − x)dt xk

594

Interpolation 1 π (−1)k−1 + ω( ) π n n−1 k=1

1 ω(αn − x) π

+

Chapter 8

π Fn (t − x)dt xk

π Fn (t − x)dt. 0

Further, by using the equalities 1 π (−1)k ω( ) π n n−1 k=1

π Fn (t + x)dt xk

ω(π/n) (−1)k = 2π

−x k

n−1 k=0

−xk+1

1 π Fn (t − x)dt − ω( ) 2π n

π Fn (t + x)dt, 0

and 1 π (−1)k−1 ω( ) π n n−1 k=1

π Fn (t − x)dt xk

ω(π/n) (−1)k−1 = 2π n−1 k=0

xk+1

1 π Fn (t − x)dt + ω( ) 2π n

xk

π Fn (t − x)dt, 0

we get xk+1 n−1 1 π k+1 ¯ Ωn (x) = (−1) Fn (t − x)dt ω( ) 2π n k=−n

xk

π 1 π 1 ) − ω( )) + (ω(x + π 2n 2 n

π Fn (t + x)dt 0

+

1 π 1 π (ω( − x) + ω( )) π 2n 2 n

π Fn (t − x)dt. 0

At the same time, n−1

xk+1

Fn (t − x)dt = 0.

(−1)k+1

k=−n

xk

(5.18)

Section 5

Summable Analog of the Favard Method

595

Hence, by virtue of (5.18), ¯ n (x) = 1 (ω(x + αn ) − 1 ω( π )) Ω π 2 n

π Fn (t + x)dt 0

1 1 π + (ω(αn − x) + ω( )) π 2 n

π Fn (t − x)dt = Ωn (x), (5.19) 0

where Ωn (x) is the right-hand side of relation (5.2). Combining relations (5.7), (5.9), and (5.19), we arrive at estimate (5.2). This completes the proof of Theorem 5.1. 5.5. We now prove inequality (5.3). In view of inequality (5.2), we can write 1 π En (x; ω) ≤ ω( ) 2π n

π (Fn (t − x) − Fn (t + x))dt 0

1 + ω(x + αn ) π

π

1 Fn (t + x)dt + ω(αn − x) π

0

1 1 π ≤ ( ω( ) π 2 n

π Fn (t − x)dt 0

π (Fn (t − x) − Fn (t + x))dt 0

π + ω( − x) 2n

π Fn (t − x)dt) 0

1 π 1 π = ω( ) + ( − x) 2 n π 2n

π Fn (t − x)dt 0

1 π 1 π ≤ ω( ) + ω( ) 2 n π 2n

π Fn (t − x)dt. 0

At the same time,

(5.20)

596

Interpolation π

π Fn (t − x)dt =

π Fn (u)du −

−x

0

Chapter 8

Fn (u)du

π−x

kπ π π (1 − (−1)k ) cot + sin kx 2 2n 2n n−1

=

k=1

=

≤ =

π π + 2 n π π + 2 n

[ n−1 ]−1 2

cot

2k + 1 π sin(2k + 1)x 2n

cos

2k + 1 π 2n

k=0 [ n−1 ]−1 2

k=0

π π π sin 2[ n−1 π 2 ] 2n < + 1. + π 2 2n sin 2n 2

Substituting this estimate in (5.20), we arrive at inequality (5.3).

9. APPROXIMATIONS IN THE SPACES OF LOCALLY SUMMABLE FUNCTIONS

As a rule, periodic functions are approximated by trigonometric polynomials of a given degree n and, in particular, by polynomials generated by linear operators (of the well-known linear processes of summation of Fourier series). In this case, the attention of the researchers is, clearly, mainly focused on the Fourier operator because this operator associates each function with the sequence of its partial Fourier sums. For the approximation of functions defined on the entire real axis (and not necessarily periodic), it is natural to use entire functions of exponential type not greater than σ. The set of these functions is denoted by Eσ . It is well known that the trigonometric polynomials of degree n belong to En . The foundations of the contemporary theory of approximation by entire functions were laid by Bernstein at the beginning of the last century. Actually, the idea of constructing the theory of approximation of functions defined on the entire real axis, including the theory of approximation of periodic functions, belongs to Bernstein. This idea proved to be extremely important and, as a result, these theories enrich each other and develop fairly rapidly and successfully for several last decades. The results presented in this chapter also deal with this concept. Thus, we ˆ p of locally integrable functions including the spaces Lp of describe the spaces L ˆ ψ N of functions defined on the 2π-periodic functions and introduce the classes L β entire real axis and including the classes of periodic functions Lψ β N studied in the ˆ 1 , we introduce Fourier operators as analogs of previous sections. In the space L the operators associating each function in L1 with its partial Fourier sums, study ˆ ψ N and, in particular, establish the deviations of these operators in the classes L β the analogs of the Lebesgue and Favard inequalities known in the periodic case. 597

598

1.

Approximations in Spaces of Locally Summable Functions

Chapter 9

ˆp Spaces L

1.1. As always, let Lp , p ≥ 1, be a set of 2π-periodic functions ϕ(·) with finite norm 2π ϕp = ( |ϕ(t)|p dt)1/p , p ∈ [1, ∞), 0

or ϕ∞ = ϕM = ess sup|ϕ(t)|, i.e., L∞ = M. ˆ p , p ≥ 1, are introduced as the sets of functions ϕ(·) defined The spaces L on the real axis R (and not necessarily periodic) with finite norm ϕpˆ, where a+2π

|ϕ(t)|p dt)1/p

ϕpˆ = sup( a∈R

(1.1)

a

for p ∈ [1, ∞) and ϕ∞ ˆ = ess sup |ϕ(t)|. t∈R

It is clear that the inclusion Lp ⊂ Lpˆ holds for all p ≥ 1. Now let ψ(v) be a function continuous for all v ≥ 0 and let β be a fixed number such that the following transformation exists for almost all t ∈ R : ˆ β) = 1 ψˆβ (t) = ψ(t, π

∞ ψ(v) cos(vt +

βπ )dv. 2

(1.2)

0

ˆ 1 , admitting, for almost ˆ ψ we denote the set of functions f ∈ L Further, by L β all x, the representations ∞ f (x) = A0 +

ϕ(x + t)ψˆβ (t)dt = A0 + (ϕ ∗ ψˆβ )(x),

(1.3)

−∞

ˆ 1 , and the integral is understood as the limit of where A0 is a constant, ϕ ∈ L ˆ ψ and, in addition, ϕ ∈ integrals over symmetric expanding segments. If f ∈ L β ψ ˆ ˆ N, where N is a subset of L1 , then we set f ∈ L N. The subsets of continuous β

ˆ ψ N are denoted by Cˆ ψ and Cˆ ψ N, respectively. ˆ ψ and L functions from L β β β β ˆ ψ and the sets Lψ introduced in 1.2. The relationship between the sets L β β Section 3.7, is established by the following assertion:

ˆp Spaces L

Section 1

599

Proposition 1.1. Let ψ(v) be a function continuous for all v ≥ 0, let β ∈ R, ˆ = ψ(t, ˆ β) of the form (1.2) be summable on the and let the transformation ψ(t) entire axis (ψˆ ∈ L(R)), namely, ∞ ˆ |ψ(t)|dt = K < ∞.

(1.4)

−∞

Then ˆ ψ L01 = Lψ , L β β

(1.5)

where L01 is a subset of functions ϕ(·) from L1 such that π ϕ(t)dt = 0, −π

i.e., ϕ ⊥ 1 and, moreover, if N is a subset of L01 , then ˆ ψ N = Lψ N L β β

and Cˆβψ N = Cβψ N.

(1.5 )

ˆ ψ N is Therefore, in view of (1.5 ), every assertion proved for the classes L β 0 automatically extended to the classes Lψ β N for N ⊂ L1 provided that condition (1.4) is satisfied.

ˆ ψ L0 (0, 2π). We now show that this yields the incluProof. First, let f ∈ L β sion f ∈ Lψ β . In this case f (·) admits a representation in the form (1.3), where ϕ ∈ L0 (0, 2π). Thus, it is 2π-periodic and summable on [−π, π]. Let us show that f (·) possesses the (ψ, β)-derivative and fβψ (·) = ϕ(·) almost everywhere. The relationship between the Fourier coefficients ak = ak (g) and bk (g), ψ k = 1, 2, . . . , of the function g ∈ Lψ β and the Fourier coefficients αk = αk (gβ ) and βk = βk (gβψ ), k = 1, 2, . . . , of its (ψ, β)-derivative is established by (3.7.4) and (3.7.5). Hence, to prove the inclusion f ∈ Lψ β , it remains to show that the coefficients ak = ak (f ) and bk = bk (f ) of the function f (·) and the coefficients αk = ak (ϕ) and βk = bk (ϕ) of the function ϕ(·) satisfy the equalities ak = ψ(k)(αk cos βπ/2 − βk sin βπ/2), bk = ψ(k)(αk sin βπ/2 + βk cos βπ/2).

(1.6)

600

Approximations in Spaces of Locally Summable Functions

Chapter 9

ˆ β) cos kx is To do this, we use Proposition 4.2.1. The product ϕ(x + t)ψ(t; summable in the strip x ∈ [−π, π], t ∈ R. Hence, by applying the Fubini theorem on the order of integration and equalities (4.2.9) and (4.2.9 ), we find 1 ak (f ) = π

π ∞ ˆ β) cos kxdtdx ϕ(x + t)ψ(t; −π −∞

∞ = −∞

ˆ β) 1 ψ(t; π

π ϕ(x + t) cos kxdxdt −π

∞

∞ ˆ β) cos ktdt + bk (ϕ) ψ(t;

= ak (ϕ) −∞

ˆ β) sin ktdt ψ(t;

−∞

∞ ψˆ+ (t) cos ktdt

= ak (ϕ) cos βπ/2 −∞

∞ − bk (ϕ) sin βπ/2

ψˆ− (t) sin ktdt

−∞

= ψ(k)(ak (ϕ) cos βπ/2 − bk (ϕ) sin βπ/2). Similarly, we obtain bk (f ) = ψ(k)(ak (ϕ) sin βπ/2 + bk (ϕ) cos βπ/2). ψ Hence, f ∈ Lψ β and, moreover, fβ (·) = ϕ(·) almost everywhere.

Now let f ∈ Lψ β . Consider a function a0 g(x) = + 2

∞

ˆ β)dt. fβψ (x + t)ψ(t;

(1.7)

−∞

Since fβψ ∈ L0 (0, 2π) and ψˆ ∈ L(R), the function g(·) is 2π-periodic summable and, as proved above, its Fourier series coincides with the series S[f ]. Thus, g(x) = f (x) almost everywhere and, in view of the fact that ˆ ψ L0 (0, 2π), we conclude that f ∈ L ˆ ψ L0 (0, 2π). g∈L β β Relation (1.5) is thus proved. Equalities (1.5 ) are now obvious.

Order Relation for (ψ, β)-Derivatives

Section 2

601

If equality (1.5) is true, then the function ϕ(·) represented in the form (1.3) coincides with the (ψ, β)-derivative of the function f (·). In this connection, it is natural to say that the function ϕ(·) given by relation (1.3) is the (ψ, β)derivative of the function f (·) also in the nonperiodic case and set ϕ(·) = fβψ (·). ˆ ψ with its (ψ, β)-derivative is The operator which associates the function f ∈ L β

denoted by Dβψ and, hence, Dβψ f (·) = fβψ (·).

2.

Order Relation for (ψ, β)-Derivatives

2.1. By analogy with the periodic case (see Section 3.10), we introduce an order relation for (ψ, β)-derivatives. Let ψ1 (v) and ψ2 (v) be functions continuous for all v ≥ 0, and let β1 and let β2 be fixed numbers. We say that a couple (ψ1 , β1 ) L-precedes a couple L

ˆ ψ2 ⊆ L ˆ ψ1 . In this case, we write (ψ1 , β1 ) ≤(ψ2 , β2 ). Indeed, if (ψ2 , β2 ), if L β2 β1 L ˆ ψ2 ⊂ L ˆ ψ1 , then we write (ψ1 , β1 ) 0 and its transformation ψ(t) and c ∈ [0, σ), the equality 1 π

∞ ψ(v)λσ (v; c) cos(vx + βπ/2)dv 0

∞ ˆ σ (v; c; 0)dv = (ψˆβ ∗ λ ˆ σ )(x). ψˆβ (v + x)λ

= −∞

holds at all points x. This equality enables us the rewrite the expression Fσ,c (f ; x) in the form ∞ Fσ,c (f ; x) = A0 +

fβψ (x

∞ ˆ σ (v; c; 0)dvdt ψˆβ (t + v)λ

+ t)

−∞

−∞

ˆ σ )(x). = A0 + fβψ ∗ (ψˆβ ∗ λ Under the assumptions of Proposition 3.1, by using equality (3.3), one can easily justify the possibility of changing the order of integration in the last expresˆ ψ . Therefore, we can write sion for any f ∈ L β ˆ 0 ) ∗ ψˆβ )(x) Fσ,c (f ; x) = A0 + ((fβψ ∗ λ ∞ ∞ = A0 +

ˆ σ (t; c)dtψˆβ (z)dz fβψ (x + t + z)λ

−∞ −∞

∞ = A0 + −∞

∞ ψˆβ (u − x)

ˆ σ (t; c)dtdu. fβψ (u + t)λ

(3.4)

−∞

3.2. In what follows, we use some facts from the theory of entire functions. We take them from the well-known monograph by Akhiezer [1].

Section 3

Approximating Functions

609

An important subset of entire functions is formed by a collection of functions f (z) satisfying the inequality |f (z)| ≤ AeB|z| ,

(3.5)

for all z , where A and B are constants independent of z. Functions of this sort are called the entire functions of exponential type. The greatest lower bound of the values of B for which inequality (3.5) is satisfied (the value of the constant A may depend on B ) is called the type of the function f (z). The type of a function can be found by using the formula σ = lim sup |z|→∞

ln |f (z)| . |z|

(3.6)

The set of all entire functions of exponential type not greater than a given number σ, σ ≥ 0, is denoted by Eσ . Let ϕ(·) be an entire function of exponential type τ (ϕ ∈ Eτ ) and let, in addition, ϕ(x) , x ∈ R1 h(x) = 1 + |x| be a function from the space L2 (R1 ), i.e., ∞ |h(x)|2 dx < ∞. −∞

The set of all functions ϕ(·) of this kind is denoted by Wτ2 . Statement 3.1. Assume that γ(u) is a function absolutely continuous on the interval [−σ, σ] and its derivative γ (u) belongs to the space L2 (−σ, σ), i.e., σ

|γ (x)|2 dx < ∞.

(3.7)

−σ

If, in addition, the function h(x) satisfies one of the conditions (a) h ∈ W 2 , i.e.,

∞ −∞

|h(x)|2 dx < ∞, (1 + |x|)2

(3.8)

610

Approximations in Spaces of Locally Summable Functions

(b) h ∈ W, i.e.,

∞ −∞

|h(x)| dx < ∞, 1 + |x|

Chapter 9

(3.9)

then the convolution ∞

σ g(x + t)h(t)dt,

γ(v)eivx dv

g(x) =

−∞

(3.10)

−σ

belongs to the space Wσ2 . ˆ 1 , we set For any function ϕ ∈ L ∞ ˆ σ,c (t)dt = (ϕ ∗ λ ˆ σ,c )(x). ϕ(x + t)λ

Sσ,c (ϕ; x) =

(3.11)

−∞

Statement 3.2. The following assertions are true: (1) if ϕ ∈ W ∪ W 2 , then Sσ,c (ϕ; ·) ∈ Wσ2 ; (2) if ϕ ∈ Wτ2 for τ ≤ c, then Sσ,τ (ϕ; x) ≡ ϕ(x); (3) if ϕ(x) is a 2π-periodic integrable function and S[ϕ] =

∞

ck eikx

k=−∞

is its Fourier series, then Sσ,c (ϕ; x) =

λσ,c (k)ck eikx ;

|k| 0 and let ψˆβ ∈ L(R1 ). If, in addition, fβψ ∈ W 2 , then Fσ,c (f ; x) is an entire function of exponential type not greater than σ and, moreover, Fσ,c (f ; ·) ∈ Wσ2 . Proof. We set γ(v) = ψ(v)λσ,c (v). In view of relation (3.2), this enables us to conclude that ⎧ ψ(v), v ≤ c, ⎪ ⎪ ⎪ ⎨σ − v ψ(v), c ≤ v ≤ σ, γ(v) = σ−c ⎪ ⎪ ⎪ ⎩ 0, v ≥ c. Hence, by virtue of relation (3.1), we obtain ∞ σ 1 ψ fβ (x + t) γ(v) cos(vt + βπ/2)dvdt. Fσ,c (f ; x) = A0 + π −∞

0

Thus, to prove the required assertion, it remains to use Statement 3.1. 3.4. In what follows, it is convenient to use somewhat modified functions λσ,c (v). Indeed, we set ⎧ v ∈ [0, c] ∪ [σ, ∞), ⎪ ⎨λσ,c (v), ∗ ∗ λσ,c (v) = λσ (v; c) = (3.19) v − c ψ(σ) ⎪ , c ≤ v ≤ σ, ⎩1 − σ − c ψ(v)

Section 3

Approximating Functions

613

and if f ∈ Lψ β , then ∗ Fσ,c (f ; x)

∞ = A0 +

∗

+ )(t, β)dt. fβψ (x + t)(ψλ σ,c

(3.20)

−∞ ∗ (f ; x) differ from each other by a quantity The operators Fσ,c (f ; x) and Fσ,c

Δσ,c (f ; x) = Fσ,c (f ; x) −

∗ Fσ,c (f ; x)

∞ =

fβψ (x + t)δˆσ,c (t)dt,

(3.21)

−∞

where 1 δˆσ,c (t) = δˆσ,c (t; β) = σ−c

σ (v − c)(ψ(v) − ψ(c)) cos(vt + βπ/2)dv. (3.22) c

The following statement is true: Proposition 3.5. Assume that β ∈ R1 and the functions ψ(·) and f (·) satisfy the conditions of Proposition 3.4. Then the following assertions are true: ∗ (f ; x), are the entire functions (1) the functions Δσ,c (f ; x), and, hence, Fσ,c of exponential type not greater than σ and, moreover,

Δσ,c (f ; 0) ∈ Wσ2

∗ and Fσ,c (f ; ·) ∈ Wσ2 ;

(3.23)

(2) if the functions ψ(·) and f (·) satisfy the conditions of Proposition 3.2 and 0 < τ ≤ c < σ, then, at any point x, ∗ Δσ,c (f ; x) = 0 and Fσ,c (f ; x) = Fσ,c (f ; x) = f (x);

(3.24)

(3) if f ∈ Lψ β and ψ(·) satisfies the conditions of Proposition 3.2, then, for any c < σ, a0 ∗ ∗ (f ; x) = λσ,c (k)(ak cos kx + bk sin kx), (3.25) Fσ,c + 2 k [σ], (σ) = σ − 1 for σ = [σ], and [σ] is the integer part of the number σ. In this connection, it is natural to say that the operators Fσ,c (f ; x) and ∗ (f ; x) are Fourier operators. Fσ,c 3.6. In this chapter, the main subject of investigation are the quantities ρσ,c (f ; x)pˆ and ρ∗σ,c (f ; x)pˆ,

(3.27)

Section 4

General Estimates

615

where and

ρσ,c (f ; x) = f (x) − Fσ,c (f ; x)

(3.28)

∗ ρ∗σ,c (f ; x) = f (x) − Fσ,c (f ; x)

(3.29)

1 ˆ ˆψL under the conditions that f ∈ L β p , p ∈ [1, ∞], β ∈ R , and ψ(v) is a function continuous for all v ≥ 0 and such that ψˆβ (·) ∈ L(R1 ), i.e.,

∞ |ψˆβ (t)|dt = K < ∞.

(3.30)

−∞

Under these assumptions, quantities (3.27) are correctly defined because ˆ p . Indeed, if f ∈ L ˆ ψ Lp , then f ψ ∈ L ˆ p . Therefore, by applying ˆψL ˆp ⊂ L L β β β the Minkowski inequality, we get a+2π

|A0 +

f pˆ = sup( a∈R

∞

fβψ (x + t)ψˆβ (t)dt|p dx)1/p

−∞

a

π ∞ ≤ 2π|A0 | + sup( | fβψ (x + a + t)ψˆβ (t)dt|p dx)1/p a∈R

−π −∞

∞ π ≤ 2π|A0 | + sup ( |fβψ (x + a + t)|p |ψβ (t)|p dx)1/p dt a∈R −∞ −π

∞ ≤ 2π|A0 | +

π |ψβ (t)| sup( a∈R

−∞

= 2π|A0 | +

|fβψ (x + a + t)|p dx)1/p dt

−π

Kfβψ pˆ,

ˆ p , which proves the required inclusion. i.e., f ∈ L

4.

General Estimates 4.1. In the present section, we establish estimates of the quantities ρσ,c (f ; x)pˆ,

ρ∗σ,c (f ; x)pˆ,

and

Δσ,c (f ; x)pˆ

616

Approximations in Spaces of Locally Summable Functions

Chapter 9

ˆ ˆψL for the functions f (·) from the sets L β p , β ∈ R, p ∈ [1, ∞], under the assumption that ψ(v) is a function continuous for all v ≥ 0 and its transformation ψˆβ (·) is summable on R1 . In the next sections, the set of admissible functions ψ(·) is restricted to get specific results. Assume that Wτ2 is the set of entire functions defined in Subsection 3.2. If ˆ p , p ∈ [1, ∞], then by Eτ (h)pˆ we denote the best approximation of the h∈L ˆ P , namely, function h by the functions from Wτ2 in the space L Eτ (h) = inf h(t) − ϕ(t)pˆ. ϕ∈Wτ2

(4.1)

First, we prove the following assertion: Proposition 4.1. Let ψ(v) be a function continuous for all v ≥ 0 and satisfying condition (1.4). Then the following equality is true for any σ > 0, ˆ ˆψL 0 ≤ c < σ, any f ∈ L β p , p ∈ [1, ∞], and any β ∈ R : Δσ,c (f ; x)pˆ ≤ Ec (fβψ )pˆδˆσ,c 1 ,

(4.2)

∞ |δˆσ,c (t)|dt.

where δˆσ,c 1 = −∞

Indeed, let ϕ(·) be an arbitrary function from Wc2 and let h(v) = fβψ (v) − ϕ(v). Then, according to Proposition 3.5, Δσ,c (ϕ; x) ≡ 0 and, hence, ∞ Δσ,c (f ; x) =

(fβψ (x + t) − ϕ(x + t))δˆσ,c (t)dt.

−∞

Therefore, by applying the Minkowski inequality, we find a+2π

Δσ,c (f ; x)pˆ = sup( a∈R

∞ h(x + t)δˆσ,c (t)dt|p dx)1/p

| a

−∞

∞

π |δˆσ,c | sup(

≤ −∞

a∈R

= hpˆδˆσ,c 1 ,

|h(x + a + t)|p dx)1/p dt

−π

(4.3)

whence, by taking the lower bound for ϕ ∈ Wc2 , we arrive at estimate (4.2).

Section 4

General Estimates

617

We now deduce an estimate similar to (4.2) for the quantity ρ∗σ,c (f ; x)pˆ. By ˆ ˆψL virtue of equalities (1.3), (3.19), and (3.20), we conclude that, for any f ∈ L β p, ρ∗σ,c (f ; x)

∞ =

fβψ (x + t)ˆ rσc (t)dt,

(4.4)

−∞

where

(4.5)

⎧ 0, 0 ≤ v ≤ c, ⎪ ⎪ ⎨ rσc (v) = (v − c)(σ − c)−1 ψ(σ), c < v ≤ σ, ⎪ ⎪ ⎩ ψ(v), v ≥ σ.

(4.6)

=

rˆσc (t; β)

1 = π

∞

βπ )dv, 2

rˆσc (t)

rσc (v) cos(vt + 0

In view of assertion (2) in Proposition 3.5, we see that, for any ϕ ∈ Wτ2 and τ ≤ c < σ, ∞ ϕ(x + t)ˆ rσc (t)dt ≡ 0. (4.7) J(x) = −∞ ∗ (Φ; x), where Φ(·) is a Indeed, J(x) represents the quantity Φ(x) − Fσ,c function whose (ψ, β)-derivative is ϕ(·). Since ϕ ∈ Wτ2 , in view of (3.24), we ∗ (Φ; x) ≡ Φ(x) and, hence, relation (4.7) is true. have Fσ,c Thus, the function rˆσc is orthogonal to all shifts of any function from Wτ2 . Therefore, by setting fβψ (v) − ϕ(v) = h(v) for any ϕ ∈ Wτ2 , according to (4.4) and (4.7), we find ∞ ∗ h(x + t)ˆ rσc (t)dt, (4.8) ρσ,c (f ; x) = −∞

whence, acting as in the proof of estimate (4.3), we get ρ∗σ,c (f ; x)pˆ = hpˆˆ rσc 1 . By analyzing the lower bound for ϕ ∈

Wc2 ,

(4.9)

we arrive at the following assertion:

Theorem 4.1. Assume that ψ(v) is a function continuous for all v ≥ 0 and satisfying condition (1.4). Then the following inequality holds for any σ > 0, ˆ ˆψL 0 ≤ c < σ, any f ∈ L β p , p ∈ [1, ∞], and any β ∈ R : ρ∗σ,c (f ; x)pˆ ≤ Ec (fβψ )pˆˆ rσc (t; β)1 ,

(4.10)

618

Approximations in Spaces of Locally Summable Functions

Chapter 9

where the quantity Ec (fβψ )pˆ is given by equality (4.1), ∞ ˆ rσc (t; β)1

|ˆ rσc (t; β)|dt

=

(4.11)

−∞

and the function rˆσc (·) is given by equality (4.5). It is natural to say that inequality (4.10) is the Lebesgue inequality for Fourier operators because, in the periodic case, it corresponds to the classical Lebesgue inequality for Fourier sums. 4.2. We now establish an analog of the well-known Favard inequality for Fourier operators. To this end, we use the operators Fσ,c (f ; x) and estimate the quantity ρσ,c (f ; x)pˆ from above. ˆ ψ Lp and ψ(v) satisfies all assumptions of Proposition Assume that f ∈ L β 3.5. Consider the convolution ((fβψ − Sσ,c (fβψ ; ·)) ∗ ψˆβ )(x) = (fβψ ∗ ψˆβ )(x) − (Sσ,c (fβψ ; ·) ∗ ψˆβ )(x) ˆ σ,c ) ∗ ψˆβ )(x). = f (x) − A0 − ((fβψ ∗ λ

(4.12)

By virtue of Lemma 2.1, we have ˆ σ,c )(x) = (ψλ + )(t, β) (ψˆβ ∗ λ σ,c and, hence (the required change of the order of integration is easily justified by using relations (3.3) and (3.15)), ˆ σ,c ) ∗ ψˆβ )(x) = (f ψ ∗ (λ ˆ σ,c ∗ ψˆβ ))(x) = Fσ,c (f ; x) − A0 . ((fβψ ∗ λ β Therefore, in view of relation (4.12), we find ∞ ˆ β)dt, H(x + t)ψ(t;

ρσ,c (f ; x) =

(4.13)

−∞

where H(v) = fβψ (v) − Sσ,c (fβψ ; v).

(4.14)

ˆ p. Now let u(x) be an arbitrary function from Wc2 ∩ L1 (R) and let fβψ ∈ L In view of relations (3.3) and (3.11), we obtain

Section 4

General Estimates

619

ˆ (0) ) ∗ u)(x) (Sσ,c (fβψ ; ·) ∗ u)(x) = ((fβψ ∗ λ σ,c ˆ (0) ∗ u))(x) = (f ψ ∗ Sσ,c (u; ·))(x). = (fβψ ∗ (λ σ,c β By the property of operators Sσ,c (·; ·) (see assertion (2) in Statement 3.2), we have Sσ,c (u; t) = u(t) and, thus, (Sσ,c (fβψ ; ·) ∗ u)(x) = (fβψ ∗ u)(x). Therefore, ∞

H(x + t)u(t)dt = (fβψ ∗ u)(x) − (Sσ,c (fβψ ; ·) ∗ u)(x) ≡ 0.

−∞

In this case, equality (4.13) may take the form ∞ ˆ β) − u(t))dt. H(x + t)(ψ(t;

ρσ,c (f ; x) =

(4.15)

−∞

Further, by applying the Minkowski inequality, we find ˆ β) − u(t)1 , ρσ,c (f ; x)pˆ ≤ Eσ,c (fβψ )pˆψ(t;

(4.16)

Eσ,c (fβψ )pˆ = fβψ (·) − Sσ,c (fβψ ; ·)pˆ,

(4.17)

where

∞ ˆ β) − u(t)1 = ψ(t;

ˆ β) − u(t)|dt. |ψ(t;

(4.18)

−∞ df ¯σ = Eσ ∩ L1 (R) ⊂ Wσ2 because it is known that if f ∈ Eσ ∩ Note that W Lp (R), p ≥ 1, then lim f (x) = 0. Thus, by setting |x|→∞

eσ (f )1 = inf f (t) − u(t)1 , ¯σ u∈W

(4.19)

we arrive at the following assertion: Theorem 4.2. Assume that ψ(v) is a function continuous for all v ≥ 0 and satisfying condition (1.4). Then the following inequality is true for any σ > 0, ˆψL ˆ 0 ≤ c < σ, any f ∈ L β p , p ∈ [1, ∞], and any β ∈ R : ρσ,c (f ; x)pˆ ≤ Eσ,c (fβψ )pêc (ψˆβ )1 .

(4.20)

620

Approximations in Spaces of Locally Summable Functions

Chapter 9

The expressions on the right-hand side of this relation are specified by equalities (4.17) and (4.19). The well-known Favard inequality corresponds to inequality (4.20) in the periodic case. Inequality (4.20) is of especial interest for p ∈ (1, ∞) because the norms of the operators Sσ,c (fβψ ; ·)pˆ are bounded in this case. ˆ p , p ≥ 1. Then, for any τ > 0, and d ∈ [0, τ ), Theorem 4.3. Let f ∈ L Sτ,d (f ; ·)pˆ ≤ Cd,τ f pˆ,

(4.21)

where Cd,τ is a quantity satisfying the inequalities τ +d 1 4 τ +d 4 ln + < Cd,τ < 2 ln + 2. 2 π τ −d 3 π τ −d

(4.22)

If, in addition, p ∈ (1, ∞), then Sτ,d (f ; ·)pˆ ≤ Cp ((τ − d)−1 + 1)f pˆ,

(4.23)

where Cp is a quantity which may depend only on p. Proof. Since the norm ·pˆ is invariant under shifts of the argument, relations (4.21) and (4.22) follow from assertion (4) in Statement 3.2. To prove inequality (4.23), we first establish the following assertion: ˆ p , Ja is the interval (a, a + 2π), and fa (x) Lemma 4.1. Assume that f ∈ L is a 2π-periodic function such that fa (x) = f (x),

x ∈ Ja .

(4.24)

Furthermore, let Ja be an interval which completely belongs to Ja . Then, for any p ≥ 1, 4f pˆ ( |Sτ,d (f ; x) − Sτ,d (fa ; x)|p dx)1/p ≤ , (4.25) δ(τ − d) Ja

where δ is the distance from Ja to the set R \ Ja . df

Without loss of generality, we can assume that Ja = (−π, π) = J and denote the function fπ (·) by f˜(·). Let J be an interval which completely belongs to J and let δ be the least of the distances between the ends of the intervals J and J .

Section 4

General Estimates

Consider the difference ˆ ˆ τ,d (t)dt R1 (x) = Sτ,d (f ; x) − f (x + t)λτ,d dt = f (x + t)λ |t|≤δ

621

(4.26)

|t|≥δ

ˆ p. for f ∈ L By applying the Minkowski inequality, in view of relations (3.3) and (4.26), we find ∞ p 1/p ˆ τ,d (t)|dt = 2f pˆ ( |R1 (x)| dx) ≤ f pˆ2 |λ δ(τ − d) J

δ

and, therefore, ( |Sτ,d (f ; x) − Sτ,d (f˜; x)|p dx)1/p J

4f pˆ ≤( | (f (x + t) − f˜(x + t))dt|p dx)1/p + . (4.27) δ(τ − d) J

|t|≤δ

However, if x ∈ J and |t| ≤ δ, then x + t ∈ J. Thus, by virtue of (4.24), the first term on the right-hand side of (4.27) is equal to zero, which yields estimate (4.25). We now pass to the proof of the theorem. Indeed, u+2π

Sτ,d (f ; ·)pˆ = sup( u∈R

|Sτ,d (f ; x)|p dx)1/p .

(4.28)

u

For any fixed, u, by f˜1 (x) and f˜2 (x) we denote the 2π-periodic functions which coincide with f (x) in the intervals (u−π/2, u+3π/2) and (u+π/2, u+ 5π/2), respectively. Then u+2π

|Sτ,d (f ; x)|p dx)1/p

( u

u+π

≤(

u+2π

|Sτ,d (f ; x)| dx) p

u

1/p

+( u+π

|Sτ,d (f ; x)|p dx)1/p . (4.29)

622

Approximations in Spaces of Locally Summable Functions

Chapter 9

By virtue of Lemma 4.1 (for a = u − π/2, we have Ja = (u, u + π) and, hence, δ = π/2), u+π

|Sτ,d (f ; x)|p dx)1/p

( u

u+π

u+π

|Sτ,d (f ; x) − Sτ,d (f˜1 ; x)| dx)

≤(

p

u

≤

1/p

|Sτ,d (f˜1 ; x)|p dx)1/p

+( u

8f pˆ + Sτ,d (f˜1 ; x)pˆ. π(τ − d)

(4.30)

Similarly, we get u+2π

|Sτd (f ; x)|p dx)1/p ≤

( u+π

8f pˆ + Sτ,d (f˜1 ; x)pˆ. π(τ − d)

(4.31)

Combining relations (4.28)–(4.31), we find u+π

|Sτ,d (f ; x)|p dx)1/p ≤

( u

16f pˆ + Sτ,d (f˜1 ; x)pˆ + Sτ,d (f˜2 ; x)pˆ. (4.32) π(τ − d)

The operator Sτ,d (·, x) associates every summable 2π-periodic function f (·) with a trigonometric polynomial whose coefficients are the Fourier coefficients of a given function multiplied by the numbers λτ,d (k). Therefore, this operator is a multiplicator generated by the sequence λτ,d , k = 0, 1, . . . , [τ ], λk = 0, k ≥ [τ ]. This sequence satisfies the conditions of Theorem 6.5.1 and, hence, we conclude that Sτ,d (f ; x)pˆ = Sτ,d (f ; ·)p ≤ Cp f p = Cp f pˆ,

(4.33)

for any p ∈ (1, ∞) and arbitrary numbers τ > d > 0, where Cp is a quantity which may depend only on p.

Section 4

General Estimates

623

In view of estimate (4.33), we obtain Sτ,d (f˜1 ; x)pˆ + Sτ,d (f˜2 ; x)pˆ ≤ 2Cp f pˆ. By comparing this relation with inequality (4.32), we arrive at estimate (4.23). Theorem 4.3 is thus proved. ˆ p, Corollary 4.1. The following inequality holds for any function ϕ ∈ L p ≥ 1, and any σ > 0 and 0 ≤ c < σ : Eσ,c (ϕ)pˆ = ϕ(·) − Sσ,c (ϕ; ·)pˆ ≤ Kσ,c Ec (ϕ)pˆ,

(4.34)

where the quantity Kσ,c satisfies the inequalities 4 σ+c 1 4 σ+c ln + < Kσ,c < 2 ln + 3. 2 π σ−c 3 π σ−c

(4.35)

If, in addition, p ∈ (1, ∞), then Eσ,c (ϕ)pˆ ≤ Kp ((σ − c)−1 + 1)Ec (ϕ)pˆ,

(4.36)

where Kp is a quantity which may depend only on p and Ec (ϕ)pˆ is given by equality (4.1). Indeed, if u(·) is an arbitrary function from Wc2 , then the following identity is true: Sσ,c (u; x) ≡ u(x). Therefore, Eσ,c (ϕ)pˆ = (ϕ(·)−u(·))+Sσ,c (u−ϕ; ·)pˆ ≤ ϕ(·)−u(·)pˆ +Sσ,c (u−ϕ; ·)pˆ, whence, by using Theorem 4.3 and passing to the lower bound for u ∈ Wσ2 , we get estimates (4.34)–(4.36). 4.3. Inequalities (4.34)–(4.36) enable us to obtain the required estimates of the first factor on the right-hand side of relation (4.20). The second factor can be estimated by using the following assertion: Proposition 4.2. Assume that the function ψ(v) is continuous for v ≥ 0 and satisfies condition (1.4). Then, for any c > d ≥ 0 and β ∈ R, ˆ 1 ≤ ˆ ec (ψ) rcd (·; β)1 ,

(4.37)

624

Approximations in Spaces of Locally Summable Functions

where rˆcd (t; β)

∞

1 = π

rcd (v) cos(vt +

βπ )dv, 2

Chapter 9

(4.38)

0

⎧ 1, 0 ≤ v ≤ d, ⎪ ⎪ ⎨ rcd (v) = (v − d)(c − d)−1 ψ(c), d ≤ v ≤ c, ⎪ ⎪ ⎩ ψ(v), v ≥ c. Indeed, let 1 u(t) = π

∞ ϕ(v) cos(vt +

βπ )dv, 2

(4.39)

(4.40)

0

where

⎧ 0, v ≥ c, ⎪ ⎪ ⎨ ϕ(v) = ψ(v) − (v − d)(c − d)−1 ψ(c), d ≤ v ≤ c, ⎪ ⎪ ⎩ ψ(v), 0 ≤ v ≤ d.

(4.41)

By the Wiener–Paley theorem, we have u ∈ Ec and, in view of equalities (4.38)–(4.41), we get ψˆβ (t) − u(t) = rˆcd (t; β). Hence, ˆ 1 ≤ ψˆβ (t) − u(t)1 = ˆ ec (ψ) rcd (·; β)1 . By comparing the assertions of Theorems 4.1 and 4.2, Corollary 4.1, and Proposition 4.2, we see that estimates (4.10) and (4.20) would become constructive if we establish constructive estimates of the quantities ˆ rστ (·; β)1 . To obtain estimates of this sort, we need several auxiliary facts.

5.

ˆψ On the Functions ψ(·) Specifying the Sets L β

5.1. In the previous section, it was assumed that the functions ψ(·) in the ˆ ψ are continuous on the positive semiaxis and satisfy definitions of the sets L β condition (1.4). To obtain the required estimates of the quantities ˆ rστ (·; β)1 , it is necessary to restrict the set of admissible functions. As before (see Subsection 3.2.1), let M be a set of functions ψ(·) convex downward for all v ≥ 1 and such that lim ψ(v) = 0.

v→∞

(5.1)

ˆψ On the Functions ψ(·) Specifying the Sets L β

Section 5

625

We extend each function ψ ∈ M onto a segment [0, 1) as a function (also denoted by ψ(·) ) continuous for all v ≥ 0 and such that ψ(0) = 0 . It is also required that the derivative ψ (v) = ψ (v + 0) be a function of bounded variation in the interval [0, ∞), namely, ∞

V ψ (t) ≤ K < ∞.

(5.2)

0

The set of functions ψ(·) of this sort is denoted by A. In the case where the restrictions of elements of the sets A to the ray t ≥ 1 belong to one of the sets M0 , MC , or M∞ , these sets are denoted by A0 , AC , and A∞ , respectively. A subset of functions ψ ∈ A such that ∞

ψ(t) dt < K < ∞, t

(5.3)

1

according to Subsection 4.3.1, is denoted by A . If ψ ∈ A and ψ ∈ F for t ≥ 1, (see Section 3.13), then we set ψ ∈ F0 . The following assertion is true: Proposition 5.1. If ψ ∈ A and β ∈ R, then the function ˆ β) = 1 ˆ = ψˆβ (t) = ψ(t; ψ(t) π

∞ ψ(v) cos(vt +

βπ )dv 2

(5.4)

0

ˆ = O(t−2 ) as |t| → ∞. is summable on R1 and, moreover, ψ(t) If, in addition, ψ ∈ A, then this is also true for the function ψˆ0 (t) = ψ(t; 0). Proof. We have 1 ψˆβ (t) =

∞

0

df

ψ(v) cos(vt+βπ/2)dv = J1 (t)+J2 (t).

ψ(v) cos(vt+βπ/2)dv + 1

If |t| < a, where a is a positive number, then |J1 (t)| < K. Thus, the function J1 (t) is summable in any neighborhood of the origin. Lemma 4.3.1 implies that the function J2 (t) is also summable in any neighborhood of the point t = 0. Hence, it remains to prove that ψˆβ (t) = O(1)t−2 , t → ∞.

(5.5)

626

Approximations in Spaces of Locally Summable Functions

Chapter 9

Integrating by parts and taking into account the fact that ψ(0) = ψ(∞) = = 0, we conclude that

ψ (∞)

ψ (0) 1 ψˆ3 (t) = 2 cos βπ/2 − 2 t t

∞

cos(vt + βπ/2)dψ (v).

0

In view of the boundedness of variation of the function ψ (v), this yields relation (5.5). 5.2. Proposition 5.1 implies that the functions ψ from the set A for any β ∈ R and the functions ψ ∈ A for β = 0 satisfy all requirements imposed on these functions in Section 4.

6.

Estimates of the Quantities ˆ rσc (t, β)1 for c = σ − h and h > 0 6.1. In view of relations (4.5) and (4.6), for any ψ ∈ A , we can write rˆσc (t, β)

ψ(σ) = π

σ c

v−c βπ 1 cos(vt + )dv + σ−c 2 π

∞ ψ(v) cos(vt +

βπ )dv 2

σ

df

= J1 (σ, t, c) + J2 (σ, t).

(6.1)

Integrating this expression by parts, we find J1 (σ, t, c) =

=

ψ(σ) sin(σt + βπ/2) ( π t 2 sin((σ + c)t + βπ/2) sin((σ − c)t/2) ) − (σ − c)t2

(6.2)

ψ(σ) (σ − c)t − sin((σ − c)t) βπ sin(σt + ( ) 2 π (σ − c)t 2 +

βπ 1 − cos((σ − c)t) cos(σt + )). (σ − c)t2 2

(6.3)

It is easy to see that the quantity J2 (σ, t) is independent of c and admits a representation

Estimates of ˆ rσc (t, β)1 for c = σ − h and h > 0

Section 6

ψ(σ) βπ 1 J2 (σ, t) = − sin(σt + )− π 2 πt

∞

ψ (v) sin(vt +

627

βπ )dv 2

σ df

=−

ψ(σ) βπ 1 sin(σt + ) − J3 (σ, t). πt 2 π

(6.4)

Now let a = a(σ) be an arbitrary function continuous for all σ > 0 and such that σa(σ) ≥ a0 > 0 for any σ > 0, ma = min(a(σ), 1), and Ma = max(a(σ), 1). By using relations (6.3) with c = σ − h and (6.4), we set ψ(σ) sin(σt + βπ/2) ψ Bσ (a) = |dt, (6.5) | π t ma ≤|t|≤Ma

|J2 (σ, t)|dt,

Pσψ (a) =

Rσψ (a) =

|t|≤a(σ)

1 π

|J3 (σ; t)|dt,

(6.6)

|t|≥a(σ)

1 |J1h (σ; t)|dt,

γσ (h) =

J1h (σ; t) = J1 (σ; t; σ − h),

−1

and

| sin((2σ + h)t/2 + βπ/2) sin(ht/2)| dt. ht2

(6.7)

ˆ rcσ−h (t, β)1 ≤ Bσψ (a) + Pσψ (a) + Rσψ (a) + γσ (h) + δσ (h).

(6.8)

δσ (h) =

2ψ(σ) πh

|t|≥1

It is easy to see that

We now estimate each term on the right-hand side of this expression. To this end, we first establish the following assertion: Proposition 6.1. If ψ ∈ A , then, for any σ > 1 and t ∈ R1 , 2 |J2 (σ, t)| ≤ π

σ+2π

ψ(v)dv

(6.9)

σ

and |J3 (σ, t)| ≤

2 2π (ψ(σ) − ψ(σ + )). |t| t

(6.10)

628

Approximations in Spaces of Locally Summable Functions

Chapter 9

Proof. In the notation used in relations (4.3.13) and (4.3.31), we can write J2 (σ; t) = cos

βπ βπ J2 (ψ; σ; t)0 − sin J2 (ψ; σ; t)1 . 2 2

(6.11)

Thus, in view of estimate (4.35), its analog for the quantity J2 (ψ; σ; t)0 , and the fact that the continuity of the parameter σ is, in this case, insignificant, we conclude that σ+ 2r t

2 |J2 (σ; t)| ≤ |J2 (ψ; σ; t)0 | + |J2 (ψ; σ; t)1 | ≤ π

ψ(v)dv.

(6.12)

σ

Similarly, by using relation (3.29), we obtain estimate (6.10). By virtue of inequality (6.9), we get

Pσψ (a)

4 ≤ π

a(σ) σ+2π

ψ(v)dvdt σ

0

4t = π

σ+2π

a(σ) ψ(v)dv|0

σ

a(σ)

+8

ψ(σ + 2π/t) dt t

0 a(σ)

≤ 8(ψ(σ) +

ψ(σ + 2π/t) dt) t

0

∞ = 8(ψ(σ) +

ψ(σ + t) dt). t

(6.13)

1/a(σ)

Further, inequality (6.10) implies that Rσψ (a)

4 ≤ π

∞

ψ(σ) − ψ(σ + t

2π t )

dt

a(σ)

∞ ≤ K1 a(σ)

t−1 (ψ(σ) − ψ(σ + 1/t))dt.

(6.14)

Estimates of ˆ rσc (t, β)1 for c = σ − h and h > 0

Section 6

629

Hence, according to (6.13) and (6.14), Pσψ (a) + Rσψ (a) ≤ K1 (ψ(σ) + Qψ σ (a)),

(6.15)

where K1 is a quantity independent of σ and ∞ Qψ σ (a)

=

ψ(t + σ) dt + t

1/a(σ)

∞

1 t−1 (ψ(σ) − ψ(σ + ))dt. t

(6.16)

a(σ)

In view of relations (6.3), for c = σ − h, we obtain 1 1 ht − sin ht) 1 − cos ht 2ψ(σ) ψ(σ) h2 γσ (h) = dt+ dt) ≤ ( ( +h) (6.17) 2 2 π ht ht π 6 −1

−1

and 4ψ(σ) δσ (h) ≤ πh

∞

4ψ(σ) dt = . 2 t πh

(6.18)

1

If a(σ) ≤ 1, then ma = a(σ) and Ma = 1. Therefore, df

sin(σt + | t

Iσ (a) =

βπ 2 )

1

| sin(σt + t

|dt = 2

ma ≤|t|≤Ma

βπ 2 )|

dt.

(6.19)

a(σ)

At the same time, if a(σ) > 1, then a(σ)

Iσ (a) = 2

| sin(σt + t

βπ 2 )|

dt.

1

Hence, in exactly the same way as in Section 5.5, we get Iσ (a) =

2 (| ln a(σ)| + O(1)) π

(6.20)

and, finally 4ψ(σ) (| ln a(σ)| + O(1)). (6.21) π2 Combining relations (6.8) and (6.13)–(6.21), we arrive at the following assertion: Bσψ (a) ≤

630

Approximations in Spaces of Locally Summable Functions

Chapter 9

Proposition 6.2. Assume that ψ ∈ A and a = a(σ) is an arbitrary function continuous for all σ > 0 and such that σa(σ) ≥ a0 > 0. Then the following inequality is true for any β ∈ R and h > 0 and all σ>0: ˆ rσσ−h (t, β)1 ≤

4ψ(σ) (| ln a(σ)| + K(h2 + h−1 )) + Qψ σ (a), π2

(6.22)

where K is a quantity which may depend only on the function ψ(·). If ψ ∈ F0 , then, for t ≥ 1, the function ψ(t) belongs to the set F. Therefore, in view of Theorem 3.13.4 (see (5.11.44)), if a∗ (σ) = μ(ψ, σ)/σ, then ∗ Qψ σ (a ) = O(1)ψ(σ),

(6.23)

and, by virtue of (3.13.18), one can find a0 such that, for all σ ≥ 1, σa∗ (σ) > a0 .

(6.24)

By using these facts and setting a(σ) = a∗ (σ) = 1/(η(σ)−σ) in Proposition 6.2, we arrive at the following assertion: Theorem 6.1. Let ψ ∈ F0 . Then the following inequality is true for any β ∈ R, σ ≥ 1, and h > 0 : ˆ rσσ−h (t, β)1 ≤

4ψ(σ) (| ln(η(σ) − σ)| + K(h2 + h−1 )), π2

(6.25)

where σ > h and K is a quantity which may depend only on the function ψ(·) and η(σ) = η(ψ; σ) = ψ −1 (ψ(σ)/2). 6.2. According to (5.4), we have Ac,∞ = Ac ∪ A∞ ⊂ F0 . It is clear that A ⊃ Ac but the set A0 also contains elements decreasing very slowly and, hence, condition (5.3) is violated. (Note that, for ψ ∈ Ac , this condition is necessarily true.) For this reason, we set A0 = A0 \ Ac and establish the following analog of Theorem 6.1 for ψ ∈ A0 : Theorem 6.1. Let ψ ∈ A0 . Then, for any σ ≥ 1 and h > 0, ˆ rσσ−h (t, 0)1 ≤

4ψ(σ) (ln σ + K(h2 + h−1 )), π2

(6.26)

where σ > h and K is a quantity which may depend only on the function ψ(·).

Estimates of ˆ rσc (t, β)1 for c = σ − h and h > 0

Section 6

631

Note that if ψ ∈ Ac , then the definition of the sets Ac implies the equality η(ψ; t) − t = θt, where θ is a quantity bounded from above and below by certain positive constants. Thus, for ψ ∈ A0 ∩Ac , estimate (6.26) coincides with estimate (6.25). To deduce estimate (6.26), we set Bσψ

ψ(σ) = π

sin σt | |dt, t

|t|≤1

Rσψ

1 = π

∞ |J3,0 (σ, t)|dt, −∞

γσ(0) (h)

h |J1,0 (σ, t)|dt,

= |t|≤1

and δσ(0) (h)

ψ(σ) = π

ht | sin ht/2| | sin(σt + )|dt, 2 t 2

|t|≤1 h (σ; t) are given by relations (6.4) and (6.3) for β = 0 where J3,0 (σ; t) and J1,0 and c = σ − h, respectively. Then, in view of relations (6.1)–(6.4), one can easily see that

ˆ rστ (t, 0)1 ≤ Bσψ + Rσψ + γσ(0) (h) + δσ(0) (h).

(6.27)

Estimates (6.17) and (6.18) are true for any β ∈ R and, hence, γσ(0) (h) + δσ(0) (h) ≤ K(h2 + h−1 )ψ(σ).

(6.28)

The quantity Rσψ is estimated in Chapter 5 (see (5.4.24)): Rσψ

1 = π

∞

(0)

|J3 (σ, t)|dt ≤ Kψ(n).

(6.29)

−∞

Therefore, in view of the fact that 1

| sin σt| 2 dt = ln σ + O(1), t π

(6.30)

0

by comparing relations (6.27)–(6.30), we immediately arrive at inequality (6.26).

632

Approximations in Spaces of Locally Summable Functions

Chapter 9

Estimates of the Quantities ˆ rσc (t, β)1 for c = θσ, 0 ≤ θ ≤ 1, and ψ ∈ Ac

7.

7.1. In view of equalities (6.1)–(6.4), by setting |J1 (σ; t; θσ)|dt, γσ (θ) =

δσ (θ) =

2ψ(σ) π

Pσψ

|t|≥1/σ

|t|≤1/σ

| sin((1 + θ)σt + βπ/2) sin((1 − θ)σt/2)| dt, (1 − θ)σt2

|J2 (σ, t)|dt,

=

and

Rσψ

1 = π

|t|≤1/σ

|J3 (σ, t)|dt, |t|≥1/σ

we conclude that rσθσ (t, β)1 ≤ γσ (θ) + δσ (θ) + Pσψ + Rσψ .

(7.1)

By virtue of equality (6.3), we find 2ψ(σ) γσ (θ) ≤ ( π

1/σ 0

|(1 − θ)σt − sin((1 − θ)σt)| dt (1 − θ)σt2

1/σ + 0

≤

|1 − cos((t − θ)σt)| dt) (1 − θ)σt2

2ψ(σ) (1 − θ)2 1 − θ ( + ). π 12 2

(7.2)

Similarly, 4ψ(σ) δσ (θ) ≤ π

∞

4ψ(σ) dt = . 2 (1 − θ)σt π(1 − θ)

(7.3)

1/σ

Further, in relations (6.15) and (6.16), we set a(σ) = 1/σ. This gives ∞ Pσψ

+

Rσψ

≤ K1 (ψ(σ) + σ

ψ(σ + t) dt + t

∞

1/σ

ψ(σ) − ψ(σ + 1/t) dt), (7.4) t

Section 7 Estimates of ˆ rσc (t, β)1 for c = θσ, 0 ≤ θ ≤ 1, and ψ ∈ Ac 633 where K1 and Ki , i = 2, . . . , are quantities which may depend on the functions ψ(·). Further, if ψ ∈ Ac,∞ = Ac ∪ A∞ , then ∞ A1 (σ) =

ψ(t + σ) dt ≤ K2 ψ(σ), t

(7.5)

η(σ)−σ

∞

ψ(σ) − ψ(σ + 1/t) dt ≤ K3 ψ(σ). t

A2 (σ) =

(7.6)

1/(η(σ)−σ)

Hence, for any ψ ∈ Ac,∞ , ∞

ψ(t + σ) dt ≤ A1 (σ) + | t

σ

η(σ)−σ

ψ(t + σ) dt| t

σ

≤ K2 ψ(σ) + ψ(σ)| ln

1 | μ(σ)

(7.7)

and ∞

ψ(σ) − ψ(σ + 1/t) dt = A2 (σ) + | t

1/σ

1/(η(σ)−σ)

ψ(σ) − ψ(σ + 1/t) dt| t

1/σ

≤ K3 ψ(σ) + ψ(σ)| ln μ(σ)|.

(7.8)

If ψ ∈ Ac , then, according to its definition, the quantity μ(σ) = μ(ψ; σ) is bounded from below and above by positive constants. Thus, in this case, in view of relations (7.4)–(7.8), we can write Pσψ + Rσψ ≤ K4 ψ(σ).

(7.9)

Combining relations (7.1)–(7.3) and (7.9), we arrive at the following assertion: Theorem 7.1. If ψ ∈ Ac , θ ∈ [0, 1), and β ∈ R, then, for any σ ≥ 1, ˆ rσθσ (t, β)1 ≤

K ψ(σ), 1−θ

where K is a quantity which may depend only on the function ψ(·).

(7.10)

634

Approximations in Spaces of Locally Summable Functions

Chapter 9

Estimates of the Quantities ˆ rσc (t, β)1 for c = 2σ − η(σ) and ψ ∈ A∞

8.

Theorem 8.1. If ψ ∈ A∞ , c∗ = max(2σ − η(σ), 0), and β ∈ R, then, for any σ ≥ 1, ∗ (8.1) ˆ rσc (t, β)1 ≤ Kψ(σ). Proof. For ψ ∈ A∞ , according to the definition of the sets A∞ , the quantity μ(t) = μ(ψ; t) = t/(η(t) − t) monotonically and infinitely increases. Thus, the quantity (2σ − η(σ))/σ increases and approaches 1 as σ → ∞. Hence, beginning with a certain σ0 , the inequality 2σ − η(σ) > 0 is always satisfied. Therefore, in what follows, without loss of generality, we can assume that c∗ = 2σ − η(σ) because inequality (8.1) can be satisfied for σ ∈ [1, σ0 ) by the proper choice of the constant K. By using equalities (6.1)–(6.4), we set ∗ |J1 (σ; t; c∗ )|dt, γσ = |t|≤1/(η(σ)−σ)

δσ∗

2ψ(σ) = πh

|t|≥1/(η(σ)−σ)

| sin((σ + c∗ )t + βπ/2) sin((σ − c∗ )t/2)| dt, (σ − c∗ )t2

Pσψ

|J2 (σ, t)|dt,

=

and

Rσψ

|t|≤(η(σ)−σ)

1 = π

|J3 (σ, t)|dt. |t|≥1/(η(σ)−σ)

Thus, it is easy to see that ∗

ˆ rσc (t, β)1 ≤ γσ∗ + δσ∗ + Pσψ + Rσψ .

(8.2)

In view of inequalities (6.15), (7.5), and (7.6), we find ∞ Pσψ

+

Rσψ

≤ K1 (ψ(σ) +

ψ(σ + 1) dt t

η(σ)−σ

∞ + 1/(η(σ)−σ)

ψ(σ) − ψ(σ + 1/t) dt) ≤ K5 ψ(σ). (8.3) t

Section 9

Estimates of ˆ rcσ (t, 0)1 for c = θσ, 0 ≤ θ < 1, and ψ ∈ A0

635

Further, we apply relation (6.3) and set η(σ) − c = z. As a result, we obtain γσ∗

2ψ(σ) = π

1/(η(σ)−σ)

0

2ψ(σ) = π

1/z

(σ − c∗ )t − sin((σ − c∗ )t) + 1 − cos((σ − c∗ )t) dt (σ − c∗ )t2

zt − sin zt + 1 − cos zt dt ≤ K6 ψ(σ) zt2

(8.4)

0

and δσ∗

4ψ(σ) = zπ

1/z

dt dt ≤ K7 ψ(σ). t2

(8.5)

0

Comparing inequalities (8.2)–(8.5), we arrive at inequality (8.1).

9.

Estimates of the Quantities ˆ r cσ (t, 0)1 for c = θσ, 0 ≤ θ < 1, and ψ ∈ A0 Theorem 9.1. If ψ ∈ A0 and 0 ≤ θ < 1, then, for any σ ≥ 1, ˆ rσθσ (t, 0)1 ≤

K ψ(σ), 1−θ

(9.1)

where K is a quantity which may depend only on the function ψ(·). (0)

(0)

Proof. By J1 (σ; t, θσ) and J2 (σ; t) we denote the values of the integrals J1 (σ; t, c) and J2 (σ; t) in the case where β = 0 and c = θσ. Then, in view of relations (6.1)–(6.3), we get (0)

(0)

rˆσθσ (t, 0) = J1 (σ; t, θσ) + J2 (σ; t), (0)

J1 (σ; t, θσ) =

ψ(σ) sin σt 2 sin((1 + θ)σt/2) sin((1 − θ)σt/2) ), ( − π t (1 − θ)t2

(0) J2 (σ; t)

1 ψ(σ) sin σt − − πt πt

1

ψ (v) sin vtdv

0 df

=−

ψ(σ) 1 (0) sin σt − J3 (σ; t). πt π

636

Approximations in Spaces of Locally Summable Functions

Chapter 9

Therefore, ∞ | sin((1 + θ)σt/2) sin((1 − θ)σt/2)| 4ψ(σ) dt) ( ˆ rσθσ (t, 0)1 ≤ − π(1 − θ)σ t2 0

+

2 π

∞

(0)

|J3 (σ; t)|dt 0

4ψ(σ) 2 = J(θ) + π π df

∞

(0)

|J3 (σ; t)|dt.

(9.2)

0

By setting z = σt/2, we conclude that 1 J(θ) = 2(1 − θ)

∞

| sin((1 + θ)z) sin((1 − θ)z)| dz. z2

0

We now split the interval of integration into the sets θ ≤ z ≤ 1 and z ≥ 1. This gives 1+θ 1 1 J(θ) ≤ + 1 and c ≥ 1, we get σ ψ(c) − ψ(σ) = −

σ

ψ (t)dt = c

|ψ (t)|dt > |ψ (σ)|(σ − c).

c

Hence, for 1 ≤ c < σ, we can write |δσ,c (t, β)| =

6 ψ(c) − ψ(σ) . πt2 σ−c

(10.2)

First, let c = σ − h, where h is a number such that σ − h > 1. Thus, by virtue of relations (10.1) and (10.2), ˆ ˆ |δσ,c (t, β)|dt + |δˆσ,c (t, β)|dt δσ,c (t, β)1 ≤ |t|≤1

≤

ψ(c) − ψ(σ) ψ(σ)

|t|>1

|J1 (σ; t, c)|dt + |t|≤1

24 ψ(c) − ψ(σ) , π σ−c

whence, in view of estimate (6.17) and the fact that, in the analyzed case, J1 (σ; t, c) = J1h (σ; t), we arrive at the following assertion:

638

Approximations in Spaces of Locally Summable Functions

Chapter 9

Proposition 10.1. If ψ ∈ A and c = σ − h > 1, then, for any β ∈ R, δˆσ,c (t, β)1 ≤

ψ(c) − ψ(σ) 2 24 (h + h + ). π h

(10.3)

10.2. We now consider the case where c = θσ, 0 < θ < 1, and ψ ∈ Ac . By using relations (10.1) and (10.2), we obtain ˆ ˆ δσ,c (t, β)1 ≤ |δσ,c (t, β)|dt + |δˆσ,c (t, β)|dt |t|≤1/σ

=

|t|>1/σ

ψ(c) − ψ(σ) ψ(σ)

|J1 (σ; t, c)|dt + |t|≤1/σ

6 ψ(c) − ψ(σ) . π 1−θ

Hence, estimate (7.2) implies the following assertion: Proposition 10.2. If ψ ∈ Ac and c = θσ, 0 < θ < 1, then, for any β ∈ R, δˆσ,c (t, β)1 ≤

8(ψ(c) − ψ(σ)) . π(1 − θ)

(10.4)

Finally, let ψ ∈ A∞ and c = 2σ − η(σ) > 1. In this case, δˆσ,c (t, β)1 = |δˆσ,c (t, β)|dt + |δˆσ,c (t, β)|dt |t|≤1/(η(σ)−σ)

=

ψ(c) − ψ(σ) ψ(σ)

|t|>1/(η(σ)−σ)

|J2 (σ; t, θσ)|dt +

6 (ψ(c) − ψ(σ)). π

|t|≤1/(η(σ)−σ)

Therefore, by using estimate (8.4), we arrive at the following statement: Proposition 10.3. If ψ ∈ A∞ , c∗ = 2σ − η(σ) > 1, and η(σ) = η(ψ; σ) = ψ −1 (1/2ψ(σ)), then, for any β ∈ R, δˆσ,c∗ (t, β)1 ≤

8 (ψ(c∗ ) − ψ(σ)). π

(10.5)

Section 11

11.

Basic Results

639

Basic Results

11.1. Combining assertions of Theorems 4.1 and 6.1, we arrive at the following theorem: Theorem 11.1. Let ψ ∈ F0 , β ∈ R, and h > 0. Then the following ˆψL ˆ inequality holds for any f ∈ L β p , p ∈ [1, ∞], and σ > 1 : ρ∗σ,σ−h (f ; x)pˆ ≤

4ψ(σ) (| ln(η(σ) − σ)| + K(h2 + h−1 ))Eσ−h (fβψ )pˆ, (11.1) π2

whenever σ > h. In this inequality, η(σ) = ψ −1 (ψ(σ)/2) and K is a quantity which may depend only on the function ψ(·). ˆ ψ we denote a subset of functions f (·) from L ˆ ˆψL By L β,p β p such that ˆ ψ and τ, we have Eτ (f ψ )pˆ ≤ 1. fβψ pˆ ≤ 1. It is clear that, for any f ∈ L β,p β Thus, by analyzing the upper bounds of both sides of inequality (11.1) in the set ˆ ψ , we get the following corollary of Theorem 11.1: L β,p Theorem 11.2. Let ψ ∈ F0 , β ∈ R, h > 0, and p ∈ [1, ∞]. Then, for any σ > 1 and σ > h, sup ρ∗σ,σ−h (f ; x)pˆ ≤

ˆψ f ∈L β,p

4ψ(σ) (| ln(η(σ) − σ)| + K(h2 + h−1 )), π2

(11.2)

where K is a quantity which may depend only on the function ψ(·). By comparing Theorems 4.1 and 6.1 , we arrive at the following assertion: ˆ ˆψL Theorem 11.3. Let ψ ∈ A0 and let h > 0. Then, for any f ∈ L 0 p, p ∈ [1, ∞], and any σ > 1 such that σ > h, sup ρ∗σ,σ−h (f ; x)pˆ ≤

ˆψ f ∈L 0,p

4ψ(σ) (ln σ + K(h2 + h−1 ))Eσ−h (f0ψ )pˆ, π2

(11.3)

where K is a quantity which may depend only on the function ψ(·). Moreover, sup ρ∗σ,σ−h (f ; x)pˆ ≤

ˆψ f ∈L 0,p

4ψ(σ) (ln σ + K(h2 + h−1 )). π2

(11.4)

640

Approximations in Spaces of Locally Summable Functions

Chapter 9

In the case where f ∈ Lψ β , σ = n ∈ N, and h ∈ (0, 1], in view of Proposition 3.5, we can write ∗ Fn,n−h (f ; x) = Sn−1 (f ; x).

Hence, the assertions of Theorems 11.1–11.3 can be regarded as the generalization of the corresponding results from Chapters 5 and 6 concerning the approximation of periodic functions by Fourier sums to the case of approximation of functions ˆ ψ Lp by Fourier operators F ∗ (f ; x). from the classes L σ,c β 11.2. The following statement enables us to establish the estimates of the best approximations of functions from the indicated classes by the entire functions of ∗ (f ; x). exponential type ≤ σ with the help of the operators Fσ,c Theorem 11.4. Let ψ ∈ A0 and θ ∈ (0, 1). Then the following inequality is ˆψL ˆ true for any f ∈ L 0 p , p ∈ [1, ∞], and σ ≥ 1 : ρ∗σ,θσ (f ; x)pˆ ≤

Kψ(σ) Eθσ (f0ψ )pˆ. 1−θ

(11.5)

ˆ ˆψL If ψ ∈ Ac and θ ∈ (0, 1), then, for any f ∈ L 0 p , p ∈ [1, ∞], β ∈ R, and σ ≥ 1, Kψ(σ) ρ∗σ,θσ (f ; x)pˆ ≤ (11.6) Eθσ (f0ψ )pˆ. 1−θ ˆ ˆψL Moreover, if ψ ∈ A∞ , then, for any f ∈ L 0 p , p ∈ [1, ∞], β ∈ R, and σ ≥ 1, ρ∗σ,c∗ (f ; x)pˆ ≤ Kψ(σ)Ec∗ (f0ψ )pˆ, c∗ = 2σ − η(σ). (11.7) In relations (11.5)–(11.7), K is a quantity which may depend only on the function ψ(·) and η(σ) = ψ −1 (ψ(σ)/2). The proof of this theorem is obtained by combining relations (4.10), (9.1), (7.10), and (8.1). ψ ˆ ˆψ ˆ ˆψL 11.3. By L 0 p we denote the set of functions f ∈ L0 Lp such that fβ ∈ ˆ , then F ∗ (f ; ·) ∈ W 2 for ˆψL W 2 . Thus, in view of Proposition 3.5, if f ∈ L 0

p

σ,c

σ

any σ > c > 0. Hence, Theorem 11.4 implies the following assertion: Theorem 11.5. If ψ ∈ A0 , then the following inequality holds for any θ ∈ ˆψL ˆ (0, 1), f ∈ L 0 p , p ∈ [1, ∞], and σ ≥ 1 : Eσ (f )pˆ ≤

Kψ(σ) Eθσ (f0ψ )pˆ. 1−θ

(11.8)

Section 11

Basic Results

641

ˆ ˆψL If ψ ∈ Ac , then, for any θ ∈ (0, 1), f ∈ L β p , p ∈ [1, ∞], β ∈ R, and σ ≥ 1, Kψ(σ) Eσ (f )pˆ ≤ (11.9) Eθσ (fβψ )pˆ. 1−θ ˆ ˆψL At the same time, if ψ ∈ A∞ , then for any f ∈ L β p , p ∈ [1, ∞], β ∈ R, and σ ≥ 1, Eσ (f )pˆ ≤ Kψ(σ)E2σ−η(σ) (fβψ )pˆ. (11.10) In relations (11.8)–(11.10) K is a quantity which may depend only on the function ψ(·) and η(σ) = ψ −1 (ψ(σ)/2). By setting

ψ ˆ ψ,1 = {f : f ∈ L ˆψL ˆ L 0 p , fβ pˆ ≤ 1} β,p

and Eσ (N)pˆ = sup Eσ (f )pˆ, f ∈N

(11.11)

we arrive a the following corollary of Theorem 11.5: Theorem 11.6. If ψ ∈ A0 , then, for any p ∈ [1, ∞] and σ ≥ 1, ˆ ψ,1 )pˆ ≤ Kψ(σ). Eσ (L 0,p

(11.12)

If ψ ∈ Ac ∪ A∞ , then, for any p ∈ [1, ∞], β ∈ R, and σ ≥ 1, ˆ ψ,1 )pˆ ≤ Kψ(σ). Eσ (L 0,p

(11.13)

In relations (11.12) and (11.13), K is a quantity which may depend only on the function ψ(·). 11.4. We now pass to the results obtained for the quantities ρσ,c (f ; x)pˆ. First, we introduce an intermediate assertion established by combining Theorem 4.2, Corollary 4.1, and Proposition 4.2. Proposition 11.1. Let ψ(v) be a function continuous for all v ≥ 0 and satisfying condition (1.4). Then the following inequality is true for any σ > 0, ˆ p , p ∈ [1, ∞], and β ∈ R (in the notation introduced ˆψL 0 ≤ d < c < σ, f ∈ L β

in Theorem 4.2, Corollary 4.1, and Proposition 4.2): ρσ,c (f ; x)pˆ ≤ Kσ,c Ec (fβψ )pˆˆ rcd (·, β)1 .

(11.14)

642

Approximations in Spaces of Locally Summable Functions

Chapter 9

If, in addition, p ∈ (1, ∞), then ρσ,c (f ; x)pˆ ≤ Kp (

1 rcd (·, β)1 . + 1)Ec (fβψ )pˆˆ σ−c

(11.15)

If ψ ∈ Ac and d = θc, θ ∈ (0, 1), then, according to (7.10), for any β ∈ R and c ≥ 1, we get K ˆ rcθc (t; β)1 ≤ ψ(c). (11.16) 1−θ By virtue of (9.1), this estimate remains true for ψ ∈ A0 and β = 0. Here, K is a quantity which may depend only on the function ψ(·). Since the quantity ρσ,c (f ; x)pˆ is independent of d, we can use inequality (11.16) with θ = 0, i.e., the inequality rcd (·, β)1 ≤ Kψ(c), in the estimates of this quantity (11.14) and (11.15) for ψ ∈ Ac and A0 . By virtue of Theorem 8.1, a similar estimate also holds for ψ ∈ A∞ . Hence, Proposition 11.1 implies the following statement: ˆ ˆψL Theorem 11.7. If ψ ∈ Ac ∪ A∞ and f ∈ L β p , then the following inequality holds for all p ∈ [1, ∞], β ∈ R, and σ > c > 0 : ρσ,c (f ; x)pˆ ≤ Kψ(c)Ec (fβψ )pˆ ln

σ+c , σ−c

(11.17)

where K is a quantity which may depend only on the function ψ(·). Further, if p ∈ (1, ∞), then ρσ,c (f ; x)pˆ ≤ Kp (

1 + 1)ψ(c)Ec (fβψ )pˆ, σ−c

(11.18)

where Kp is a quantity which may depend only on the function ψ(·) and the number p. If, in addition, ψ ∈ A0 , then estimates (11.17) and (11.18) are true for β = ˆψL ˆ 0, i.e., for f ∈ L 0 p. According to the definition of the set M0 , its elements ψ(·) satisfy the inequality t μ(ψ; t) = ≤ K < ∞, t ≥ 1, (11.19) η(t) − t where η(t) = ψ −1 (ψ(t)/2) and K is a quantity which may depend only on the function ψ(·). Therefore, if we set η1 (t) = η(η(t)) and ην (t) = η(ην−1 (t)),

Section 11

Basic Results

643

ν = 1, 2, . . . , then, by virtue of (11.19), for any θ ∈ (0, 1) and σ > 1 such that θσ > 1, we can write ην (θσ) ≥ (1 +

1 1 1 )ην−1 (θσ) > . . . > (1 − )ν η(θσ) > (1 + )ν+1 (θσ). K K K

Hence, for any ψ ∈ M0 , there exists a natural number ν such that ην (θσ) is greater than σ and, thus, ψ(σ) ≥ ψ(ην (θσ)). In every interval (ηi (·), ηi+1 (·)), the function ψ(·) becomes exactly half as large. Consequently, ψ(ην (θσ)) = ψ(θσ)/2ν +1 and, as a result, we obtain

ψ(θσ) ≤ 2ν +1 ψ(σ), θσ > 1.

(11.20)

By virtue of Theorem 11.7, this yields the following analog of Theorem 11.4 for the quantities ρσ,θσ (f ; x) : ˆ ˆψL Corollary 11.1. If ψ ∈ Ac , then, for any θ ∈ (0, 1), f ∈ L 0 p , p ∈ [1, ∞], σ ≥ 1, β ∈ R, and all σ such that θσ > 1, ρσ,θσ (f ; x)pˆ ≤ Kψ(σ)Eθσ (fβψ )pˆ,

(11.21)

where K is a quantity which may depend only on the function ψ(·) and the ˆ ˆψL number θ. If ψ ∈ A0 , then estimate (11.21) holds for the classes L 0 p. If ψ ∈ M+ ∞ , then relation (11.20) is violated for all ν0 . Nevertheless, in this case, one can replace the quantity ψ(c) by the quantity ψ(σ) in relations (11.17) and (11.18) due to the proper choice of the parameter c. Thus, as c, we can take the quantity c∗ = 2σ − η(σ) for σ such that c∗ ≥ 1. Hence, we arrive at the following assertion: Corollary 11.2. If ψ ∈ A∞ , then the following inequality is true for any ˆψL ˆ f ∈L 0 p , p ∈ [1, ∞], β ∈ R, and all σ such that σ ≥ η(ψ; 1) : ρσ,c∗ (f ; x)pˆ ≤ Kψ(σ)Ec∗ (fβψ )pˆ ln

σ + c∗ , σ − c∗

(11.22)

where K is a quantity which may depend only on the function ψ(·) and the number c∗ = 2σ − η(σ) ≥ 1. If, in addition, p ∈ (1, ∞), then ρσ,c∗ (f ; x)pˆ ≤ Kp (

1 + 1)ψ(σ)Ec∗ (fβψ )pˆ, σ − c∗

(11.23)

644

Approximations in Spaces of Locally Summable Functions

Chapter 9

where Kp is a quantity which may depend only on the function ψ(·) and the number p. To establish estimates (11.22) and (11.23), by virtue of (11.17) and (11.18), it suffices to show that there exists an absolute constant K such that ψ(c∗ ) ≤ Kψ(σ) for any σ > 1. To this end, we specify the points ci , i = 0, 1, . . . , df

by the equalities σ = c1 , η(c1 ) = c0 , and η(c2 ) = c1 , . . . , η(cν ) = cν−1 and show that, for any ψ ∈ M+ ∞ , there exists a number ν independent of σ and such that cν ≤ c∗ . By applying the Lagrange theorem on finite increments, we obtain ci−1 − ci η(η(ci+1 )) − η(ci+1 ) = = η (ζi ), ci − ci+1 η(ci+1 ) − ci+1

ζi ∈ (ci , ci−1 ), i = 1, 2, . . . .

Therefore, Δ1 = c1 − c2 ≤

Δ0 , η (ζ1 )

Δ2 = c2 − c3 =

Δ0 = c0 − c1 = η(σ) − σ,

Δ1 η (ζ2 )

=

η(σ) − σ η (ζ1 )η (ζ2 ) ν $

,...,

η (ζi ))−1

Δν−1 = cν−1 − cν = (η(σ) − σ)(

i=1

and, hence, for any ν > 1, σ − cν =

ν−1 i=1

j ν−1 $ Δi = (η(σ) − σ) ( η (ζi ))−1 . j=1 i=1

If ψ ∈ M+ ∞ , then, by definition, the quantity μ(ψ; t) monotonically tends to ∞. This means that, for all t ≥ 1, we have η(t) = t(1 + α(t)), where α(t) is a nonnegative differentiable monotonically vanishing function. Thus, for all t ≥ 1, 1 ψ (t) ≤ = η (t) = 1 + α(t) + tα (t) < 1 + α(t). 2 2ψ (η(t)) Consequently, for any δ ∈ (0, 1), beginning with a certain tδ , we have 1/2 ≤ η (t) ≤ 2−δ for t > tδ and, thus, there exists σδ such that 3σ−2η(σ) > tδ for σ > σδ and η (t) < 2 − δ in the interval I = (3σ − 2η(σ), η(σ)). Hence, if cν ∈ I, then ν−1 (2 − δ)−i . σ − cν = (η(σ) − σ) i=1

Section 11

Basic Results

645

As ν increases, the value of the last sum monotonically approaches the number (1 − δ)−1 > 1 . This means that there exists a number ν0 such that either σ − cν ≥ η(σ) − σ or cν ≤ 2σ − η(σ) = c∗ . In every interval [ci+1 , ci ], the values of the function ψ(·) become exactly half as large. Therefore, if σ > σδ , then ψ(c∗ ) ≤ ψ(cν ) ≤ 2ν ψ(σ), i.e., ψ(c∗ ) ≤ Kψ(σ). At the same time, if σ ∈ [1, σδ ], then the last inequality can also be satisfied by the proper choice of the constant K. The quantity σ − c∗ for ψ ∈ A∞ can be bounded from below. It is also possible that this quantity vanishes as σ → ∞. In this connection, we set M∞ = {ψ : ψ ∈ M+ ∞ , η(ψ; t) − t ≥ K > 0 ∀t ≥ 1}.

(11.24)

It is easy to see that the function ψ(t) = exp(−αtr ), t ≥ 1, belongs to M∞ for any α > 0 and r ∈ (0, 1]. Further, if ψ ∈ A∞ , then, by virtue of (11.24), σ − c∗ = η(ψ; σ) − σ ≥ K > 0. Hence, the following assertion is true: Corollary 11.3. If ψ ∈ A∞ , then the following inequality holds for any ˆψL ˆ f ∈L β p , p ∈ [1, ∞], β ∈ R, and all σ ≥ η(ψ; 1) : ρσ,c∗ (f ; x)pˆ ≤ Kψ(σ)Ec∗ (fβψ )pˆ ln σ,

(11.25)

where K is a quantity which may depend only on the function ψ(·). If, in addition, p ∈ (1, ∞), then ρσ,c∗ (f ; x)pˆ ≤ Kp ψ(σ)Ec∗ (fβψ )pˆ,

(11.26)

where Kp is a quantity which may depend only on the function ψ(·) and the number p. 2 ˆψL ˆ By virtue of Proposition 3.4, if f ∈ L β p , then Fσ,c (f ; ·) ∈ Wσ for any σ > c > 0 and, hence, Corollaries 11.1–11.3 imply the following statement:

Theorem 11.8. If ψ ∈ Ac , then the following inequality holds for any θ ∈ ˆψL ˆ (0, 1), f ∈ L 0 p , p ∈ [1, ∞], and σ ≥ 1 : Eσ (f )pˆ ≤ Kψ(σ)Eθσ (fβψ )pˆ.

(11.27)

ˆ ψ,1 )pˆ ≤ Kψ(σ). Eσ (L β,p

(11.28)

Furthermore,

646

Approximations in Spaces of Locally Summable Functions

Chapter 9

Estimates (11.26) and (11.27) also hold for ψ ∈ A0 and β = 0. ˆ ˆψL In addition, if ψ ∈ A∞ , then, for any f ∈ L β p , p ∈ (1, ∞), β ∈ R, and all σ ≥ η(ψ; 1), Eσ (f )pˆ ≤ Kp ψ(σ)Ec∗ (fβψ )pˆ,

c∗ = 2σ − η(σ).

(11.29)

Moreover, ˆ ψ,1 )pˆ ≤ Kp ψ(σ). Eσ (L β,p

(11.30)

In relations (11.27)–(11.30), K and Kp are quantities which may depend only on the function ψ(·) and the number p. By comparing Theorems 11.6 and 11.8, we conclude that relations (11.27)– (11.30) and, hence, estimates (11.17), (11.18), (11.21), (11.24), and (11.26) of the quantities ρσ,c (f ; x)pˆ are exact in order as σ → ∞. At the same time, inequalities (11.22) and (11.25) can be strengthened. Indeed, by combining relaˆ p , p ∈ [1, ∞], ψ ∈ A∞ , ˆψL tions (3.21), (4.2), (10.4), and (11.7), for any f ∈ L β

and β ∈ R, we get

ρσ,c∗ (f ; x)pˆ ≤ ρ∗σ,c∗ (f ; x)pˆ + Δσ,c∗ (f ; x)pˆ ≤ Kψ(σ)Ec∗ (fβψ )pˆ +

8 (ψ(c∗ ) − ψ(σ))Ec∗ (fβψ )pˆ. π

(11.31)

Thus, by using the fact of existence of an absolute constant K such that ψ(c∗ ) ≤ Kψ(σ) for any ψ ∈ A∞ established in the proof of Corollary 11.2, we arrive at the following assertion: ˆ ˆψL Theorem 11.8. If ψ ∈ A∞ , then, for any f ∈ L 0 p , p ∈ [1, ∞], β ∈ R, and σ ≥ 1 : ρσ,c∗ (f ; x)pˆ ≤ Kψ(σ)Ec∗ (fβψ )pˆ

(11.32)

ˆ ψ,1 )pˆ ≤ Kψ(σ), Eσ (L β,p

(11.33)

and, in addition, where K is a quantity which may depend only on the function ψ(·). It is clear that this theorem strengthens estimates (11.22), (11.25) and generalizes estimates (11.29) and (11.30) to all p ≥ 1. It is also clear that inequalities (11.32) and (11.33) are exact in order as σ → ∞.

Section 11

Basic Results

647

We now present a corollary of Theorem 11.7 for c = σ − h with fixed h > 0 such that σ − h > 1. ˆ ˆψL Corollary 11.4. If ψ ∈ Ac ∪ A∞ , then, for any f ∈ L β p , p ∈ [1, ∞], and all σ and h > 0 such that σ − h > 1, ρσ,σ−h (f ; x)pˆ ≤ Kψ(σ)Eσ−h (fβψ )pˆ ln σ,

(11.34)

where K is a quantity which may depend only on the function ψ(·). If, in addition, p ∈ (1, ∞), then ρσ,σ−h (f ; x)pˆ ≤ Kp (1 +

1 )ψ(σ)Eσ−h (fβψ )pˆ, h

(11.35)

where Kp is a quantity which may depend only on the function ψ(·) and the number p. Further, if ψ ∈ A0 , then estimates (11.34) and (11.35) are true for β = 0 ˆ ˆψL and f ∈ L β p. Estimates (11.34) and (11.35) are obtained from inequalities (11.17) and (11.18) by using the inequality ψ(σ − h) ≤ Kψ(σ).

(11.36)

For ψ ∈ A0 ⊃ Ac , this inequality is deduced from (11.19) as follows: In view of relation (11.19), we have 1 h 1 η(σ − h) ≥ (σ − h)(1 + ) = σ(1 − )(1 + ). k σ k Therefore, there exists σ0 such that η(σ − h) ≥ σ for σ > σ0 . On the interval [σ − h, η(σ − h)], the function ψ(·) becomes half as large. Hence, ψ(σ − h) = 1/2ψ(η(σ − h)) ≤ 1/2ψ(σ). This proves (11.36) for σ ≥ σ0 . Clearly, inequality (11.36) also holds for σ < σ0 . If ψ ∈ A∞ , then, by virtue of (11.24), one can find K > 0, such that η(t) − t > K for any t ≥ 1. Thus, by setting (as in the proof of relation (11.20)) η1 (t) = η(η(t)) and ην (t) = η(ην−1 (t)), ν = 1, 2, . . . , we conclude that ην (σ − h) > σ for ν > h/k. Consequently, there exists ν such that

648

Approximations in Spaces of Locally Summable Functions

Chapter 9

ην (c − h) > σ. This yields inequality (11.36) because, as indicated above, the function ψ(·) becomes exactly half as large in the intervals (ηi (·), ηi+1 (·)). In view of Proposition 10.1, inequality (11.36) implies that δˆσ,σ−h (t; β)pˆ ≤ K(h2 +

1 )ψ(σ) h

(11.37)

for any ψ ∈ Ac ∪ A∞ . Hence, combining relations (3.21), (4.2), (11.35), and (11.37), for any f ∈ ψ ˆ ˆ Lβ Lp , p ∈ (1, ∞), we get ρ∗σ,σ−h (f ; x)pˆ ≤ ρσ,σ−h (f ; x)pˆ + Δσ,σ−h (t; β)pˆ ≤ Kp (h2 +

1 )ψ(σ)Eσ−h (fβψ )pˆ. h

This enables us to conclude that, for p ∈ (1, ∞), the assertions of Theorems 11.1, 11.2, and 11.3 can be made more precise. ˆ ˆψL Theorem 11.9. If ψ ∈ Ac ∪ A∞ , then, for any f ∈ L β p , p ∈ [1, ∞], and all σ and h > 0 such that σ − h > 1, ρ∗σ,σ−h (f ; x)pˆ ≤ Kp (h2 +

1 )ψ(σ)Eσ−h (fβψ )pˆ, h

(11.38)

and, in addition, sup ρ∗σ,σ−h (f ; x)pˆ ≤ Kp (h2 +

ˆψ f ∈L β,p

1 )ψ(σ). h

(11.39)

In relations (11.38) and (11.39), Kp is a quantity which may depend only on the function ψ(·) and the number p. If ψ ∈ A0 , then estimates (11.38) and (11.39) are true for β = 0 and ˆ ˆψL f ∈L β p.

12.

Upper Bounds of the Deviations ρσ (f ; ·) ˆ ψ Hω ˆ ψ and C in the Classes C β,∞ β

∗ (f ; x) defined ˆ ψ . The operators Fσ,c 12.1. Assume that ψ ∈ A and f ∈ L β by relation (3.20) for c = σ − h, where h is a positive number (clearly, such that c ≥ 0), are denoted by Fσ (f ; x). Therefore,

Section 12

ψ Upper Bounds of ρσ (f ; ·) in Classes Cˆβ,∞ and Cˆβψ Hω

∞

+ )(t, β)dt, fβψ (x + t)(ψλ σ

Fσ (f ; x) = A0 +

649

(12.1)

−∞

where

⎧ 1, 0 ≤ v ≤ c, ⎪ ⎪ ⎪ ⎪ ⎨ v − c ψ(σ) , c ≤ v ≤ σ, λσ (v) = 1 − ⎪ σ − c ψ(v) ⎪ ⎪ ⎪ ⎩ 0, v ≥ σ, c = σ − h

(12.2)

and, according to relations (4.4)–(4.6), we find ∞ ρσ (f ; x) =

fβψ (x + t)ˆ rσ (t)dt,

(12.3)

−∞

where 1 rˆσ (t) = rˆσ (t, β) = π

∞

βπ )dv, 2

(12.4)

⎧ 0, 0 ≤ v ≤ c, ⎪ ⎪ ⎨ rσ (t) = (v − c)(σ − c)−1 ψ(σ), c ≤ v ≤ σ, ⎪ ⎪ ⎩ ψ(v), v ≥ σ, c = σ − h.

(12.5)

rσ (v) cos(vt + 0

In this section, we consider the quantities ρσ (f ; x) for the sets Cˆβψ N in the case where N is the unit ball S∞ in the space M, i.e., df

S∞ = SM = {ϕ : ess sup |ϕ(t)| ≤ 1}, ψ and also in the case where and we set Cˆβψ N = Cˆβ,∞

N = Hω = {ϕ : |ϕ(t) − ϕ(t )| ≤ ω(|t − t |), t, t ∈ R1 } and ω(t) is a fixed modulus of continuity (here, we write Cˆβψ N = Cˆβ Hω ). 12.2. First, we prove the following assertion: Lemma 12.1. Let ψ ∈ A and let a = a(σ) be an arbitrary function continψ uous for all σ > 0 and such that σa(σ) ≥ a0 > 0 for any σ > 0. If f ∈ Cˆβ,∞ , then the following equality is true at any point x for all σ ≥ h :

650

Approximations in Spaces of Locally Summable Functions

ρσ (f ; x) = νa

ψ(σ) π

fβψ (x + t)

Chapter 9

sin(σt + βπ/2) dt + bψ σ (a; f ; x), (12.6) t

ma ≤|t|≤Ma

where ma = min(a(σ), h−1 ), Ma = max(aσ , h−1 ), νa = sgn (a(σ) − h−1 ) , and (12.7) |bψ σ (a; f )| ≤ K1 (ψ(σ) + Qσ (a, ψ)). If, in addition, f ∈ Cˆβψ Hω , then, for any x and σ ≥ h, ρσ (f ; x)

ψ(σ) = −νa π

ϕ(x; t)

sin(σt + βπ/2) dt + dψ σ (a; f ; x), (12.8) t

ma ≤|t|≤Ma

where ϕ(x; t) = fβψ (x) − fβψ (x + t)

(12.9)

|dψ σ (a; f ; x)| ≤ K2 (ψ(σ) + Qσ (a; ψ))ω(1/σ).

(12.10)

and

The quantity Qσ (a; ψ) is given by the formula Qσ (a; ψ) =

ψ(t + σ) dt + t

1/a(σ)

∞

t−1 (ψ(σ) − ψ(σ + 1/t))dt

(12.11)

a(σ)

and K1 and K2 are quantities which may depend only on the function ψ(·) and the number h. Proof. In the periodic case, this lemma is in fact proved in Chapter 5. The scheme of the proof proposed there is, generally speaking, applicable to the proof of the analyzed lemma. The main difference is connected with the fact that, in the periodic case, the principal values of integrals of the form |t|≥a>0

fβψ (x

+ t)

sin t cos t

dt t

ψ Upper Bounds of ρσ (f ; ·) in Classes Cˆβ,∞ and Cˆβψ Hω

Section 12

651

are finite. In the general case, this is clearly not true and, hence, the required estimates should be obtained in a different way. According to relation (4.7), we have ∞ rˆσ (t)dt = 0. (12.12) −∞

Thus, by virtue of (12.3), ∞ ρσ (f ; x) = −

Δ(x; t)ˆ rσ (t)dt.

(12.13)

−∞

To use this formula, we now set Bσψ (a; f ; x)

ψ(σ) = −νa π

ϕ(x; t)

sin(σt + βπ/2) dt, t

(12.14)

ma ≤|t|≤Ma

Pσψ (a; f ; x)

=−

ϕ(x; t)J2 (σ; t)dt,

|t|≤a(σ)

Rσψ (a; f ; x)

1 = π

ϕ(x; t)J3 (σ; t)dt,

(12.15)

|t|≥a(σ) h−1

γσ (f ; x) = −

ϕ(x; t)J1h (σ; t)dt,

J1h (σ; t) = J1 (σ; t; σ−h), (12.16)

−h−1

and δσ (f ; x) =

2ψ(σ) πh

ϕ(x; t) |t|≥h−1

sin((2σ − h)t/2 + βπ/2) sin(ht/2) dt. (12.17) ht2

Here, the quantities J1 (σ; t; c), J2 (σ; t), and J3 (σ; t) are given by relations (6.1)–(6.4). In this notation, equality (12.13) takes the form ρσ (f ; x) = Bσψ (a; f ; x) + Pσψ (a; f ; x) + Rσψ (a; f ; x) + γσ (f ; x) + δσ (f ; x).

(12.18)

652

Approximations in Spaces of Locally Summable Functions

Chapter 9

This equality holds for any f ∈ Cˆβψ M, ψ ∈ A , β ∈ R, σ ≥ h, and x ∈ R. The upper estimates of the last four terms of this equality are established in ψ the classes Cˆβ,∞ and Cˆβψ Hω . To this end, we first use equality (6.11), a similar representation of the quantity J3 (σ; t), i.e., βπ βπ J3 (σ; t)0 + sin J3 (σ; t)1 , 2 2

J3 (σ; t) = cos

(12.19)

estimates (5.11.29), (5.11.41), and their corresponding analogs for the integrals containing the functions J2 (σ; t)1 and J3 (σ; t)1 . In view of the fact that these inequalities have been deduced actually without using the periodicity of the function f (·) and the fact that σ ∈ N, they must be true, respectively, for the funcψ and f ∈ Cˆβψ Hω for any σ ≥ h. As a result, we arrive at the tions f ∈ Cˆβ,∞ formulas ψ(t + σ) ψ ψ |Pσ (a; f ; x)| ≤ O(1)(ψ(σ) + , (12.20) dt) ∀f ∈ Cˆβ,∞ t 1/a(σ)

|Pσψ (a; f ; x)|

ψ(τ + σ) dt)ω(1/σ) ∀f ∈ Cˆβψ Hω , (12.21) t

≤ O(1)(ψ(σ) + 1/a(σ)

and |Rσψ (a; f ; x)| ∞ ≤ O(1)(ψ(σ) +

ψ t−1 (ψ(σ) − ψ(σ − 1/t))dt) ∀f ∈ Cˆβ,∞ , (12.22)

a(σ)

∞ |Rσψ (a; f ; x)|

≤ O(1)(ψ(σ) +

t−1 (ψ(σ) − ψ(σ − 1/t))dt)ω(1/σ) (12.23)

a(σ)

∀f ∈ Cˆβψ Hω . Therefore, ψ |Pσψ (a; f ; x)| + |Rβψ (a; f ; x)| ≤ O(1)(ψ(σ) + Qσ (a; ψ)) ∀f ∈ Cˆβ,∞ (12.24)

ψ Upper Bounds of ρσ (f ; ·) in Classes Cˆβ,∞ and Cˆβψ Hω

Section 12

653

and |Pσψ (a; f ; x)| + |Rβψ (a; f ; x)| ≤ O(1)(ψ(σ) + Qσ (a; ψ))ω(1/σ) ∀f ∈ Cˆβψ Hω . (12.25) 12.3. We now deduce the required estimates of the quantities γσ (f ; x) and δσ (f ; x). In view of equality (6.3), we find |γσ (f ; x)| ≤

ψ(σ) (| π

h−1

ϕ(x; t)

−h−1

ht − sin ht sin(σt + βπ/2)dt| ht2

h−1

+|

ϕ(x; t)

−h−1

1 − cos ht cos(σt + βπ/2)dt| ht2

ψ(σ) (|J4 (f ; σ)| + |J5 (f ; σ)|). π First, we estimate the integral df

=

(1) J4 (f ; σ)

(12.26)

h−1 = ϕ(x; t)g(t) sin(σt + βπ/2)dt, 0

g(t) = (ht − sin ht)/ht2 .

(12.27)

For any fixed x, the function ϕ(x; t) belongs to the class Aω and, hence, to Hω (0, 1). Consequently, Lemma 5.1.3 can be applied to the integral in (12.27). To do this, we set x G(x) = g(t) sin(σt + βπ/2)dt. (12.28) 0

Under the assumption that the value of σ is sufficiently large, by t1 , . . . , tN we denote all zeros of the function sin(σt + βπ/2) lying in the interval [0, 1] and numbered in the order of increasing. For t ∈ (0, h−1 ), the function g(t) is nondecreasing. Hence, in each interval [0, t1 ], [t1 , t2 ], . . . , [tN −1 , tN ], the function G(x) turns into zero exactly once at a certain point xk , k = 0, 1, . . . , N, and, moreover, x0 = 0. It is also clear that xk+1

g(t) sin(σt + βπ/2)dt = 0, xk

k = 0, 1, . . . , N − 2.

654

Approximations in Spaces of Locally Summable Functions

Chapter 9

By using Lemma 5.1.3, we get (1)

|J4 (f ; σ)| h−1

xN −1

≤ ω(Δ)

|g(t)|dt + 0

max

xN −1 ≤t≤h−1

|ϕ(x; t)|

|g(t)|dt. (12.29)

xN −1

In the analyzed case, we have Δ = max (xk+1 − xk ) ≤ max(tk+2 − tk ) ≤ 2π/σ, k≤N −1

k

h−1 − xN −1 ≤ π/σ,

and ϕ(x; 0) = 0. Therefore, in view of the inequality ω(λt) ≤ (λ + 1)ω(t) , for any λ > 0, we obtain (1) |J4 (f ; σ)|

h−1 ≤ Kω(1/σ) g(t)dt + ω(h−1 )g(h−1 )π/σ 0

= O(1)ω(1/σ).

(12.30)

Clearly, a similar estimate also holds for the integral (2) J4 (f ; σ)

0 =

ϕ(x; t)g(t) sin(σt + βπ/2)dt. −h−1

Hence, for the entire integral J4 (f ; σ), we obtain |J4 (f ; σ)| ≤ O(1)ω(1/σ).

(12.31)

The function g1 (t) = (1 − cos ht)/ht2 is nonincreasing in the interval (0, h−1 ) and, thus, the function h−1 g1 (t) cos(σt + βπ/2)dt G1 (x) = x

has exactly one zero xk in each interval [t1 , t2 ], . . . , [tN −1 , tN ], where ti , i = 1, N , are the zeros of the function cos(σt + βπ/2) from the interval [0, h−1 ]

Section 12

ψ Upper Bounds of ρσ (f ; ·) in Classes Cˆβ,∞ and Cˆβψ Hω

655

and, moreover xN = h−1 . This enables us to apply Lemma 5.1.3 to the integral J5 (f ; σ). As a result, we conclude that |J5 (f ; σ)| ≤ O(1)ω(1/σ).

(12.32)

Combining relations (12.26), (12.31), and (12.32), we find |γσ (f ; x)| ≤ O(1)ψ(σ)ω(1/σ),

f ∈ Cˆβψ Hω .

(12.33)

ψ If f ∈ Cˆβ,∞ , then

|ϕ(x, t)| = |fβψ (x) − fβψ (x + t)| ≤ 2. The functions g(t) and g1 (t) are also bounded on the interval [−h−1 , h−1 ]. Therefore, by virtue of (12.26), |γσ (f ; x)| ≤ O(1)ψ(σ),

ψ f ∈ Cˆβ,∞ .

(12.34)

In relations (12.33) and (12.34), O(1) are quantities which may depend only on ψ(·) and the number h. To obtain the required estimate of the quantity δσ (t; x), we first represent it in the form cos((σ − h)t + βπ ψ(σ) 2 ) δσ (f ; x) = ϕ(x; t) dt ( 2 π t |t|≥h−1

−

ϕ(x; t)

|t|≥h−1 df

=

cos(σt + t2

βπ 2 )

dt)

ψ(σ) (J6 (f ; σ − h) − J6 (f ; σ)) π

and obtain an estimate of the quantity (1) J6 (f ; σ)

h−1 cos(σt + βπ/2) = ϕ(x; t) dt. t2 0

Let

∞ C(x) = x

cos(σt + βπ/2) dt. t2

(12.35)

656

Approximations in Spaces of Locally Summable Functions

Chapter 9

Due to the monotonicity of the function t−2 there exists exactly one zero of the function C(x) between any two positive zeros of the function cos(σt + βπ/2) . Assume that x1 , x2 , . . . is the set of all zeros of this sort from x ≥ h−1 numbered in the order of increasing. Since xk+1

cos(σt + βπ/2) dt = 0, t2

xk

we can use these points as points appearing in Lemma 5.1.3 and apply this lemma. As a result, we get (1) |J6 (f ; σ)|

x1 ≤

max

h−1 ≤t≤x1

|ϕ(x, t)| h−1

dt + ω(Δ) t

∞

h−1

dt , t2

whence it follows that the inequality (1)

|J6 (f ; σ)| ≤ Kω(1/σ) holds for any function f ∈ Cˆβψ Hω . It is clear that a similar estimate is true for the integral J6 (f ; σ) taken over the interval (−∞, −h−1 ), and, hence, for the integral J6 (f ; σ). Therefore, by virtue of equality (12.35), for any σ ≥ h−1 , we obtain |δσ (f ; x)| ≤ O(1)ψ(σ)ω(1/σ), f ∈ Cˆβψ Hω . (12.36) (2)

ψ It is clear that if f ∈ Cˆβ,∞ , then

|δσ (f ; x)| ≤ O(1)ψ(σ).

(12.37)

Comparing relations (12.18), (12.24), (12.25), and (12.33)–(12.37), we complete the proof of the lemma. 12.5. We now use Lemma 12.1 to establish asymptotic (as σ → ∞ ) relations for the quantities ψ ψ Eσ (Cˆβ,∞ ) = sup{|ρσ (f ; x)| : f ∈ Cˆβ,∞ }

(12.38)

Eσ (Cˆβψ Hω ) = sup{|ρσ (f ; x)| : f ∈ Cˆβψ Hω }.

(12.38 )

and

ψ Upper Bounds of ρσ (f ; ·) in Classes Cˆβ,∞ and Cˆβψ Hω

Section 12

657

Theorem 12.1. Let ψ ∈ A and let a = a(σ) be an arbitrary function continuous for all σ > 0 and such that σa(σ) ≥ a0 > 0 for any σ > 0. Then the quantities Eσ (Cˆ ψ ) and Eσ (Cˆ ψ Hω ) are independent of the point x and β,∞

β

the following equalities hold as σ → ∞ :

4 ψ Eσ (Cˆβ,∞ ) = 2 ψ(σ)| ln a(σ)| + bψ σ (a), π

(12.39)

where bψ σ (a) = O(1)(ψ(σ) + Qσ (a; ψ)), 2 Eσ (Cˆβψ Hω ) = 2 ψ(σ)eσ (ω)| ln a(σ)| + dψ σ (a), π

(12.40) (12.39 )

and dψ σ (a; ψ) = O(1)(ψ(σ) + Qσ (a; ψ))ω(1/σ). The quantities O(1) are uniformly bounded in σ and β, the quantity Qσ (a; ψ) has the same meaning as in Lemma 12.1, and π/2 ω(2t/σ) sin tdt, eσ (ω) = θω

2/3 ≤ θω ≤ 1,

0

where θω = 1 if ω = ω(t) is a convex modulus of continuity. Proof. The classes Cˆβψ N, where N is SM or Hω , are invariant under shifts of the argument: If f ∈ Cˆ ψ N, then, for any τ, the function f1 (x) = f (x + τ ) β

also belongs to Cˆβψ N. Therefore, if f ∈ Cˆβψ N, then wee can find a function f1 ∈ Cˆβψ N such that ρσ (f ; x) = ρσ (f ; 0). Consequently, Eσ (Cˆβψ N) = sup{|ρσ (f ; 0)| : f ∈ Cˆβψ N}. This immediately implies that the quantities Eσ (Cˆβψ N) are indeed independent of x. Further, if f ∈ Cˆβψ N, then, by definition fβψ ∈ N. On the other hand, for any y ∈ N, one can find a function f (·) in the class Cˆ ψ N such that β

fβψ (·) = y(·) almost everywhere. Thus, by using equalities (12.6)–(12.10), we obtain ψ Eσ (Cˆβ,∞ )

ψ(σ) = sup | ϕ∞ ≤1 π

ma ≤|t|≤Ma

ϕ(t)

sin(σt + βπ/2) dt| + r1 (a), (12.41) t

658

Approximations in Spaces of Locally Summable Functions

Chapter 9

where |r1 (a)| = O(1)(ψ(σ) + Qσ (a; ψ)), Eσ (Cˆβψ Hω ) = sup ϕ∈Hω

ψ(σ) π

(ϕ(t) − ϕ(0))

(12.42)

sin σt + βπ/2 dt + r2 (a, ω), (12.41 ) t

ma ≤|t|≤Ma

and |r2 (a; ω)| ≤ O(1)(ψ(σ) + Qσ (a, ψ))ω(1/σ).

(12.42 )

Comparing relations (12.39)–(12.42 ), we conclude that, in order to prove the theorem, it suffices to establish the equalities sin(σt + βπ/2) 4 sup | ϕ(t) dt| = | ln a(σ)| + O(1) (12.43) t π ϕ∞ ≤1 ma ≤|t|≤Ma

and sup |

ϕ∈Hω

(ϕ(t) − ϕ(0))

sin(σt + βπ/2) dt| t

ma ≤|t|≤Ma

=

2 eσ (ω)| ln a(σ)| + O(1)ω(1/n). (12.44) π

To prove these equalities, we first follow the proof of Lemma 5.4.7. Namely, we set xk = (kπ − βπ/2)/σ, tk = xk − π/2σ, k = 0, ±1, . . . , σ ∈ R1 .

(12.45)

Let k0 be a value of k for which tk0 is the point from the collection of points such that sin(σt + βπ/2) = 1 closest to the point (ma + π)/σ and let k1 be the maximum value of k such that tk < Ma . Further, let k2 be such that tk2 is the point from the collection of points such that sin(σt + βπ/2) = 1 closest to the point −(ma + π)/σ from the left , let k3 be the lowest value of k satisfying the condition tk > −Ma , and, finally, let lσ (t) = xk , t ∈ [tk , tk+1 ], k = k0 , . . . , k1 − 1,

k = k3 , k3 + 1, . . . , k2 − 1.

(12.46)

Section 12

ψ Upper Bounds of ρσ (f ; ·) in Classes Cˆβ,∞ and Cˆβψ Hω

659

In this notation, the following assertion is true: Lemma 12.2. If ψ ∈ SM , then ϕ(t)

sin(σt + βπ/2) dt t

ma ≤|t|≤Ma

ϕ(t)

=

sin(σt + βπ/2) dt + O(1). (12.47) lσ (t)

i3,1

At the same time, if ϕ ∈ Hω , then (ϕ(t) − ϕ(0)) ma ≤|t|≤Ma

sin(σt + βπ/2) dt t

(ϕ(t) − ϕ(0))

=

sin(σt + βπ/2) dt + O(1)ω(1/σ). (12.48) lσ (t)

i3,1

In equalities (12.47) and (12.48), i3,1 = [t3 , t2 ] ∪ [t0 , t1 ]. Proof. The proof of this lemma repeats the proof of the corresponding assertions in Subsection 5.4.10. 12.5. If ϕ ∈ SM , then I3,1 (ϕ, β) = |

ϕ(t)

sin(σt + βπ/2) dt| lσ (t)

i3,1

|

≤

sin(σt + βπ/2) df |dt = I3,1 (β). lσ (t)

(12.49)

i3,1

On the other hand, if ϕ∗ (t) is a function from SM which coincides with the function sgn (sin(σt + βπ/2)/lσ (t)) on the set i3,1 , then I3,1 (ϕ∗ ) = I3,1 . This means that sup I3,1 (ϕ; β) = I3,1 (β). (12.49 ) ϕ∞ ≤1

660

Approximations in Spaces of Locally Summable Functions

Chapter 9

Without loss of generality, we can assume that β ∈ [0, 2). As a result, by using relations (12.46) and (5.5.10), we find (β) I3,1

=

k 2 −1 k=k3

= 2(

tk+1

1 |xk |

| sin(σt + βπ/2)|dt +

k=k0

tk

|k3 |

k=|k2 |+1

k 1 −1

1 xk

tk+1

| sin(σt + βπ/2)|dt tk

k 1 −1 1 1 + ), β ∈ [0, 2). kπ − βπ/2 kπ + βπ/2

(12.50)

k=k0

Hence, in view of the definition of the numbers ki , i = 0, 3, we get k 1 −1

I3,1 (β) = 4

k=k0

4 = π

k1

k1 −1 1 1 4 + O(1) = + O(1) kπ π k k=k0

dt 4 Ma + O(1). + O(1) = ln t π ma

(12.51)

k0

Further, if a(σ) > h−1 , then ma = h−1 , Ma = aσ and, in view of relation (12.51), 4 I3,1 (β) = ln a(σ) + O(1). (12.52) π At the same time, if a(σ) < h−1 , then ma = a(σ), Ma = h−1 and, thus, I3,1 (β) =

4 h−1 4 ln = | ln a(σ)| + O(1). π a(σ) π

(12.53)

This means that we always have I3,1 (β) =

4 | ln a(σ)| + O(1), π

(12.54)

where O(1) is a quantity uniformly bounded in σ. Combining relations (12.47), (12.49), and (12.54), we arrive at equality (12.43). 12.6. We now prove (12.44). By virtue of (12.46), for any ϕ ∈ Hω and β ∈ [0, 2), we find

Section 12

ψ Upper Bounds of ρσ (f ; ·) in Classes Cˆβ,∞ and Cˆβψ Hω

I3,1 (ϕ, β) = |

(ϕ(t) − ϕ0 )

661

sin(σt + βπ/2) dt| ln(t)

i3,1

≤

k 2 −1 k=k3

where

k 1 −1 1 1 ek (ω), ek (ω) + |xk | xk

(12.55)

k=k0

tk+1

ek (ω) = sup | f ∈Hω

ϕ(t) sin(σt + βπ/2)dt|

(12.56)

tk

and, hence (see (5.5.17)), π/2σ

1 ω(2t) sin σtdt = σ

ek (ω) ≤ 0

π/2 2t 1 ω( ) sin tdt = eσ (ω). σ σ

(12.57)

0

Thus, according to (12.55)–(12.57) and (12.51), I3,1 (ϕ, β) ≤ eσ (ω)(

|k3 |

k=|k2 |+1

k 1 −1 1 1 + ). kπ − βπ/2 kπ + βπ/2

(12.58)

k=k0

Further, let (see Subsection 5.5.5) ⎧ 1 ⎪ ⎪ t ∈ [tk , xk ], ⎪ ⎨ 2 ω(2xk − t), ϕk (t) = ⎪ ⎪ 1 ⎪ ⎩− ω(2(tk − xk )), t ∈ [xk , tk+1 ], k = k0 , k1 − 1, k = k3 , k2 − 1. 2 and ϕ∗ (t) =

⎧ ⎨(−1)k ϕk (t), t ∈ I = [xk3 , xk2 −1 ] ∪ [xk0 , xk2 −2 ], ⎩ 0,

(12.59) t ∈ R1 \ I.

By direct calculations, we can show that the quantity I3,1 (ϕ∗ , β) coincides with the right-hand side of inequality (12.58) to within the quantities O(1)ω(1/σ) . If ω(t) is a convex modulus of continuity, then ϕ∗ ∈ Hω . Thus, in view of relations (12.50) and (12.54), we can write

662

Approximations in Spaces of Locally Summable Functions

sup I3,1 (ϕ, β) = eσ (ω)(

ϕ∈Hω

|k3 |

k=|k2 |+1

=

Chapter 9

k 1 −1 1 1 + ) kπ − βπ/2 kπ + βπ/2 k=k0

2 eσ (ω)| ln a(σ)| + O(1)ω(1/σ). π

(12.60)

If ω(t) is not necessarily a convex modulus of continuity, then we can conclude that the function 2ϕ∗ (t)/3 belongs to Hω (see Subsection 5.1.2). Hence, for arbitrary moduli of continuity, relation (12.60) remains true if its right-hand side is multiplied by a certain factor θ ∈ [2/3, 1], i.e., we always have sup I3,1 (ϕ, β) =

ϕ∈Hω

2θω eσ (ω)| ln a(σ)| + O(1)ω(1/σ). π

(12.61)

Combining relations (12.55) and (12.61), we arrive at equality (12.44). This completes the proof of Theorem 12.1. 12.7. If ψ ∈ F0 and a∗ (σ) = μ(ψ; σ)/σ, then relations (6.23) and (6.24) are true. Therefore, by setting a(σ) = a∗ (σ) in Theorem 12.1, we arrive at the following assertion: Theorem 12.2. Let ψ ∈ F0 . Then, for any β ∈ R, 4 ψ ) = 2 ψ(σ)| ln(η(σ) − σ)| + O(1)ψ(σ), Eσ (Cˆβ,∞ π

(12.62)

2 ψ Eσ (Cˆβ,∞ ) = 2 ψ(σ)eσ (ω)| ln(η(σ) − σ)| + O(1)ψ(σ)ω(1/σ), π

(12.63)

and

where O(1) are the quantities uniformly bounded in σ and β, η(σ) = η(ψ; σ) = ψ −1 (ψ(σ)/2), and eσ (ω) is the same quantity as in Theorem 12.1. By comparing the assertions of Theorems 6.1 and 12.2, we see that equality (6.25) is exact in order and, moreover, asymptotically exact. ψ Assume that Cβ,∞ and Cβψ Hω are subsets of 2π-periodic functions from the classes Cˆ ψ and Cˆ ψ Hω , respectively. For the classes C ψ and C ψ Hω β,∞

β

β,∞

β

with σ = n ∈ N, Theorem 5.10.1 is an analog of Theorem 12.2. Comparing equalities (5.10.44) and (5.10.45) with equalities (12.62) and (12.63), we see that

Section 12

ψ Upper Bounds of ρσ (f ; ·) in Classes Cˆβ,∞ and Cˆβψ Hω

663

the latter contain the quantity | ln(η(n) − n)| instead of ln+ (η(n) − n). If there exists a constant K such that 0 < η(ψ; σ) − σ ≤ K,

(12.64)

then the quantity ln+ (η(σ) − σ) is also bounded and, thus, the approximations ψ by Fourier sums in the classes Cβ,∞ and Cβψ Hω have the orders ψ(σ) and ψ(σ)ω(1/n), respectively, i.e., (see Section 7.4) coincide (in order) with the best polynomial approximations in these classes. At the same time, as follows from equalities (12.62) and (12.63), in the case where η(ψ; σ) − σ is not bounded from below by a positive number, condition (12.64) does not guarantee that the ψ quantities Eσ (Cˆβ,∞ ) and Eσ (Cˆβψ Hω ) vanish with the same rates as the quantities ψ(σ) and ψ(σ)ω(1/σ), respectively. In particular, this is true in the case where lim (η(σ) − σ) = 0.

σ→0

Note that this condition is satisfied, e.g., for the function ψ(t) = exp(−αtr ), t ≥ 1, α > 0 generating the classes of entire functions for all r > 1 . 12.8. We now prove the analogs of Lemma 12.1 and Theorem 12.2 in the case where ψ ∈ A0 and β = 0. ψ . Then the equality Lemma 12.3. Assume that ψ ∈ A0 and f ∈ Cˆ0,∞

ψ(σ) ρσ (f ; x) = π

h−1

f0ψ (x + t)

−h−1

sin σt dt + O(1)ψ(σ) t

(12.65)

holds at all points x ∈ R1 , . At the same time, if f ∈ Cˆ0ψ Hω , then, at any point x, ψ(σ) ρσ (f ; x) = − π

h ϕ(x; t) −h−1

sin σt dt + O(1)ψ(σ)ω(1/σ), t

where ϕ(x, t) = f0ψ (x) − f0ψ (x + t) and O(1) are quantities uniformly bounded in σ.

(12.66)

664

Approximations in Spaces of Locally Summable Functions

Chapter 9

Proof. In view of relations (6.2) and (6.3), we have J1 (σ; t; σ − h)0 = J1 (σ; t; σ − h)β=0 =

ψ(σ) sin σt 2 sin(2σ − h)t/2 sin ht/2 ) ( − π t ht2

(12.68)

and J2 (σ; t)0 = J2 (σ; t)β=0 = −

ψ(σ) sin σt 1 − J3 (σ; t)0 , π t π

where J3 (σ; t)0 = J3 (σ; t)β=0

1 = πt

∞

ψ (v) sin vtdv.

(12.69)

(12.70)

σ

Thus, by virtue of relation (12.13), the quantities ρσ (f ; x), admit a representation ∞ ρσ (f ; x) = −

ϕ(x; t)(J1 (σ; t; σ − h)0 + J2 (σ; t)0 )dt

−∞

2ψ(σ) = πh

∞ ϕ(x, t) −∞

1 + π

sin(2σ − h)t/2 sin ht/2dt t2

∞ ϕ(x; t)J3 (σ; t)0 dt,

(12.71)

−∞

where ϕ(x; t) is either f ψ (x) − f0ψ (x + t) or −f0ψ (x + t). Further, we set σ0 = σ − h/2, ∞ 1 ψ Rσ (f ; x) = ϕ(x; t)J3 (σ; t)0 dt, π −∞

2ψ(σ) δσ (f ; x)0 = πh J7 (f ; x) =

ψ(σ) πh

ϕ(x; t) |t|≥h−1

ϕ(x; t) |t|≤h−1

sin σ0 t sin ht/2dt; t2

2 sin ht/2 − ht sin σ0 tdt. t2

Section 12

ψ Upper Bounds of ρσ (f ; ·) in Classes Cˆβ,∞ and Cˆβψ Hω

665

Hence, relation (12.71) can be rewritten in the form ψ(σ) ρσ (f ; x) = π

h−1

ϕ(x; t) −h−1

sin σ0 t dt t + Rσψ (f ; x) + δσ (f ; x)0 + J7 (f ; x). (12.72)

If f (·) is a 2π-periodic function and σ ∈ N, then, according to (5.4.26) and (5.4.26 ), we get ψ O(1)ψ(σ), f ∈ C0,∞ , |Rσ (f ; x)| ≤ (12.73) O(1)ψ(σ)ω(1/σ), f ∈ C0ψ Hω . In proving this inequality, we actually do not use the periodicity of the function f (·) and the fact that σ ∈ N . Therefore, it remains valid in the analyzed case. By virtue of relations (12.36) and (12.37), a similar estimate holds for the quantity |δσ (f ; x)0 |. Further, in view of the fact that the function 2 sin ht/2 − ht t2 is monotonic in the interval (0, h−1 ), as in the proof inequality (12.31), we conclude that an estimate of the form (12.73) is also true for the quantity |J7 (f ; x)|. Hence, by combining these estimates with equality (12.72), and taking into account the fact that the error in the integral appearing in (12.72) caused by the substitution of σ for σ0 can also be estimated by using relations of the form (12.73), we arrive at the assertion of Lemma 12.3. 12.9. In exactly the same way as in Section 5.6 for the quantity En (Hω ) (see also the proofs of equalities (12.43) and (12.44)), we deduce the equalities ψ(σ) sup | π ˆψ f ∈C 0,∞

h−1

f0ψ (x + t)

−h−1

sin σt 4 dt| = 2 ψ(σ) ln σ + O(1)ψ(σ) t π

and ψ(σ) sup | π ˆ0 Hω f ∈C

h−1

ϕ(x; t) −h−1

sin σt 2 dt| = 2 ψ(σ)eσ (ω) ln σ + O(1)ψ(σ)ω(1/σ), t π

666

Approximations in Spaces of Locally Summable Functions

Chapter 9

where eσ (ω) is the same quantity as in Theorem 12.1. Therefore, by analyzing the upper bounds of both sides of equalities (12.65) and (12.66), we arrive at the following assertion: ψ Theorem 12.3. Let ψ ∈ A0 . Then the quantities Eσ (Cˆ0,∞ ) and Eσ (Cˆ0ψ Hω ) are independent of the point x and the following equalities hold as σ → ∞ :

4 ψ Eσ (Cˆ0,∞ ) = 2 ψ(σ) ln σ + O(1)ψ(σ) π and

2 ψ(σ)eσ (ω) ln σ + O(1)ψ(σ)ω(1/σ), π2 where O(1) are quantities uniformly bounded in σ. Eσ (Cˆ0ψ Hω ) =

13.

Some Remarks on the Approximation of Functions of High Smoothness

13.1. In the present section, we also consider the quantities ρσ (f ; x) defined by relation (12.3) for the functions f ∈ Cˆβψ M but under an additional assumption that their (ψ; β)-derivatives ϕ(·) satisfy he inequality ∞ −∞

|ϕ(t)| dt ≤ K < ∞. |t| + 1

(13.1)

The subset of functions ϕ ∈ M satisfying inequality (13.1) is denoted by M1 . We set S∞ = SM = {ϕ : ϕ ∈ M, ess sup |ϕ(t)| ≤ 1}, SM1 = {ϕ : ϕ ∈ M1 ∩ SM }. In this notation, we can formulate the following lemma: Lemma 13.1. Let ψ ∈ A and let a(σ) be an arbitrary function continuous for all σ ≥ 0 and such that σa(σ) ≥ a0 > 0 for any σ > 0. If, in addition, f ∈ Cˆβψ SM1 , then, for all x ∈ R, β ∈ R, and σ ≥ h, ρσ (f ; x) ψ(σ) =− π

ma ≤|t|≤h−1

fβψ (x + t)

sin(σt + βπ/2) dt + ¯bψ σ (a; f ; x), (13.2) t

Section 13

Remarks on Approximation of Functions of High Smoothness

667

where ma = min(a(σ), h−1 ), |¯bψ σ (a; f ; x)| ≤ K1 (ψ(σ) + Qσ (a; ψ)),

(13.3)

and Qσ (a; ψ) is the quantity defined by relation (12.11). ψ Proof. Since Cˆβψ SM1 ⊂ Cˆβ,∞ equality (12.6) holds by virtue of Lemma 12.1. Therefore, if a(σ) ≤ h−1 , then (13.2) follows from (12.6). It remains to consider the case where a(σ) > 1. In this case, ma = h−1 , Ma = a(σ) and, according to (12.6) and (13.1), we have ψ(σ) sin(σt + βπ/2) dt + bψ fβψ (x + t) ρσ (f ; x) = σ (a; f ; x) π t h−1 ≤|t|≤a(σ)

= O(1)(ψ(σ)) + bψ σ (a; f ; x).

(13.4)

This completes the proof of Lemma 13.1. By using equality (13.2), we follow the proof of Theorem 12.1 and arrive at the following assertion: Theorem 13.1. Under the conditions of Lemma 13.1, the quantity Eσ (Cˆβψ SM1 ) = sup{|ρσ (f ; x)| : f ∈ Cˆβψ SM1 } is independent of x ∈ R and the following equality holds as σ → ∞ : Eσ (Cˆβψ SM1 ) =

1 4 ψ(σ) ln+ + bψ σ (a), 2 π a(σ)

(13.5)

where ln+ t = max (ln t, 0), bψ σ (a) = O(1)(ψ(σ) + Qσ (a; ψ)),

(13.6)

and O(1) is a quantity uniformly bounded in σ and β. 13.2. Now let ψ ∈ F0 and let a(σ) = a∗ (σ) = μ(ψ; σ)/σ. Then Theorem 13.1 implies the following analog of Theorem 12.2: Theorem 13.2. Let ψ ∈ F0 . Then Eσ (Cˆβψ SM1 ) =

4 ψ(σ) ln+ (η(σ) − σ) + O(1)ψ(σ), π2

(13.7)

668

Approximations in Spaces of Locally Summable Functions

Chapter 9

as σ → ∞, where O(1) is a quantity uniformly bounded in σ and β and η(σ) = η(ψ; σ) = ψ −1 (ψ(σ)/2). By comparing equalities (13.7) and (12.62), we conclude that if the quantities η(σ) − σ are bounded below by a positive number, then their right-hand sides, in fact, coincide and, hence, it is natural to use a more general equality (12.62). However if, e.g., lim (η(σ) − σ) = 0, (13.8) σ→∞

then, beginning with some σ0 , we have ln+ (η(σ) − σ) = 0 and, therefore, Eσ (Cˆβψ SM1 ) = O(1)ψ(n), i.e., the orders of the quantities in relations (12.62) and (13.7) are, in this case, noticeably different. Note that condition (13.8) is satisfied only for rapidly decreasˆ ψ of functions f (·) of high smoothness. ing functions ψ(·) specifying the sets L As indicated in Subsection 12.7, the function exp(−αtr ), t ≥ 1, α > 1 can be regarded as an example of a function ψ(·) of this sort for any r > 1. 13.3. We also note that, as follows from the theorems proved in the present section and Section 12, the principal terms of the upper bounds of the quantities |ρσ (f ; x)| in the analyzed classes of functions are independent of the quantities h appearing in the construction of the operators Fσ (f ; x). Therefore, in analyzing the problems of asymptotic equalities for the upper bounds of the quantities |ρσ (f ; x)| in these classes, we can always assume that h = 1.

14.

Strong Means of Deviations of the Operators Fσ (f ; x)

14.1. In the present section, we continue our investigation of the deviations ρσ (f ; x) = f (x) − Fσ (f ; x), where Fσ (f ; x) are defined by relation (12.1) (or (3.20)) for c = σ−1. However, in this case, the approximating properties of the operators Fσ (f ; x) are characterized by the following functionals: (p) Hd (f ; x; λ)

∞ λ(σ)|ρσ (f ; x)|p dσ,

=

d ≥ h, p > 0,

d

where λ = λ(σ) is a nonnegative function continuous for all σ ≥ d.

(14.1)

Section 14

Strong Means of Deviations of the Operators Fσ (f ; x)

669

(p)

The quantity Hdp (f ; x; λ) is an integral analog of the functional Hn (f ; x; λ) studied in Section 5.24 in analyzing the problems of strong summability of Fourier series on the sets Cβψ . As in Section 3, let Wσ2 be the set of entire functions ϕ(·) of exponential type ≤ σ for which the function (ϕ(t)/(1 + |t|))2 is summable on R and let Eσ (f ) = inf f (x) − ϕ(x) = inf esssup|f (x) − ϕ(x)|, ϕ∈Wσ2

ϕ∈Wσ2

M = {ϕ : ϕM < ∞}.

(14.2) (14.3)

In this notation, the following statements are true: Theorem 14.1. Let ψ ∈ F0 and, in addition, there exist a number a0 = a0 (ψ) > 0 such that η(t) − t = ψ −1 (ψ(t)/2) − t ≥ a0 ∀t ≥ 1.

(14.4)

Further, assume that p is an arbitrary positive number and the function λ(σ) is such that the product λ(σ)ψ p (σ) is nonincreasing for all σ > 1. If, in addition, f ∈ Cˆβψ M, then the following inequality is true for any β ∈ R and d ≥ 1 : (p)

Hd (f ; x; λ)C ≤ K(λ(d)ψ p (d)(η(t) −

p t)Ed−1 (fβψ )

∞ +

p λ(σ)ψ p (σ)Eσ−1 (fβψ )dσ), (14.4)

d

where · C = max | · | and K is a quantity independent of f (·), d, and β. x

Theorem 14.2. Assume that ψ ∈ A0 , p > 0, d ≥ 1, and λ(σ) is such that the product λ(σ)ψ p (σ) is nonincreasing for all σ > 1. In this case, if f ∈ Cˆ0ψ M, then (p) Hd (f ; x; λ)C

≤ K(λ(d)dψ

p

p (d)Ed−1 (f0ψ )

∞ +

p λ(σ)ψ p (σ)Eσ−1 (fβψ )dσ).

d

Prior to proving these theorems we first establish several auxiliary facts. Lemma 14.1. Assume that ψ ∈ F0 and a∗ (σ) = η(σ) − σ. If f ∈ Cˆβψ M, then, for any x ∈ R, β ∈ R, and σ ≥ 1, ∗ ρσ (f ; x) = Bσψ (f ; a∗ ; x) + bψ σ (f ; a ; x),

(14.5)

670

Approximations in Spaces of Locally Summable Functions

where

ψ ∗ |bψ σ (f ; a ; x)| ≤ Kψ(σ)Eσ−1 (fβ ), νψ(σ) sin(σt + βπ/2) ψ ∗ h(x + t) dt, Bσ (f ; a ; x) = π t

Chapter 9

(14.6) (14.7)

m≤|t|≤M

m = min(a∗ (σ), 1),

M = max(a∗ (σ), 1),

ν = sign(a∗ (σ) − 1),

h(v) = fβψ (v) − ϕ(v), 2 and ϕ(v) is a function from Wσ−1 such that

Eσ−1 (f ) = fβψ (x) − ϕ(x)C .

(14.8)

Proof. In view of equality (4.8), we can write ∞ ρσ (f ; x) =

h(x + t)ˆ rσ (t)dt,

(14.9)

−∞

where rˆσ (t) is given by relations (4.5) and (4.6) with c = σ − 1. Further, we set (see relations (12.14)–(12.17)) ψ Pσ (f ; x) = − h(x + t)J2 (σ; t)dt, |t|≤a∗ (σ)

Rσψ (f ; x) =

1 π

h(x + t)J3 (σ; t)dt, |t|≥a∗ (σ)

1 γσ (f ; x) = −

h(x + t)J1 (σ; t; σ − 1)dt,

−1

and 2ψ(σ) δσ (f ; x) = π

h(x + t)

sin((2σ − 1)t/2 + βπ/2) sin(t/2) dt. t2

|t|≥1

Here, J1 (σ; t; c), J2 (σ; t), and J3 (σ; t) are given by relations (6.1)–(6.4). In this notation, we have ∗ ψ ψ ψ bψ σ (f ; a ; x) = Bσ (f ; x) + Pσ (f ; x) + Rσ (f ; x) + γσ (f ; x) + δσ (f ; x)

Section 14

Strong Means of Deviations of the Operators Fσ (f ; x)

671

and, hence, ∗ ψ ψ |bψ σ (f ; a ; x)| ≤ (|Pσ (f ; x)| + |Rσ (f ; x)|

+ |γσ (f ; x)| + |δσ (f ; x)|)Eσ−1 (fβψ ).

(14.10)

By using estimates (12.20)–(12.25) and (12.37) and relations (6.2.3) and (14.10), we arrive at inequality (14.6). Lemma 14.2. Let ψ ∈ A0 . If f ∈ Cˆ0ψ M, then, for any x ∈ R and σ ≥ 1, ρσ (f ; x) = Bσψ (f ; x) + bψ σ (f ; x), where Bσψ (f ; x) = −

ψ(σ) π

h(x + t)

sin σt dt, t

(14.11)

|t|≤1 ψ |bψ σ (x; t)| ≤ Kψ(σ)Eσ−1 (fβ )

(14.12)

2 and, as earlier, h(v) = fβψ (v) − ϕ(v), where ϕ(v) is a function from Wσ−1 satisfying equality (14.8).

Proof. By using equality (14.9), we represent fσ (f ; x) in the form (12.72) with h = 1 and ϕ(x; t) = −h(x + t). Further, by virtue of estimate (12.73) and similar estimates of the quantities δσ (f ; x) and J7 (f ; x), we obtain inequality (14.12). 14.2. Assume that d ≥ 1 and −1 ψ (ψ(d)/2), ψ ∈ F0 , γ(d) = d, ψ ∈ A0 , Vdp (f ; x)

1 =( γ(d)

(14.13)

d+γ(d)

|ρd (f ; x)|p dσ)1/p ,

p > 0.

(14.14)

d

Lemma 14.3. Let ψ ∈ F0 . Assume that there exists a number a0 = a0 (ψ) > 0 such that condition (14.4) is satisfied. If f ∈ Cˆβψ M, then, for any x ∈ R, d ≥ 1, p > 0, and β ∈ R, (p)

Vd (f ; x) ≤ Kψ(d)Ed−1 (fβψ ).

(14.15)

672

Approximations in Spaces of Locally Summable Functions

Chapter 9

Further, if ψ ∈ A0 , then, for every f ∈ Cˆ0ψ M and any x ∈ R, d ≥ 1, and p > 0, (p) Vd (f ; x) ≤ Kψ(d)Ed−1 (f0ψ ). (14.16) In equalities (14.15) and (14.16), K is a quantity which may depend only on the function ψ(·). (p)

Proof. It follows from the Hölder inequality that the quantity Vd (f ; x) does not decrease as a function of the parameter p. Hence, it suffices to prove inequalities (14.15) and (14.16) for p ≥ 2. First, we prove inequality (14.15). Substitut(p) ing the quantity ρσ (f ; x) from (14.5) in Vd (f ; x) and applying the Minkowski inequality, we get 1 (p) Vd (f ; x) ≤ ( γ(d)

η(d) |Bσψ (f ; a∗ ; x)|p dσ)1/p d

1 + K( γ(d)

η(d) (ψ(σ)Eσ−1 (fβψ ))p dσ)1/p d

df

= u1 (d) + u2 (d).

(14.17)

where K and, in what follows, Ki , i = 1, 2, . . . , are quantities uniformly bounded in d ≥ 1, f ∈ Cˆβψ M, and β ∈ R. The function ψ(σ)Eσ−1 (fβψ ) is nonincreasing and, therefore, u2 (d) ≤ K2 ψ(d)Ed−1 (fβψ ).

(14.18)

To estimate the quantity u1 (d), we first note that ψ(σ) |Bσψ (f ; a∗ ; x)| = | π

a∗ (σ)

h(x + t) sin(σt + βπ/2)dt| t

1

ψ(σ) |( = π

a∗ (d)

a∗ (σ)

+ 1

a∗ (d)

)

h(x + t) sin(σt + βπ/2)dt. t

Hence, by applying the Minkowski inequality once again, we find

Strong Means of Deviations of the Operators Fσ (f ; x)

Section 14

673

∗

η(d) a (d) h(x + t) | sin(σt + βπ/2)dt|p dσ)1/p t

ψ(d) 1 ( u1 (d) ≤ π γ(d)

1

d

∗

η(d) a (σ) ψ(d) 1 h(x + t) + | ( sin(σt + βπ/2)dt|p dσ)1/p π γ(d) t a∗ (d)

d df

(1)

(2)

= u1 (d) + u1 (d).

(14.19)

It is clear that (2)

u1 (d) ≤

ψ(d) | σ∈[d,η(d)] π

a∗ (σ)

max

a∗ (d)

≤K

h(x + t) dt| t

γ(d) ψ(d) Ed−1 (fβψ ) max | ln |. π γ(σ) σ∈[d,η(d)]

(14.20)

If ψ ∈ F0 , then, for t ≥ 1, the function ψ(t) belongs to the set F (see Section 3.13). Hence (see (7.9.32)), 0 < K3 ≤

γ(d) η(d) − d = ≤ K4 , γ(σ) η(σ) − σ

σ ∈ [d, η(d)].

(14.21)

It follows from relations (14.20) and (14.21) that (2)

u1 (d) ≤ Kψ(d)Ed−1 (fβψ ).

(14.22)

(1)

To estimate the quantity u1 (d) we apply the well-known Hausdorff–Young inequality, namely,

((2π)

−1

∞

−∞

−1

∞

|(2π)

f (t)e−ixt dt|q dx)1/q

−∞

≤ ((2π)

−1

∞ |f (x)|q dx)1/q , f ∈ Lq (R), (14.23)

−∞

1 < q ≤ 2,

q = q/(1 − q).

674

Approximations in Spaces of Locally Summable Functions

Chapter 9

For this purpose, we set ⎧ ⎨h(x + t)/t, t ∈ [m, M ], ϕx (t) =

⎩

¯ [m, M ]. t∈

0,

Then

(1) u1 (d)

ψ(d) 1 = ( π γ(d)

η(d) ∞ | ϕx (t) sin(σt + βπ/2)dt|p dσ)1/p −∞

d

−1/p

≤ Kψ(d)(γ(d))

∞ ∞ ( | ϕx (t) sin(σt + βπ/2)dt|p dσ)1/p −∞ −∞

≤ Kψ(d)(γ(d))−1/p Ed−1 (fβψ )|(a∗ (d))1−p − 1|1/p . The cases where γ(d) < 1 and γ(d) ≥ 1 are analyzed separately. As a result, we conclude that −1/p

(γ(d))−1/p |(a∗ (d))1−p − 1|1/p ≤ max(1, a0

)

and, therefore, (1)

u1 (d) ≤ Kψ(d)Ed−1 (fβψ ).

(14.24)

Combining relations (14.17)–(14.19), (14.22), and (14.24), we establish estimate (14.15). To prove inequality (14.16), we represent Bσψ (f ; x) from relation (14.11) in the form Bσψ (f ; x)

ψ(σ) = ( π

|t|≤1/d

+

)

h(x + t) sin σtdt. t

1/d≤|t|≤1

By applying the Minkowski inequality, in view of relation (14.11), we find

Strong Means of Deviations of the Operators Fσ (f ; x)

Section 14

(p) Vd (f ; x)

1 ≤( d

2d

ψ(σ) | π

675

h(x + t) sin σtdt|p dσ)1/p t

|t|≤1/d

d

1 +( d

2d d

1 + K( d

ψ(σ) | π

h(x + t) sin σtdt|p dσ)1/p t

1/d≤|t|≤1

2d

|ψ(σ)Eσ−1 (f0ψ )|p dσ)1/p

d df

= u1 (d) + u2 (d) + u3 (d).

(14.25)

The function ψ(σ)Eσ−1 (f0ψ ) is nonincreasing and, thus, u3 (d) ≤ Kψ(d)Ed−1 (f0ψ ).

(14.26)

Further, we get 1 u1 (d) ≤ ( d

2d ψ(σ) ( π d

σ|h(x + t)|dt)p dσ)1/p |t|≤1/d

≤ Kψ(d)Ed−1 (f0ψ ). By setting ϕx (t) =

(14.27)

h(x + t)/t, |t| ∈ [1/d, 1],

¯ [1/d, 1], 0, |t|∈ and following the proof of estimate (14.24), we arrive at the inequality h(x + t) 1/p 1/p u2 (d) ≤ Kψ(d)d−1/p ( | | dt) t 1/d≤|t|≤1

≤ Kψ(d)Ed−1 (f0ψ ).

(14.28)

Combining relations (14.25)–(14.28), we obtain inequality (14.16). 14.3. We now pass directly to the proof of the theorem. By setting d0 = d, di = η(di−1 ), i ∈ N, for any f ∈ Cˆβψ , we conclude that (p) Hd (f ; x; λ)

∞

d ∞ i+1 λ(σ)|ρσ (f ; x)| dσ = λ(σ)|ρσ (f ; x)|p dσ. p

= d

i=0 d i

676

Approximations in Spaces of Locally Summable Functions

Chapter 9

A sequence of numbers σi ∈ [di , di+1 ], i = 0, 1, . . . , is chosen from the condition λ(σi ) = max λ(σ). σ∈[di ,di+1 ]

Thus, by virtue of relation (14.15), we get di+1

di+1

λ(σ)|ρσ (f ; x)| dσ ≤ λ(σi )

|ρσ (f ; x)|p dσ

p

di

di

≤ Kλ(σi )ψ p (di )Edi −1 (fβψ )γ(di ) and, hence, (p)

Hd (f ; x; λ) ≤ K

∞

λ(σi )ψ p (di )Edpi −1 (fβψ )γ(di )

i=1 p (fβψ ) = K(λ(σ0 )ψ p (d)γ(d)Ed−1

+

∞

λ(σi )ψ p (di )Edpi −1 (fβψ )γ(di−1 ).

(14.29)

i=1

At the same time, if ψ ∈ F, then, according to (14.21), η(di ) − di γ(di ) ≤ K4 . = γ(di−1 ) η(di−1 ) − di−1

(14.30)

Therefore, relation (14.29) implies that (p)

p Hd (f ; x; λ) ≤ K(λ(σ0 )ψ p (d)Ed−1 (fβψ )

+

∞

λ(σi )ψ p (di )Edpi−1 (fβψ )γ(di ). (14.31)

i=1

λ(σ)ψ p (σ)

The function is nonincreasing and, moreover, ψ(di ) 2ψ(di+1 ). Thus, if σi ∈ [di , di+1 ], then λ(σi )ψ p (di ) = λ(σi )2p ψ p (di+1 ) ≤ 2p λ(σi )ψ p (σi ) ≤ 2p λ(di )ψ p (di ). Therefore, in view of (14.31), we arrive at inequality (14.4), i.e., (p)

p (fβψ ) Hd (f ; x; λ) ≤ K(λ(d)ψ p (d)γ(d)Ed−1

+

∞ i=1

λ(di )ψ p (di )Edpi−1 (fβψ )γ(di )

=

Section 14

Strong Means of Deviations of the Operators Fσ (f ; x)

677

p ≤ K(λ(d)ψ p (d)γ(d)Ed−1 (fβψ )

+

∞ di

p λ(σ)ψ p (σ)Eσ−1 (fβψ )dσ)

i=1 d

i−1

= K(λ(d)ψ

p

p (d)γ(d)Ed−1 (fβψ )

∞ +

p λ(σ)ψ p (σ)Eσ−1 (fβψ )dσ).

d

14.4. Proof of Theorem 14.2. By using relation (14.16), we find Hdp (f ; x; λ) =

∞

i+1 ∞ 2 d λ(σ)|ρσ (f ; x)|p dσ = λ(σ)|ρσ (f ; x)|p dσ

i=0

d

≤

∞

i+1 d 2

|ρσ (f ; x)|p dσ

λ(σi )

i=0

≤K

2i d

∞

2i d

λ(σi )ψ p (2i d)E2pi d−1 (f0ψ )2i d,

(14.32)

i=0

where the numbers σi ∈ [2i d, 2i+1 d] are such that λ(σi ) =

max

σ∈[2i d,2i+1 d]

λ(σ).

According to the definition of the class M0 , for any ψ ∈ M0 , one can find a number α > 1 such that (see Section 3.16) η(t) = η(ψ; t) = ψ −1 (ψ(t)/2) > αt, t ≥ 1.

(14.33)

By choosing the number ν ∈ N such that the inequality αν > 2 holds and applying relation (14.33) ν times, we find η(. . . (η(2i d))) > 2i αν d > 2i+1 d. By using the properties of the function ψ(·), we conclude that ψ(2i d) = 2ψ(η(2i d)) = . . . = 2ν ψ(η(. . . (η(2i d)) . . .)) ≤ 2ν ψ(2i+1 d). The function λ(σ)ψ p (σ) is nonincreasing and, therefore, λ(σi )ψ p (2i d) ≤ 2pν λ(σi )ψ p (σi ) ≤ 2pν λ(2i d)ψ p (2i d).

(14.34)

678

Approximations in Spaces of Locally Summable Functions

Chapter 9

Relations (14.32) and (14.34) now imply the required estimate. Indeed, p Hdp (f ; x; λ) ≤ K4 (dλ(d)ψ p (d)Ed−1 (f0ψ )

+

∞

λ(2i d)ψ p (2i d)E2pi d−1 (f0ψ )2i−1 d)

i=1

= K(dλ(d)ψ

p

p (d)Ed−1 (f0ϕ )

+

2i d ∞ i=1

= K(dλ(d)ψ

p

p (d)Ed−1 (f0ψ )

∞ + d

Theorem 14.2 is proved.

p λ(σ)ψ p (σ)Eσ−1 (f0ψ )dσ)

2i−1 d

p λ(σ)ψ p (σ)Eσ−1 (f0ψ )dσ).

10. APPROXIMATION OF CAUCHY-TYPE INTEGRALS

In this chapter, we study the approximations of Cauchy-type integrals in the regions bounded by closed rectifiable Jordan curves by algebraic polynomials constructed on the basis of the Faber polynomials by using fixed Λ-methods of summation of the Faber series. The proposed approach has rich history with numerous profound results (see, e.g., the monographs by Dzyadyk [5], Smirnov and Lebedev [2], Suetin [1], etc). The Cauchy-type integrals 1 Kf (z) = 2πi

∂Ω

f (ζ) dζ ζ −z

are determined by their density f and the boundary ∂Ω of the region Ω. If Ψ(w) is a function which conformally and univalently maps the domain |w| > 1 onto the complement of the domain Ω, then the function f ∗ (w) = f (Ψ(w)) (as a function of the argument w (|w| = 1) ) reflects both the properties of the function f (·) and specific features of the structure of the boundary ∂Ω. Therefore, the approximating properties of Kf (z)-type integrals also depend on f ∗ (·). This fact was emphasized and frequently used by numerous authors (in particular, in the cited monographs). However, as far as we know, it most cases, the role played by this fact was insignificant: The results obtained in approximation theory were, as a rule, formulated in terms of the regions Ω and functions f (·). As a distinctive feature of the material presented in this section, we can mention, in particular, the fact that the analyzed classes of functions are determined by the conditions imposed on the functions f ∗ , and the estimates of approximations of the functions Kf (z) are expressed in the explicit form via the values of the best approximations of the generalized ψ-derivatives of the functions f ∗ by trigonometric polynomials of a given degree. 679

680

Approximation of Cauchy-Type Integrals

Chapter 10

The functions f ∗ (w) are defined on a circle T = {w : |w| = 1} and, hence, they are periodic in the argument θ = arg w with period 2π. This enables us to define, on the curves Γ = ∂Ω, the classes Lψ N(Γ) of functions f (ζ), ζ ∈ Γ, adequate to the classes Lψ N introduced by the author in the periodic case. This analogy is basic throughout the chapter. The results established for the functions f ∈ Lψ N(Γ) are analogs of the corresponding results obtained for the functions from Lψ N in the periodic case. It should also be emphasized that the basic methods used in the present chapter are similar to the methods used for the investigation of approximations of periodic functions from the sets Lψ N and - ψ N defined on the entire axis. functions from the sets L

1.

Definitions and Auxiliary Statements

1.1. The domains and Jordan curves in the complex plane are the principal geometric objects studied in this chapter. We now recall their definitions. Definition 1.1. A continuous bijective image of a segment [a, b] ⊂ R into C is called a Jordan curve in the plane C. A continuous bijection φ mapping [a, b] onto a Jordan curve is called a parametrization of this curve. If, in addition, φ(a) = φ(b), then the Jordan curve is called closed. Definition 1.2. A domain in the complex plane C is defined as an open set any two points of which can be connected by a polygonal line all points of which belong to this set. All points of the complex plane C that do not belong to a given domain but are its limiting points are called boundary points of this domain. The collection of all boundary points of a domain is called its boundary. A domain is called simply connected either if its boundary is formed by a continuum (defined as a closed set containing more than one point which cannot be decomposed into two closed sets without common points) or if this domain is a complete plane. Closed Jordan curves Γ possess the following remarkable property: Every - = C ∪ {∞} into two simply connected curve Γ splits the extended plane C domains with common boundary Γ; one of these domains is bounded and called the interior of Γ and the other contains the point ∞ and is called the exterior of Γ. This fact reflects the principal result of the well-known Jordan theorem.

Section 1

Definitions and Auxiliary Statements

681

1.2. Assume that z(t) is a continuous complex-valued function defined on [a, b] ⊂ R and Γ = {z ∈ C : z = z(t), t ∈ [a, b]} is a continuous curve. For any n ∈ N, by Tn we denote the collection of all possible sets τn = {t1 , . . . , tn } of n points from the interval [a, b] and set Sn = sup

n−1

τ ∈Tn k=1

|z(tk+1 ) − z(tk )|.

Definition 1.3. A curve Γ is called rectifiable if S = sup Sn < ∞. n∈N

In this case, the number S is called the length of the curve Γ. This definition implies that, in order that a Jordan curve Γ be rectifiable, it is necessary and sufficient that the functions x(t) = Re z(t) and y(t) = Im z(t) have bounded variations. 1.3. We say that an analytic (meromorphic) function f in a domain Ω univalently maps this domain onto a domain Ω if this function establishes a one-to-one correspondence between the points of the domains Ω and Ω . If a function f is univalent in the domain Ω, then f (z) = 0 for any z ∈ Ω. Hence, a univalent mapping is also called a univalent conformal mapping. In the case of simply connected domains, this mapping is called conformal. The possibility of conformal mapping of domains is established by the following theorem: Riemann Theorem. Assume that Ω is a simply connected domain, Ω = C, and z0 ∈ Ω. Then, for any α ∈ [0, 2π), there exists a unique univalent function f mapping the disk D = {w : |w| < 1} onto Ω and satisfying the conditions f (0) = z0 and arg f (0) = α. In what follows, Ω is a simply connected domain in the complex plane C whose boundary is a closed rectifiable Jordan curve Γ (sometimes, we write Γ = - \ Ω is the exterior of ∂Ω ), Ω = Ω ∪ Γ is the closure of the domain Ω, Ω∞ = C - : |w| > 1} is the curve Γ (complement to the domain Ω ), and D∞ = {w ∈ C the exterior of the unit circle T = {w ∈ C : |w| = 1}. By using the Riemann theorem, one can easily establish the existence of a conformal mapping of the domain D∞ onto the domain Ω∞ . We now prove this fact.

682

Approximation of Cauchy-Type Integrals

Chapter 10

Let c ∈ Ω. Then the mapping z ∗ = 1/(z − c) transforms the domain Ω∞ into a domain Ω∗ C containing the origin of coordinates. This enables us to apply the Riemann theorem to the domain Ω∗ . Assume that f1 : D −→ Ω∗ is a conformal mapping from the Riemann theorem normalized by the conditions f1 (0) = 0, f1 (0) = a1 , a1 > 0. Then the function a1 a2 f2 (w) = f1 (w−1 ) = + 2 + . . . , |w| > 1 w w univalently maps D∞ onto Ω∗ , and, moreover, f2 (∞) = f1 (0) = 0 and lim wf2 (w) = f1 (0) = a1 . Thus, it is to see that the function w→∞

1 w +c= +c f2 (w) a1 + a2 w−1 + . . . a2 w + (c − 2 ) + . . . , |w| > 1 = a1 a1

g(w) =

conformally and univalently maps D∞ onto Ω∞ so that g(w) > 0. w→∞ w Therefore, by virtue of the Riemann theorem, there exists a unique function Ψ(w) meromorphic in the domain D∞ and expandable in the Laurent series in a neighborhood of the infinitely remote point w = ∞, i.e., g(∞) = ∞

and

Ψ(w) = γw + α0 +

lim

∞

αk w−k ,

γ > 0,

(1.1)

k=1

which conformally and univalently maps the domain D∞ onto the domain Ω∞ . Moreover, there exists a unique function w = Φ(z) inverse to the function Ψ(w) whose Laurent expansion in the vicinity of the infinitely remote point z = ∞ has the form ∞ 1 Φ(z) = z + β0 + βk z −k (1.2) γ k=1

which conformally and univalently maps the domain Ω∞ onto the domain D∞ . The number γ is called the transfinite diameter of the domain Ω . In what follows, without loss of generality, we set γ = 1. Definition 1.4. A Faber polynomial of degree n for the domain Ω is defined as an algebraic polynomial Fn (z) in the form of the sum of terms with nonnegative powers in the Laurent expansion of the function [Φ(z)]n (i.e., the regular part of the Laurent expansion), namely,

Section 1

Definitions and Auxiliary Statements [Φ(z)]n = [z + β0 +

∞

683

βk z −k ]n

k=1 n

=z +

n−1

k

cn,k z +

k=0

cn,k z k

k=−∞

and, thus, n

−1

Fn (z) = z +

n−1

cn,k z k .

(1.3)

k=0

We now present examples of Faber polynomials for some domains of simple form. Example 1.1. Let Ω = {z ∈ C : |z − z0 | < R} be a disk of radius R > 0 centered at the point z0 ∈ C. Then the mapping function Φ(z) has the form Φ(z) =

z − z0 R

and, consequently, Fn (z) =

1 (z − z0 )n , n = 0, 1, 2, . . . . Rn

Example 1.2. Let Ω be an ellipse in the plane C with foci at the points ±1 and semiaxes 1 1 1 1 a = (R + ) and b = (R − ), 2 R 2 R where R > 1. It is known that the Zhukovskii function 1 1 ) z = Ψ(w) = (Rw + 2 Rw conformally and univalently maps the domain D∞ onto Ω∞ . Therefore, the function √ 1 w = Φ(z) = (z + z 2 − 1), R where we use a branch of the square root such that √ lim

z→∞

is inverse to the function Ψ.

z2 − 1 = 1, z

(1.4)

684

Approximation of Cauchy-Type Integrals

Chapter 10

By virtue of condition (1.4), we conclude that the following expansion can be constructed in the neighborhood of the point z = ∞ : √

∞

√ k z 2 − 1 = z 1 − z −2 = z + z (−1)k C1/2 z −2k , k=1

where

− 1) . . . ( 12 − k + 1) . k! Thus, we see that the regular parts of the √Laurentnexpansions of the functions n n −n [Φ(z)] and gn (z) = [Φ(z)] + R (z − z 2 − 1) coincide. Moreover, since 1 gn (z) = n ((z + z 2 − 1)n + (z − z 2 − 1)n ) R k C1/2 =

=

1 1 2(2

[n/2] 1 2k n−2k 2 Cn z (z − 1)k Rn

(1.5)

k=0

([α] is the integer part of a number α), the following equality is true for the Faber polynomials: Fn (z) = gn (z) =

[n/2] 1 2k n−2k 2 Cn z (z − 1)k . Rn

(1.6)

k=0

Now let z = x, x ∈ [−1, 1]. By setting x = cos θ, θ ∈ [0, π], in view of equalities (1.5) and (1.6), we get Fn (cos θ) =

1 inθ 2 (e + e−inθ ) = n cos nθ Rn R

and, hence,

2 2 cos(n arccos x) = n Tn (x), n R R where Tn (x) are the well-known Chebyshev polynomials. Since the polynomials Fn (z) and 2R−n Tn (z) coincide on the segment [−1, 1] ⊂ Ω, by virtue of the theorem on uniqueness for analytic functions, we conclude that Fn (x) =

Fn (z) =

2 Tn (z) ∀z ∈ Ω. Rn

Some other examples of Faber polynomials in domains of more complex shape and their basic properties are discussed in detail, e.g., in the cited monographs by Smirnov and Lebedev, Dzyadyk, Suetin, etc.

Section 1

Definitions and Auxiliary Statements

685

1.4. Consider a Cauchy-type integral with bounded density f in a domain Ω 1 f (ζ)dζ Kf (z) = , z ∈ Ω, Γ = ∂Ω, 2πi ζ −z Γ

where the integration along the curve Γ is carried out in the positive direction, i.e., in tracing the domain Ω along the boundary Γ, the indicated domain lies to the left of Γ. It is known that the integral Kf (z) is a function analytic in the domain C \ Γ. We now analyze the possibility of expansion of the function Kf (z), z ∈ Ω, in a series in Faber polynomials. For this purpose, we need the following definitions and well-known facts: Definition 1.5. The set Hp of functions f analytic in the disk D and satisfying, for given p ∈ (0, ∞), the condition ||f ||Hp

1 = sup ( 0 0, then 2π

2π |f (ρe )| dt =

|f (eit )|p dt

it p

lim

ρ→1−0 0

0

686

Approximation of Cauchy-Type Integrals

and

Chapter 10

2π |f (ρeit ) − f (eit )|p dt = 0.

lim

ρ→1−0 0

Similar equalities also hold for the functions from Hp (D∞ ). Theorem 1.1. If a function Ψ univalently maps the domain D∞ onto the - \ Ω of the domain Ω bounded by a closed rectifiable complement Ω∞ = C Jordan curve, then (i) Ψ is continuous in D∞ and absolutely continuous on the circle T; (ii) Ψ ∈ H1 (D∞ ); (iii)

dΨ(eit ) dt

= ieit Ψ (eit ) almost everywhere on the circle T;

(iv) the length s(t1 , t2 ) of an arc specified by the equation z = Ψ(eit ), t ∈ [t1 , t2 ], is given by the formula t2 s(t1 , t2 ) =

|Ψ (eit )|dt.

t1

Theorem 1.2. Let Ω be a simply connected domain whose boundary is a closed rectifiable Jordan curve. Then, for any fixed z ∈ Ω, the function Kz (w) = K(w, z) =

Ψ (w) Ψ(w) − z

belongs to the space H1 (D∞ ) and admits an expansion in the Laurent series Kz (w) =

∞

Fk (z)w−k−1

k=0

absolutely and uniformly convergent in D∞ . Proof. Since, for any fixed z ∈ Ω, we have |Kz (w)| ≤

|Ψ (w)| d(z, Γ)

∀w ∈ D∞ ,

Section 1

Definitions and Auxiliary Statements

687

where d(z, Γ) = min |ζ − z|, ζ∈Γ

in view of assertion (iv) in Theorem 1.1, we find 2π

−1

|Kz (Re )|dt ≤ [d(z, Γ)] it

sup 1 A0 − (

(Ak−1 − Ak )|z|k + An |z|n+1 )

k=1 n ≥ A0 − ( (Ak−1 − Ak ) + An ) = 0 k=1

and, for z ∈ (0, 1], ln (z) =

n

Ak z k > 0.

k=0

Consequently, all zeros of the polynomial ln (z) lie outside the circle T. Hence, the function Ln (z) = z

n ln (1/z)

1·3...(2n−1) 2·4...(2n)

=

ln (z)

+ · · · + 12 z n−1 + z n

1 + 12 z + · · · +

1·3...(2n−1) n 2·4...(2n) z

is analytic in the closed disk D and, on the circle T, we have |Ln (eit )| = |

ln (e−it ) | = 1. ln (eit )

Therefore, f ∈ B. It remains to compute the quantity Sn (Ln ; 1) : 1 Sn (Ln ; 1) = 2πi

2π

Ln (eit ) 2 it it l (e )de ei(n+1)t n

0

1 = 2π

2π

−it

ln (e

1 )ln (e )dt = 2π

0

=1+

n k=1

which proves (4.1).

2π |ln (eit )|2 dt

it

0

[

(2k − 1)!! 2 ] = Gn , (2k)!!

716

Approximation of Cauchy-Type Integrals

Chapter 10

To prove (4.2), we note that (2k − 1)!! (2k)! . = 2k (2k)!! 2 (k!)2 Further, by using the Stirling formula Γ(t + 1) =

√

1 2πtt+1/2 e−t (1 + O( )), t

t → ∞,

where Γ(·) is the Euler gamma function, we obtain ln

(2k − 1)!! 1 1 = − ln k − ln π + o(1), (2k)!! 2 2

(4.4)

whence it follows that n n (2k − 1)!! 2 11 1 [ ] = + O(1) = ln n + O(1). (2k)!! π k π k=1

5.

k=1

Asymptotic Equalities

5.1. In the present section, by using Theorem 3.3 and Lemma 3.2, we establish asymptotic equalities for the quantities ψ Rn (C∞ (T)+ ) =

sup ψ f ∈C∞ (T)+

f (·) − Sn−1 (f ; ·)C(T) ,

where ψ (T)+ = {f ∈ C ψ (T)+ : f ψ L∞ (T) ≤ 1}. C∞

The following assertion is true for ψi ∈ M0 , i = 1, 2 : Theorem 5.1. Assume that ψ = ψ1 + iψ2 and ψ1 , ψ2 ∈ M0 . Then the following asymptotic equality holds as n → ∞ : ψ Rn (C∞ (T)+ ) =

1 |ψ(n)| ln n + O(1)|ψ(n)|, π

(5.1)

where O(1) is a quantity uniformly bounded in n. Proof. By virtue of Corollary 3.3, ψ (T)+ ) = |ψ(n)| Rn (C∞

sup ψ f ∈C∞ (T)+

Sn−1 (f ψ ; ·)C(T) + O(1)|ψ(n)|.

(5.2)

Section 5

Asymptotic Equalities

717

ψ If f ∈ C∞ (T)+ , then f ψ (ρeit ) ∈ B and, moreover,

lim f ψ (ρeit ) = f ψ (eit )

ρ→1−0

almost everywhere in the circle T. On the other hand, for any function g ∈ B, there exists a function f from ψ the class C∞ (T)+ such that f ψ coincides with the angular boundary values of the function g almost everywhere in the circle T. Thus, by virtue of Theorem 4.1, sup ψ f ∈C∞ (T)+

Sn−1 (f ψ ; ·)L∞ (T) = sup max |Sn−1 (f ; z)| f ∈B z∈D

= Gn =

1 ln n + O(1) π

(5.3)

as n → ∞. Combining relations (5.2), (5.3), we arrive at relation (5.1). 5.2. Lemma 3.2 yields the following analog of Theorem 5.10.1: Theorem 5.2. Assume that ψ = ψ1 + iψ2 , ψ1 , ψ2 ∈ F, and, moreover, the following conditions are satisfied: 0 < K1 ≤

η(ψ1 ; t) − t ≤ K2 < ∞ ∀t ≥ 1. η(ψ2 ; t) − t

(5.4)

Then the following asymptotic equality is true: ψ (T)+ ) = Rn (C∞

1 |ψ(n)| ln+ (η(n) − n) + O(1)|ψ(n)|, π

(5.5)

where η(n) is either η(ψ1 ; t) = ψ1−1 ( 12 ψ1 (n)) or η(ψ2 ; t) = ψ2−1 ( 12 ψ2 (n)) and O(1) is a quantity uniformly bounded in n. Proof. We choose the quantity a(n) from relation (3.22) in the form a(n) =

1 . η(n) − n

(5.6)

Then, in view of (5.11.45), we have na(n) > a0 > 0 and, in view of (5.11.44) and (5.11.44 ), by virtue of (5.4), we obtain bψ n (a; f ; ·)C(T) ≤ O(1)|ψ(n)|.

(5.7)

718

Approximation of Cauchy-Type Integrals

Chapter 10

In this case, it follows from equality (3.22) that ψ Rn (C∞ (T)+ )

1 = |ψ(n)| 2π

max |

sup ψ f ∈C∞ (T)+

f ψ (ei(t+θ) )

t∈[0,2π]

e−inθ dθ| θ

|θ|≥a(n)

+ O(1)|ψ(n)|.

(5.8)

ψ (T)+ , there exists a function g ∈ C∞ f ψ = g almost everywhere on T. Hence,

Note that, for any function f ∈ L∞ (T)+ such that gL∞ (T) ≤ 1 and in view of the invariance of the set L∞ (T) under shifts of the argument, we conclude that e−inθ sup max | f ψ (ei(t+θ) ) dθ| θ t∈[0,2π] f ∈C ψ (T) ∞

+

|θ|≥a(n)

=

sup

ϕ∈L∞ (T)+ :ϕL∞ (T) ≤1

|

ϕ(eiθ )

e−inθ dθ| = Mn . (5.9) θ

|θ|≥a(n)

The following estimate is true for any fixed p ∈ (0, 1] (see relation (5.11.46)): ∞ |

ϕ(eiθ )

e−inθ dθ| = O(1)ϕL∞ (T) , θ

n → ∞.

pπ

Therefore, Mn =

sup

ϕ∈L∞ (T)+ :ϕL∞ (T) ≤1

|

ϕ(eiθ )

e−inθ dθ| + O(1) θ

(5.10)

a(n)≤|θ|≤pπ

and, consequently, Mn ≤ 2 ln+

pπ + O(1) = 2 ln+ (η(n) − n) + O(1), a(n)

n → ∞,

(5.11)

whence, by virtue of relations (5.8)–(5.10), we arrive at the required upper bound ψ (T)+ ). To obtain equality (5.4), it remains to show that of the quantity Rn (C∞ (5.11) cannot be a strict inequality. To this end, we consider the sequence of functions Ln (z) = z n

ln (1/z) , ln (z)

n = 1, 2, . . . .

Section 5

Asymptotic Equalities

719

In the proof of Theorem 4.1, it is shown that the functions Ln (z) are analytic in the closed disk D and, in addition, |Ln (ζ)| = |

¯ ln (ζ) | = 1 ∀ζ ∈ T, ln (ζ)

i.e., Ln ∈ B. Thus, by using (5.10), we get the following lower bound of the quantity Mn : e−inθ Ln (eiθ ) Mn ≥ | dθ| + O(1) θ a(n)≤|θ|≤pπ

=|

ln (e−iθ ) dθ · | + O(1), θ ln (eiθ )

n → ∞,

(5.12)

a(n)≤|θ|≤pπ

for any positive number p ≤ 1/2. Further, we replace the integral in (5.12) by the integral ln (e−iθ ) ieiθ dθ · . ln (eiθ ) eiθ − 1 a(n)≤|θ|≤pπ

The error of this substitution is equal to ln (e−iθ ) 1 eiθ − i ( )dθ|. qn = | ln (eiθ ) θ eiθ − 1 a(n)≤|θ|≤pπ

To estimate this quantity, we use the identities 1 2 1 e−iθ 1 = + = . + iθ iθ i2θ iθ i sin θ e + 1 e −1 e −1 e +1 In view of the fact that θ − sin θ ≤ θ sin θ (θ ∈ [0, π/2]), we find ln (e−iθ ) 1 ln (e−iθ ) eiθ 1 qn ≤ | ( · dθ| − )dθ| + | ln (eiθ ) θ sin θ ln (eiθ ) eiθ + 1 a(n)≤|θ|≤pπ

≤ a(n)≤|θ|≤pπ

|θ − sin θ| dθ + θ sin θ

a(n)≤|θ|≤pπ

a(n)≤|θ|≤pπ

1 ≤ 2(pπ − a(n))(1 + √ ) < 2π 2

dθ + 1|

|eiθ ∀n ∈ N.

720

Approximation of Cauchy-Type Integrals

Chapter 10

Hence, by virtue of (5.12), we can write ln (e−iθ ) eiθ dθ Mn ≥ | · | + O(1) ln (eiθ ) eiθ − 1 a(n)≤|θ|≤pπ

ln (e−iθ ) eiθ dθ | + O(1). · ln (eiθ ) 1 − eiθ

≥ | Re

(5.13)

a(n)≤|θ|≤pπ

At the same time, since ln (e−iθ ) eiφ(θ) eiθ eiθ ln (e−iθ ) iφ(θ) = | |e = · · , ln (eiθ ) 1 − eiθ ln (eiθ ) 1 − eiθ |1 − eiθ | where φ(θ) = θ − 2 arg ln (eiθ ) − arg(1 − eiθ ), we get Re

ln (e−iθ ) eiθ dθ = · ln (eiθ ) 1 − eiθ

a(n)≤|θ|≤pπ

Re

eiφ(θ) dθ |1 − eiθ |

a(n)≤|θ|≤pπ

=

cos φ(θ) dθ. |1 − eiθ |

a(n)≤|θ|≤pπ

As a result, relation (5.13) takes the form cos φ(θ) dθ| + O(1). Mn ≥ | |1 − eiθ |

(5.14)

a(n)≤|θ|≤pπ

Now let s be an arbitrary fixed number and let s > 1. Since cos φ(θ) dθ a(n)(s − 1) dθ| ≤ ≤2 | iθ iθ |1 − e | |1 − e | |1 − eia(n) | a(n)≤|θ|≤sa(n)|

a(n)≤|θ|≤sa(n)|

=

a(n)(s − 1) = O(1), sin a(n)/2

in view of relation (5.14), we find Mn ≥ | sa(n)≤|θ|≤pπ

n → ∞,

cos φ(θ) dθ| + O(1). |1 − eiθ |

(5.15)

Section 5

Asymptotic Equalities

721

We now establish estimates for the values of the function cos φ(θ) in the intervals γn = {θ : sa(n) ≤ |θ| ≤ pπ}, n ∈ N. By using the expansions ∞

√ and

(2k − 1)!! 1 = zk , (2k)!! 1−z k=0

z ∈ D,

∞

1 = z k sin2k x, 1 − z sin2 x k=0

z ∈ D, x ∈ R,

and the Wallis formula for the coefficients of the first expansion (2k − 1)!! 2 = (2k)!! π

π/2 sin2k xdx,

(5.16)

0

we obtain 2 1 √ − ln (z) = z n+1 π 1−z

π/2 0

sin2(n+1) x dx 1 − z sin2 x

∀z ∈ D.

Hence, if θ ∈ γn , then, by virtue of relations (5.16) and (4.4), we get |√

1 1 − eiθ

2 − ln (e )| = | lim ρ→1− π

π/2

iθ

0

2 =| π

π/2 0

sin2(n+1) x dx| 1 − ρeiθ sin2 x

sin2(n+1) x dx| 1 − eiθ sin2 x

2 1 ≤ 2 iθ π minx∈[0,π/2] |1 − e sin x| = as n → ∞.

π/2 sin2n xdx 0

1 1 √ (1 + o(1)) minx∈[0,π/2] |1 − eiθ sin2 x| nπ

(5.17)

722

Approximation of Cauchy-Type Integrals

Chapter 10

It is easy to see that min |1 − eiθ sin2 x| = sin θ

x∈[0,π/2]

∀θ ∈ [0, π/2].

Thus, in view of the facts that na(n) > a0 > 0, n ∈ N and sin x ≥ 2x/π ∀x ∈ [0, π/2], relation (5.17) implies that |1 − eiθ | 1 iθ · √ (1 + o(1)) |1 − ln (e ) 1 − eiθ | ≤ | sin θ| πn 2| sin θ/2| 1 · √ (1 + o(1)) = | sin θ| πn =

1 1 1 · √ (1 + o(1)) · cos θ/2 πn 2| sin θ/2|

1 1 ≤ · √ (1 + o(1)) πn | sin θ/2| 1 1 ≤ (1 + o(1)) ≤ (1 + o(1)) |θ|n sna(n) 4/a0 . Then it follows from relation (5.18) that there exists a number n0 ∈ N such that the following inequality is true for all n ≥ n0 : 2 < 1 ∀θ ∈ γn , |1 − ln (eiθ ) 1 − eiθ | < √ sa0 whence, by using simple geometric reasoning, we conclude that the following relation holds for θ ∈ γn as n → ∞ : 1 | arg(ln (eiθ ) 1 − eiθ )| = | arg ln (eiθ ) + arg(1 − eiθ )| 2 2 . < arcsin √ sa0 We choose a number p from the equality p = p(s) =

2 1 . arcsin √ π sa0

(5.19)

Section 6

Lebesgue–Landau Inequalities

723

Thus, in view of relation (5.19), we obtain |φ(θ)| = |θ − 2 arg ln (eiθ ) − arg(1 − eiθ )| 1 arg(1 − eiθ )| 2 2 2 = 3 arcsin √ ≤ pπ + 2 arcsin √ sa0 sa0

≤ |θ| + 2| arg ln (eiθ ) +

∀θ ∈ γn ,

n → ∞.

Further, we fix an arbitrary number ε > 0 and choose s sufficiently large in order that cos φ(θ) > 1 − ε for all θ ∈ γn and sufficiently large n. Hence, we can continue inequality (5.15). As a result, we get dθ + O(1) Mn ≥ (1 − ε) |1 − eiθ | sa(n)≤|θ|≤pπ

pπ = (1 − ε)

dθ + O(1) = 2(1 − ε) sin θ/2

sa(n)

pπ

dθ + O(1) θ

sa(n)

pπ = 2(1 − ε) ln + O(1) sa(n) = 2(1 − ε) ln(η(n) − n) + O(1),

n → ∞.

(5.20)

In view of the arbitrariness of ε, by virtue of (5.11), (5.15), and (5.20), we find Mn = 2 ln+ (η(n) − n) + O(1), n → ∞. Substituting the established asymptotic value of the quantity Mn in relation (5.8), we arrive at relation (5.5).

6.

Lebesgue–Landau Inequalities 6.1. Assume that f ∈ A(D), n ∈ N. Then, according to Theorem 4.1,

∗ ∗ f − Sn−1 (f )A(D) ≤ f − Pn−1 A(D) + Sn−1 (f − Pn−1 )A(D)

1 ≤ (Gn−1 + 1)En (f )A(D) = ( ln n + O(1))En (f )A(D) , π ∗ where Pn−1 is a polynomial of degree n − 1 of the best approximation of the function f.

724

Approximation of Cauchy-Type Integrals

Chapter 10

Thus, for every function f ∈ A(D) and any n ∈ N, 1 f − Sn−1 (f )A(D) ≤ ( ln n + O(1))En (f )A(D) . π The last inequality is an analog of the well-known Lebesgue inequality and, therefore, it is natural to call it the Lebesgue–Landau inequality for functions analytic in the disk D. In this section, we establish analogs of the Lebesgue–Landau inequality for the classes C ψ (T)+ . Theorem 6.1. Let ψ = ψ1 + iψ2 and let ψ1 , ψ2 ∈ M0 . Then the following inequality holds for any function f ∈ C ψ (T)+ and all n ∈ N : 1 rn (f )C(T) ≤ ( ln n + O(1))|ψ(n)|En (f ψ )C(T) , π

(6.1)

where O(1) is a quantity uniformly bounded in n. For any function f ∈ C ψ (T)+ and any n ∈ N, one can find a function F (eit ) = F (eit ; n) such that En (F ψ )C(T) = En (f ψ )C(T) , and relation (6.1) turns for this function into the equality. Remark 6.1. Theorem 6.1 remains true also in the case ψ ≡ 1. ∗ (eit ) is a polynomial of the best approximation of Proof. Assume that Pn−1 ψ it the function f (e ) of degree n − 1. Then, according to Corollary 3.3, we get

rn (f )C(T) ∗ = |ψ(n)|Sn−1 (f ψ − Pn−1 )C(T) + O(1)|ψ(n)|En (f ψ )C(T) . (6.2) ∗ Since the function f ψ − Pn−1 is a nontangential boundary value of the funcψ ∗ tion f (z) − Pn−1 (z) bounded and analytic in the disk D, by virtue of the Landau theorem, we can write

1 ∗ Sn−1 (f ψ − Pn−1 )C(T) ≤ ( ln n + O(1))En (f ψ )C(T) . π Substituting this estimate in (6.2), we obtain inequality (6.1). To prove the second part of the theorem, we construct a function F (eit ) such that F ψ (eit ) = En (f ψ )C(T) Ln (eit ).

Section 6

Lebesgue–Landau Inequalities

725

For this purpose, it is necessary to establish the following assertion: Lemma 6.1. For any n ∈ N, the function Ln (z) = z n

ln (1/z) ln (z)

belongs to the space A(D) and the role of the polynomial of its best approximation by polynomials of degree n − 1 is played by the polynomial identically equal to zero. Therefore, En (Ln )A(D) = Ln A(D) = Ln C(T) = 1, where En (f )A(D) = inf f (z) − Pn (z)A(D) . p

Proof. In the proof of Theorem 4.1, it is shown that the function Ln (z) is analytic in the closed disk D (and, hence, in Ln ∈ A(D) ). Moreover, Ln A(D) = Ln C(T) = 1, and the function Ln (z) has exactly n zeros in the disk D . To prove the lemma, it remains to show that the polynomial of the best uniform approximation of the function Ln of degree n − 1 is identically equal to zero. To this end, we use the well-known Kolmogorov criterion: In order that the polynomial Pn∗ (z) of the best uniform approximation of the function g ∈ A(D) of degree not greater than n be identically equal to zero, it is necessary and sufficient that the inequality min Re{Pn (z)g(z)} ≤ 0

z∈e(Pn∗ )

(6.3)

hold for any polynomial Pn (z) of degree ≤ n in the set e(Pn∗ ) of points z0 ∈ D such that |g(z0 ) − Pn∗ (z0 )| = g − Pn∗ A(D) . The function Ln (z) has exactly n zeros in the disk D. Thus, according to the principle of argument, in tracing the unit circle T by a point z in the positive direction, the argument of the number Ln (z) decreases by 2πn. At the same time, as a result of the same tracing by the point z, the argument of the number Pn−1 (z) increases by at most 2π(n − 1) (because Pn−1 (z) has at most

726

Approximation of Cauchy-Type Integrals

Chapter 10

n − 1 roots in the disk D ). Thus, in this tracing, the resulting increment of the argument of the product Pn−1 (z)Ln (z) is not smaller than 2π. Hence, in view of the continuity of the function Ln (z), the real part of the product Pn−1 (z)Ln (z) is nonpositive at at least one point z0 ∈ T . Therefore, the following inequality holds for any polynomial Pn−1 (z) : min Re{Pn−1 (z)Ln (z)} ≤ 0. z∈T

This implies the validity of a condition of the form (6.3) because, in the ana∗ ) = T. lyzed case, e(Pn−1 Returning to the proof of Theorem 6.1, we conclude that, by virtue of Lemma 6.1, En (F ψ )C(T) = En (f ψ )C(T) En (Ln )C(T) = En (f ψ )C(T) . Let us show that relation (6.1) turns into the equality for the function F (eit ) = J ψ F ψ (eit ) = En (f ψ )C(T) J ψ Ln (eit ) . For this purpose, it suffices to prove that 1 |J ψ Ln (1) − Sn−1 (J ψ Ln )(1)| ≥ |ψ(n)|( ln n + O(1)). π

(6.4)

According to relation (3.20), for any polynomial Pn−1 (eit ) of degree ≤ n − 1, we have |ψ(n)||Sn−1 (Ln − Pn−1 ; 1)| ≤ |J ψ Ln (1) − Sn−1 (J ψ Ln )(1)| + O(1)|ψ(n)|Ln − Pn−1 C(T) . By choosing the polynomial of the best approximation of the function Ln as the polynomial Pn−1 and taking into account the fact that, by virtue of Lemma 6.1, this polynomial is identically equal to zero, we get |J ψ Ln (1) − Sn−1 (J ψ Ln )(1)| ≥ |ψ(n)||Sn−1 (Ln ; 1)| − O(1)|ψ(n)|,

n ∈ N.

By virtue of Theorem 4.1, this yields relation (6.4). 6.2. By analogy with the proof of estimate (5.11) based on the use of Lemma 3.2 in Subsection 5.2, we can establish the following assertion:

Section 7

Approximation of Cauchy-Type Integrals

727

Theorem 6.2. Assume that ψ = ψ1 + iψ2 , ψ1 , ψ2 ∈ F, and, in addition, conditions (5.4) are satisfied. Then the following inequality is true for any function f ∈ C ψ (T)+ and any n ∈ N : 1 rn (f )C(T) ≤ ( ln+ (η(n) − n) + O(1))|ψ(n)|En (f ψ )C(T) , π

(6.5)

where O(1) is a quantity uniformly bounded in n and the quantity η(n) has the same meaning as in Theorem 5.2.

7.

Approximation of Cauchy-Type Integrals

7.1. We now study the problem of approximation of Cauchy-type integrals from the classes Kpψ (Ω)+ at internal points of the domain Ω in the case where the derivative Ψ of the mapping function belongs to the space Hq (D∞ ), q = p/(p − 1). As a starting point, we use Proposition 2.3, according to which, in the case where Ψ ∈ Hq (D∞ ), for any function f ∈ Kpψ (Ω)+ (f = Kϕ) and any fixed z ∈ Ω, the Cauchy-type integral ϕ(ζ[eit ]) ϕ(Ψ(Φ(ζ)eit )) 1 1 f (z[eit ]) = dζ = dζ, 2πi ζ −z 2πi ζ −z Γ

Γ

regarded as a function of a point eit on the circle T belongs to the class C ψ (T)+ and, in addition, (f (z[·]))ψ = f ψ (z[·]). By virtue of this proposition, the Lebesgue–Landau inequalities established in Subsections 6.1 and 6.2 immediately imply the following statements: Theorem 7.1. Assume that 1 ≤ p ≤ ∞, Ψ ∈ Hq (D∞ ), p−1 + q −1 = 1, ψ = ψ1 + iψ2 , and ψ1 , ψ2 ∈ M0 . Then, for any function f ∈ Kpψ (Ω)+ , the following inequality holds for all n ∈ N and z ∈ Ω : |f (z) −

n−1

fk Fk (z)| ≤ f (z[·]) − Sn−1 (f (z[eit ]); ·)C(T)

k=0

1 ≤ ( ln n + O(1))|ψ(n)|En (f ψ (z[·]))C(T) , π where 1 En (f (z[·]))C(T) = inf Tn−1,z ∈Tn−1 2πi

ψ

Γ

(7.1)

ϕψ (ζ[·]) dζ − Tn−1,z (·)C(T) , ζ −z

728

Approximation of Cauchy-Type Integrals

Chapter 10

Tn−1 is a set of functions Tn−1,z of the form it

Tn−1,z (e ) =

n−1

ck (z)eikt ,

k=0

and O(1) is a quantity uniformly bounded in n. Theorem 7.2. Assume that 1 ≤ p ≤ ∞, Ψ ∈ Hq (D∞ ), p−1 + q −1 = 1, ψ = ψ1 + iψ2 , ψ1 , ψ2 ∈ F and, in addition, conditions (5.4) are satisfied. Then, for any function f ∈ Kpψ (Ω)+ , the following inequality holds for all n ∈ N and z∈Ω: |f (z) −

n−1

fk Fk (z)|

k=0

1 ≤ ( ln+ (η(n) − n) + O(1))|ψ(n)|En (f ψ (z[·]))C(T) , (7.2) π where η is either η(ψ1 ; n) or η(ψ2 ; n) and O(1) is a quantity uniformly bounded in n. 7.2. Inequalities (7.1) and (7.2) contain the quantities En (h(z[·]))C(T) depending on the point z. Therefore, to obtain uniform estimates in the closed domain Ω, it is reasonable to investigate these quantities in more details. First, we note that, in the case where ϕ ∈ Lp (Γ)+ , we can substantially modify the Cauchy kernel of the Cauchy-type integral ϕ(ζ[eit ]) 1 it Kϕ(z[e ]) = dζ 2πi ζ −z Γ

without changing the value of the integral. Namely, the following assertion is true: Lemma 7.1. Assume that Ψ ∈ H1 (D∞ ) and GΩ is the set of functions g(ζ, z), ζ ∈ Γ, z ∈ Ω, satisfying the condition that, for any z ∈ Ω, the function gz (·) = g(Ψ(·), z)Ψ (·) belongs to L1 (T)+ . Then, for any function f ∈ L1 (Γ, Φ )+ and any function g ∈ GΩ , the following equality holds at all points z ∈ Ω almost everywhere in the variable w on the circle T : 1 1 Kf (z[w]) = f (ζ[w])( + g(ζ, z))dζ. (7.3) 2πi ζ −z Γ

Proof. It is clear that, in order to prove (7.3), it suffices to show that, for all z ∈ Ω,

Section 7

Approximation of Cauchy-Type Integrals

729

(f ◦ Ψ)(τ w)gz (τ )dτ = 0

f (ζ[w])g(ζ, z)dζ = T

Γ

at almost all points w ∈ T. This equality follows from the fact that the last integral, as a convolution of functions from L1 (T)+ , is a function summable on the circle T. Therefore, in view of the Parseval equality, (f ◦ Ψ)(τ w)gz (τ )dτ T

2π = i (f ◦ Ψ)(ei(t+θ) )gz (eiθ )eiθ dθ 0

∼ 2πi

∞

(f ◦ Ψ)(k) · 0 · eikt + 2πi

k=0

∞

0 · g-z (k)e−ikt ≡ 0.

k=1

If the conditions of Lemma 7.1 are satisfied, then, for any function h ∈ K1 (Ω)+ (h = Kϕ), the following relation holds at all points z ∈ Ω : En (h(z[·]))C(T) ≤

inf

Tn−1,z ∈Tn−1

1 2πi

ϕ(ζ[·])Q(ζ, z)dζ − Tn−1,z (·)L∞ (T) , (7.4) Γ

where Q(ζ, z) =

1 + g(ζ, z), ζ −z

g ∈ GΩ .

Let τn−1 (eit ) be an arbitrary trigonometric polynomial of degree n − 1 of the form n−1 τn−1 (eit ) = τ-k eikt (7.5) k=−(n−1)

and let Tn−1,z (eit ) =

n−1 k=0

Then, for any fixed z ∈ Ω,

ck (z)eikt .

(7.6)

730

Approximation of Cauchy-Type Integrals

(ϕ ◦ Ψ)(weit )Tn−1,z (w) ¯ T

+

Chapter 10

dw w

τn−1 (weit )(Q(Ψ(w), z)Ψ (w)w − Tn−1,z (w)) ¯

T

=

n−1

dw w

ak (z)eikt ∈ Tn−1 , (7.7)

k=0

¯ is the number conjugate to the number w. where ak (z) are functions of z and w Therefore, in view of (7.4) and (7.7), the following estimate is true for any fixed z ∈ Ω : En (h(z[·]))C(T) 1 ≤ inf 2πi

((ϕ ◦ Ψ)(w·) − τn−1 (w·))(Q(Ψ(w), z)Ψ (w)w

T

¯ − Tn−1,z (w))

dw , (7.8) w L∞ (T)

where the lower bound is taken over all trigonometric polynomials τn−1 and all functions g ∈ GΩ and Tn−1,z ∈ Tn−1 . We now set GΩ,p = {g ∈ GΩ : |gz (·)|p ∈ L1 (T)},

p > 0,

and assume that h ∈ Kp (Ω)+ (h = Kϕ), 1 ≤ p ≤ ∞, and Ψ ∈ Hq (D∞ ), p−1 + q −1 = 1. By applying the Hölder inequality to the right-hand side of (7.8), for any z ∈ Ω, we get En (h(z[·]))C(T) ≤ inf ϕ ◦ Ψ − τn−1 Lp (T) τn−1

×

inf

gz ∈G0Ω,q ,Tn−1,z ∈Tn−1

Kz∗ (·) + gz (·) − Tn−1,z (¯·)Lq (T) ,

where G0Ω,q

= {g ∈ GΩ,q :

gz (w)dw = 0 ∀z ∈ Ω} T

(7.9)

Section 7

Approximation of Cauchy-Type Integrals

and Kz∗ (w) = Kz (w)w =

731

Ψ (w)w . Ψ(w) − z

Further, we set Em (ϕ)p = inf (ϕ ◦ Ψ)(·) − τk

τk ek (·)Lp (T) ,

|k|≤m−1

where τk are arbitrary numbers, ek (w) = wk (hence, Em (ϕ)p is the value of the best approximation of the function ϕ ◦ Ψ by trigonometric polynomials of degree m − 1 in the space Lp (T) ), and E−m,∞ (Ω, z)q =

inf

g∈G0Ω,q ,Tm−1,z ∈Tm−1

Kz∗ (·) + gz (·) − Tm−1,z (¯·)Lq (T) .

Thus, we arrive at the following statement: Theorem 7.3. Assume that 1 ≤ p ≤ ∞, Ψ ∈ Hq (D∞ ), and p−1 +q −1 = 1. Then, for any function h ∈ Kp (Ω)+ (h = Kϕ) and all n ∈ N, the following inequality holds at any point z ∈ Ω : En (h(z[·]))C(T) ≤ En (ϕ)p E−n,∞ (Ω, z)q .

(7.10)

7.3. According to Lemma 7.1, in the case where ϕ ∈ Lp (Γ)+ , the Cauchy kernel of the Cauchy-type integral Kϕ(z[eit ]) can be modified without changing the value of the integral. Note that, in this integral, we can also modify the function ϕ without changing the value of the integral. Lemma 7.2. Assume that Ψ , g ∈ H1 (D∞ ), and g(∞) = 0. Then, for any function ϕ ∈ L1 (Γ, Φ )+ , the following relation holds at any point z ∈ Ω almost everywhere in w on the circle T : dζ 1 (ϕ(ζ[w]) + (g ◦ Φ)(ζ[w])) Kϕ(z[w]) = . (7.11) 2πi ζ −z Γ

Proof. Since g ∈ H1 (D∞ ) and g(∞) = 0, the circle T contains angular boundary values g(eit ) and, moreover, it

S[g(e )] =

∞ k=1

g-(−k)e−ikt .

732

Approximation of Cauchy-Type Integrals

Chapter 10

In view of the fact that S[Kz∗ (eit )] =

∞

Fk (z)e−ikt

∀z ∈ Ω,

k=0

by virtue of the theorem on convolution, the following equality holds at any point z ∈ Ω for almost all w ∈ T : (g ◦ Φ)(ζ[w]) Ψ (eit )eit dζ = i g(weit ) dt = 0. ζ −z Ψ(eit ) − z T

Γ

This implies equality (7.11). If the conditions of Lemma 7.2 are satisfied, then, by virtue of (7.11), for any function h ∈ K1 (Ω)+ (h = Kϕ) and all z ∈ Ω , we obtain En (h(z[·]))C(T) 1 ≤ inf Tn−1,z ∈Tn−1 2πi

(ϕ(ζ[·]) + (g ◦ Φ)(ζ[·])) Γ

dζ − Tn−1,z (·)L∞ (T) , (7.12) ζ −z

where g ∈ H10 (D∞ ) = {g ∈ H1 (D∞ ) : g(∞) = 0}. Now let τn−1 (eit ) and Tn−1,z (eit ) be functions of the form (7.5) and (7.6), respectively. Thus, in view of the fact that (ϕ◦Ψ)(·ω) ∈ Lp (T)+ , for any ω ∈ T, we can write dw ((ϕ ◦ Ψ)(weit ) + g(weit ))Tn−1,z (w) ¯ w T

+

τn−1 (weit )(Kz∗ (w) − Tn−1,z (w)) ¯

T

=

n−1

ak (z)eikt ∈ Tn−1 ,

dw w

(7.13)

k=0

where ak (z) are functions of z. Hence, by virtue of relations (7.12) and (7.13), for any fixed z ∈ Ω , we obtain

Section 7

Approximation of Cauchy-Type Integrals En (h(z[·]))C(T) 1 ≤ inf 2πi

733

((ϕ ◦ Ψ)(w·) + g(w·) − τn−1 (w·)) T

× (Kz∗ (w) − Tn−1,z (w)) ¯

dw , w L∞ (T)

(7.14)

where the lower bound is taken over all functions g ∈ H10 (D∞ ) and all polynomials τn−1 and Tn−1,z of the form (7.5) and (7.6), respectively. If we now assume that ϕ ∈ Lp (Γ, Φ )+ for some p ≥ 1 and Ψ ∈ Hq (D∞ ), −1 p + q −1 = 1, and apply the Hölder inequality to the right-hand side of (7.14), then we get the following analog of inequality (7.9): En (h(z[·]))C(T) ≤

inf

g∈Hp0 (D∞ ), τn−1

ϕ ◦ Ψ + g − τn−1 Lp (T)

×

inf

Tn−1,z ∈Tn−1,z

Kz∗ (·) − Tn−1,z (¯·)Lq (T) .

By setting em (ϕ)p =

inf

g∈Hp0 (D∞ ), τm−1

ϕ ◦ Ψ + g − τm−1 Lp (T)

and E|m|,∞ (Ω, z)q =

inf

Tm−1,z ∈Tm−1,z

Kz∗ (·) − Tm−1,z (¯·)Lq (T) ,

we arrive at the following analog of Theorem 7.3: Theorem 7.4. Assume that 1 ≤ p ≤ ∞, Ψ ∈ Hq (D∞ ), and p−1 +q −1 = 1. Then, for any function h ∈ Kp (Ω)+ (h = Kϕ) and all n ∈ N, the following estimate holds at any point z ∈ Ω : En (h(z[·]))C(T) ≤ en (ϕ)p E|n|,∞ (Ω, z)q .

(7.15)

Combining Theorems 7.1–7.4, we arrive at the following assertions: Theorem 7.5. Assume that 1 ≤ p ≤ ∞, Ψ ∈ Hq (D∞ ), p−1 + q −1 = 1, ψ = ψ1 + iψ2 , and ψ1 , ψ2 ∈ M0 . Then, for any function f ∈ Kpψ (Ω)+ (f = Kϕ) and all n ∈ N, the following inequalities are true for any z ∈ Ω : |f (z) −

n−1 k=0

1 fk Fk (z)| ≤ ( ln n + O(1))|ψ(n)|En (ϕ)p E−n,∞ (Ω, z)q π

(7.16)

734

Approximation of Cauchy-Type Integrals

Chapter 10

and |f (z) −

n−1 k=0

1 fk Fk (z)| ≤ ( ln n + O(1))|ψ(n)|en (ϕ)p E|n|,∞ (Ω, z)q , π

(7.17)

where O(1) is a quantity uniformly bounded in n. Theorem 7.6. Assume that 1 ≤ p ≤ ∞, Ψ ∈ Hq (D∞ ), p−1 + q −1 = 1, ψ = ψ1 + iψ2 , ψ1 , ψ2 ∈ F and, moreover, conditions (5.4) are satisfied. Then, for any function f ∈ Kpψ (Ω)+ (f = Kϕ), the following inequalities are true for any n ∈ N and z ∈ Ω, : |f (z) −

n−1

fk Fk (z)|

k=0

1 ≤ ( ln+ (η(n) − n) + O(1))|ψ(n)|En (ϕ)p E−n,∞ (Ω, z)q , (7.18) π and |f (z) −

n−1

fk Fk (z)|

k=0

1 ≤ ( ln+ (η(n) − n) + O(1))|ψ(n)|en (ϕ)p E|n|,∞ (Ω, z)q , (7.19) π where η(n) is either η(ψ1 ; n) or η(ψ2 ; n) and O(1) is a quantity uniformly bounded in n. 7.4. Inequalities (7.16)–(7.19) are proved under the assumption that z ∈ Ω, i.e., they are true for any closed subset K of points of the domain Ω. However, any function f from the class Kpψ (Ω)+ is always analytic on K. Therefore, it seems likely that the appearance of the logarithmic factor in inequalities (7.16)– (7.19) is not connected with the inherent properties of the problem. Moreover, it is quite natural to expect that the quantity E−n,∞ (Ω, z)q approaches zero as n → ∞ with the rate of decrease of a geometric progression. To confirm this observation, we present the following statement: Proposition 7.1. Assume that 1 ≤ q ≤ ∞. Then the following relation holds for any point z ∈ D and all n ∈ N : E−n,∞ (D, z)q =

|z|n , (1 − |z|2 )1/p

p−1 + q −1 = 1.

(7.20)

Section 7

Approximation of Cauchy-Type Integrals

735

To prove relation (7.20), we use the duality relation formulated in the form of the following assertion: Proposition 7.2. Assume that X is a Banach space, Y is a closed subspace of X, X ∗ and Y ∗ are their dual spaces, and Y ⊥ = {φ ∈ X ∗ : φ(y) = 0 ∀y ∈ Y } is a closed subspace of X ∗ (the annihilator of the space Y ). Then the following assertions are true: (a) the quotient space X ∗ /Y ⊥ is isometrically isomorphic to the space Y ∗ and, for any fixed functional φ ∈ X ∗ , min φ + ψ =

ψ∈Y ⊥

sup y∈Y, y≤1

|φ(y)|;

(7.21)

(b) the space (X/Y )∗ is isometrically isomorphic to the space Y ⊥ and, for any fixed x ∈ X, inf x + y =

y∈Y

max

ψ∈Y ⊥ , ψ≤1

|ψ(x)|.

(7.22)

First, we prove equality (7.20) for q ∈ (1, ∞]. As X, we take the Banach space Lp (T), 1 ≤ p < ∞. As Y, we take the subspace Hp,n (T) of functions f given on the circle T and representing the angular boundary values of the functions from Hp such that f (0) = f (0) = . . . = f (n−1) (0) = 0. The space dual to X is isometrically isomorphic to the space Lq (T), p−1 + q −1 = 1. Thus, we set X ∗ = Lq (T). As an element φ ∈ Lq (T), we use the function eit /(eit − z). We set Lp,n (T)+ = {g ∈ Lp (T) : g-(k) = 0, k = −n, −∞},

0 < p ≤ ∞.

Since the set Hp,n consists of functions representable in the form eint f (eit ), where f ∈ Hp (T), it is easy to see that Y ⊥ = Lq,n (T)+ , p−1 + q −1 = 1. This equality is understood in a sense that any function g ∈ Lq,n (T)+ generates a continuous linear functional A (the annihilator of the space Hp,n (T) ) by the formula 2π 1 A(f ) = f (eit )g(eit )dt = 0. 2π 0

736

Approximation of Cauchy-Type Integrals

Chapter 10

In this case, according to (7.21), for 1 < q ≤ ∞ and any fixed z0 ∈ D, we have 1 E−n,∞ (D, z0 )q = min { g∈Lq,n (T)+ 2π 1 = sup | f ∈Bp,n (T) 2π

2π | 0

eit + g(eit )|p dt}1/p eit − z0

2π f (eit ) 0

eit dt| eit − z0

= |z0 | sup |f (z0 )| (p−1 + q −1 = 1), n

f ∈Bp

(7.23)

where Bp = {f ∈ Hp : f Hp ≤ 1} and Bp,n = {f ∈ Bp : f (0) = f (0) = . . . = f (n−1) (0) = 0}. In the last equality in (7.23), we have used the Cauchy formula (see the F. Riesz–M. Riesz theorem). It remains to show that, for p ∈ [1, ∞), sup |f (z0 )| =

f ∈Bp

1 . (1 − |z0 |2 )1/p

(7.24)

To this end, we first consider the case p = 2. Representing the function f from B2 according to the Cauchy formula and using the Cauchy–Buniakowski inequality, for any fixed z0 ∈ D, we obtain 1 |f (z0 )| = | 2π * =

2π

dt 1 f (e ) |≤{ −it 1 − e z0 2π

2π

it

0

1 . 1 − |z0 |2

(7.25)

It is easy to see that the function * f0 (z) =

0

dt }1/2 |1 − e−it z0 |2

1 − |z0 |2 , (1 − z¯0 z)2

Section 7

Approximation of Cauchy-Type Integrals

737

where we take an arbitrary branch of the square root, belongs to B2 . Moreover, for this function, inequality (7.25) turns into the equality, which proves relation (7.24) for p = 2. Now let f ∈ Bp and let p ≥ 1, p = 2. By virtue of the F. Riesz theorem, the function f can be represented in the form f = gBf , where Bf is the Blaschke function constructed according to the zeros of the function f (in this case, it is worth noting that |B(z)| ≤ 1 for all z ∈ D ), g ∈ Bp , and g(z) = 0 for all z ∈ D. Since g p/2 ∈ B2 , by virtue of (7.25), we obtain * |f (z0 )|

p/2

≤ |g(z0 )|

p/2

≤

1 , 1 − |z0 |2

and, furthermore, the function f0 (z) = (

1 − |z0 |2 1/p ) (1 − z¯0 z)2

turns the last relation into the equality. Combining (7.23) and (7.24), we obtain equality (7.20) for 1 < q ≤ ∞. To prove relation (7.20) for q = 1, we use relation (7.22) for X = L1 (T) and Y = L1,n (T)+ . In this case, X ∗ = L∞ (T) and Y ⊥ = H∞,n (T). Since any function f from H∞,n (T) such that f L∞ (T) ≤ 1 is representable in the form f (eit ) = eint g(eit ), where g(eit ) are the angular boundary values of a certain function g from B∞ , for any fixed z0 ∈ D, we can write E−n,∞ (D, z0 )1 =

inf

g∈L1,n (T)+

1 2π

1 = max | f ∈B∞,n 2π

2π | 0

eit + g(eit )|dt eit − z0

2π f (eit ) 0

eit dt| = |z0 |n . eit − z0

7.5. The results obtained in Section 3 show that, for Ω = D, inequalities (7.16) and (7.18) are exact both in order and in a sense of constant coefficients of the principal terms on the right-hand sides for z ∈ T and p = ∞ (in this case, supz∈D E(D, z)1 = 1 and the essential supremum is taken on the left-hand sides of the inequalities). Equality (7.20) implies that, for every simply connected

738

Approximation of Cauchy-Type Integrals

Chapter 10

domain Ω with closed rectifiable Jordan boundary, the following equality holds for any q ∈ (1, ∞] : df

E−n,∞ (Ω, q) = sup E−n,∞ (Ω, z)q = ∞. z∈Ω

At the same time, in numerous cases, the quantity E−n,∞ (Ω, 1) is bounded. It is clear that, for these domains, inequalities (7.16) and (7.18) with p = 1 are exact in order and exact in a sense of constant coefficients of all z ∈ Γ for properly modified left-hand sides. Let us now clarify the conditions guaranteeing that the quantities E−n,∞ (Ω, 1) are bounded. First, we note that the set Sn = {Ω C : sup E−n,∞ (Ω, z)1 < ∞}, z∈Ω

n ∈ N,

coincides with the set S1 . Thus, we write S = S1 = Sn . As a result, the problem reduces to the investigation of the quantity 1 E−1,∞ (Ω, z)1 = inf g∈GΩ 2π

2π | 0

Ψ (eit )eit + gz (eit )|dt. Ψ(eit ) − z

(7.26)

For fixed z ∈ Ω, any function g(z, w) generates a function gz (w) from the space H1 . Therefore, the lower bound in (7.26) coincides with the lower bound taken over H1 . By applying the duality relation (see relation (7.22)), where Y = H1 and Y ⊥ = H∞,1 , we get 1 sup E−1,∞ (Ω, z)1 = sup sup | z∈Ω z∈Ω f ∈B∞,1 2π

2π f (eit ) 0

Ψ (eit )eit dt|. Ψ(eit ) − z

(7.27)

Finally, we introduce two definitions: Definition 7.1. An operator T = TΩ defined on H∞ and acting according to the rule 1 T (f )(z) = TΩ (f )(z) = 2π

2π f (eit ) 0

is called the Faber operator.

Ψ (eit )eit dt Ψ(eit ) − z

Section 7

Approximation of Cauchy-Type Integrals

739

The norm of the operator T is understood as follows: T = TΩ = sup sup |T (f )(z)|. f ∈B∞ z∈Ω

(7.28)

Definition 7.2. A domain Ω C is called a Faber domain if the operator T is bounded, i.e., T < ∞. The set of all Faber domains is denoted by F. Comparing relations (7.27) and (7.28), we conclude that the quantity E−1,∞ (Ω, 1) is bounded if and only if Ω is a Faber domain. In this case, the following inequalities are true: E−1,∞ (Ω, 1) ≤ T ≤ E−1,∞ (Ω, 1) + 1.

11. APPROXIMATIONS IN THE SPACES S p 1.

Spaces Sϕp

1.1. Let X be an arbitrary complex vector space and let ϕ = {ϕk }∞ k=1 be a fixed countable system in it. We assume that, for any pair x, y ∈ X in which at least one of the vectors belongs to ϕ, the scalar product (x, y) is defined and the following conditions are satisfied: (1) (x, y) = (y, x), where z¯ is the number complex conjugate to z; (2) (λx1 + μx2 ; y) = λ(x1 , y) + μ(x2 , y), where λ and μ are arbitrary numbers; 0, k = l, (3) (ϕk , ϕl ) = 1, k = l. We associate every element f ∈ X with a system of numbers fˆ(k) = fˆϕ (k) according to the equalities fˆ(k) = fˆϕ (k) = (f, ϕk ), k ∈ N, and, for a given fixed p ∈ (0, ∞), we set Sϕp

=

Sϕp (X)

= {f ∈ X :

∞

|fˆϕ (k)|p < ∞}.

(1.1)

k=1

Elements x, y ∈ Sϕp are identified if x ˆϕ (k) = yˆϕ (k) for all k ∈ N. Thus, the set Sϕp is generated by the space X, system ϕ, and number p. We define the distance between vectors x, y ∈ X according to the equality df

ρ(x, y)p = x − yp = x − yϕ,p = (x − y)ϕ (·)lp ∞ =( |ˆ xϕ (k) − yˆϕ (k)|p )1/p . k=1

741

742

Approximations in the Spaces S p

Chapter 11

The zero in the space Sϕp is a vector θ such that θˆϕ (k) = 0 for all k ∈ N. The distance ρ(θ, x), x ∈ Sϕp , is called the norm of the vector x and is denoted by xp . Therefore, ∞ |ˆ xϕ (k)|p )1/p . xp = xϕ,p = ρ(θ, x) = (

(1.2)

k=1

The set Sϕp is a linear space: the operations of addition of vectors and multiplication of vectors by numbers defined in the entire space X remain valid for any pair x, y ∈ Sϕp and any numbers λ and μ : λx + μy = z ∈ Sϕp . Indeed, since z ∈ X, we have zˆ(k) = λˆ x(k) + μˆ y (k), and if p ≥ 1, then, by virtue of the Minkowski inequality, we get zp ≤ |λ|xp + |μ|yp ; furthermore, if p ∈ (0, 1), then, in view of the fact that, for any two numbers a and b, one has |a + b|p ≤ |a|p + |b|p , 0 ≤ p < 1, we get ∞ zp = ( |λˆ x(k) + μˆ y (k)|p )1/p k=1

≤ (|λ|p

∞ k=1

|ˆ x(k)|p + |μ|p

∞

|ˆ y (k)|p )1/p

k=1

1 p

≤ 2 p(|λ|xp + |μ|yp ), i.e., we always have z ∈ Sϕp . It is clear that, for p ≥ 1, the norm introduced by equality (1.2) satisfies all necessary axioms, and, therefore, for p ≥ 1, Sϕp is a linear normed space that contains the orthonormal system ϕ. It is also clear that, for p = 2, the space Sϕ2 , under the condition of its completeness, is a Hilbert space. For all other p ∈ (0, ∞), the spaces Sϕp possess very important properties of Hilbert spaces, namely a Parseval equality of the form (1.2) and the minimal property of partial Fourier sums, which is formulated as follows:

Spaces Sϕp

Section 1

743

Proposition 1.1. Let f ∈ Sϕp , p ∈ (0, ∞), let S[f ] = S[f ]ϕ =

∞

fˆ(k)ϕk

(1.3)

k=1

be the Fourier series of the element f in the system ϕ, and let Sn (f ) = Sn (f )ϕ =

n

fˆ(k)ϕk , k ∈ N,

k=1

be the partial sums of this series. Among all sums of the form Φn =

n

αk ϕk

k=1

for a given n ∈ N, the partial sum Sn (f ) has the least deviation from f, i.e., inf f − Φn p = f − Sn (f )p . αk

Moreover, f − Sn (f )pp = f pp −

n

|fˆ(k)|p .

(1.4)

k=1

The proof of this statement follows from equality (1.2), according to which f − Φn pp =

n k=1

|fˆ(k) − αk |p +

∞

|fˆ(k)|p .

k=n+1

The right-hand side of (1.4) tends to zero as n → ∞. This fact implies that, for any element f from Sϕp , its Fourier series (1.3) converges to f, i.e., the system ϕ is complete in Sϕp , and Sϕp is separable. Note one more important property of the spaces Sϕp : if a system ϕ = ∞ {ϕk }∞ k=1 is obtained from the system {ϕk }k=1 by an arbitrary permutation of its elements, then Sϕp = Sϕp

and f ϕ,p = f ϕ ,p ∀f ∈ Sϕp .

This fact follows immediately from (1.1) and (1.2). This remark enables us to generalize Proposition 1.1 as follows:

(1.5)

Approximations in the Spaces S p

744

Chapter 11

Proposition 1.2. Let {gα } be a family of bounded subsets dependent on a parameter α and such that an arbitrary number n ∈ N belongs to all sets gα with sufficiently large indices α, let f ∈ Sϕp , and let fˆ(k)ϕk Sα (f ) = Sgα (f ) = k∈gα

be the partial sum of the series S[f ]ϕ corresponding to the set gα . Then, among all sums of the form Φα = ck ϕk , k∈gα

the partial sum Sα (f ) has the least deviation from f, i.e., inf f − Φα p = f − Sα (f )p . ck

Moreover, f − Sα (f )pp = f pp −

|fˆ(k)|p

k∈gα

and lim f − Sα (f )p = 0.

α→∞

Below, we present one of possible realizations of these constructions, which will be used in what follows. 1.2. Assume that Rm is an m-dimensional Euclidean space (m ≥ 1), x = (x1 , . . . , xm ) are its elements, Z m is the integer lattice in Rm (the set of vectors k = (k1 , . . . , km ) with integer coordinates), xy = x1 y1 +. . .+x m ym , and |x| = 2 . (xx); in particular, kx = k1 x1 + . . . + km xm and |k| = k12 + . . . + km m Also assume that L = L(R ) is the set of all functions f (x) = f (x1 , . . . , xm ) 2π-periodic in each variable and summable in the cube of periods Qm , i.e., in the domain Qm = {x : x ∈ Rm , −π ≤ xk ≤ π, k = 1, . . . , m}. If f ∈ L, then we denote by S[f ] the Fourier series of the function f in the trigonometric system (2π)−m/2 eikx , k ∈ Z m , (1.6) i.e., −m/2

S[f ] = (2π)

k∈Z m

fˆ(k)eikx , fˆ(k) = (2π)−m/2

Qm

f (t)e−ikt dt. (1.7)

Section 2

ψ-Integrals and Characteristic Sequences

745

If we assume that functions equivalent with respect to the Lebesgue measure are indistinguishable, then we can take the space L(Rm ) as X, and the trigonometric system τ = {τs }, s ∈ N, where τs = (2π)−m/2 eiks x ,

ks ∈ Z m , s = 1, 2, . . . ,

(1.8)

as the system ϕ. The system τ = {τs }, s ∈ N, is obtained from system (1.6) by an arbitrary fixed enumeration of its elements. In this case, the scalar product is introduced as follows: f (t)e−iks t dt = fˆ(ks ) = fˆτ (ks ). (1.9) (f, τs ) = (2π)−m/2 Qm

According to (1.5), the sets Sτp obtained as a result are independent of the enumeration of system (1.6); in what follows, we denote them by S p .

2.

ψ-Integrals and Characteristic Sequences

2.1. Let ψ = {ψk }∞ k=1 be an arbitrary system of complex numbers. If, for a given element f ∈ X whose Fourier series has the form (1.3), there exists an element F ∈ X such that S[F ] =

∞

ψk fˆ(k)ϕk ,

(2.1)

k=1

i.e., Fˆϕ (k) = ψk fˆ(k), k ∈ N,

(2.2)

then F is called the ψ-integral of the vector f, and we write F = J ψ f. If N is a certain subset of X, then by ψN we denote the set of ψ-integrals of all elements of N. In particular, ψSϕp is the set of the ψ-integrals of all vectors belonging to Sϕp . If f and F satisfy relation (2.1) (or (2.2)), then it is sometimes convenient to call f the ψ-derivative of F and write f = Dψ F = F ψ . In what follows, we assume that the system ψ satisfies the condition lim |ψk | = 0.

k→∞

(2.3)

It is clear that this condition guarantees the inclusion ψSϕp ⊂ Sϕp . It is obvious that a necessary and sufficient condition for the validity of this inclusion is the boundedness of the set of numbers |ψk |, k ∈ N.

Approximations in the Spaces S p

746

Chapter 11

The structure of aggregates used for the approximation of f ∈ ψSϕp is determined by the characteristic sequences ε(ψ), g(ψ), and δ(ψ) of the system ψ, which are defined as follows: Let ψ = {ψk }∞ k=1 be an arbitrary system of complex numbers satisfying condition (2.3). Denote by ε(ψ) = ε1 , ε2 , . . . the set of the values of |ψk | arranged in decreasing order and by g(ψ) = g1 , g2 , . . . the system of sets gn = gnψ = {k ∈ N :

|ψk | ≥ εn }.

Further, let δ(ψ) = δ1 , δ2 , . . . be the sequence of numbers δn = |gn |, where |gn | is the number of numbers k ∈ N belonging to the set gn . Taking into account condition (2.3), we can determine the sequences ε(ψ) and g(ψ) using the following relations: ε1 = sup |ψk |, g1 = {k ∈ N : |ψk | = ε1 }, k∈N

(2.4) εn = sup |ψk |, gn = gn−1 ∪ {k ∈ N : |ψk | = εn }. k∈gn−1

Note that any number n∗ ∈ N belongs to all sets gnψ with sufficiently large numbers n, and we always have lim δk = ∞.

k→∞

(2.5)

In what follows, for convenience, we denote the empty set by g0 = g0ψ and assume that δ0 = 0. 2.2. Assume that the set Sϕp is generated by a space X, system ϕ, and number p, p > 0, and ψ = {ψk }∞ k=1 is an arbitrary system of complex numbers satisfying condition (2.3). As approximating aggregates for elements f ∈ ψSϕp , we consider the polynomials fˆ(k)ϕk , n = 1, 2, . . . , S0 (f )ϕ,ψ = θ, (2.6) Sn (f )ϕ,ψ = Sgψ (f ) = n

ψ k∈gn

where gnψ are elements of the sequence g(ψ) and θ is the zero element of the space Sϕp . We set En (f )ψ,p = f − Sn−1 (f )ϕ,ψ p .

Section 3

Best Approximations and Widths of p-Ellipsoids

Also let

En (f )ψ,p = inf f − αk

αk ϕk p

747

(2.7)

ψ k∈gn−1

be the best approximation of an element f ∈ ψSϕp by polynomials constructed in ψ the domains gn−1 and let sup inf x − uY

dn (M; Y ) = inf

Fn ∈Fn x∈Mu∈Fn

be the Kolmogorov width of the set M in the space Y with norm · Y . Here, Fn is the set of all n-dimensional (n ∈ N ) subspaces of the space Y. We introduce one more definition. Assume that n ∈ N, γn is an arbitrary collection of n natural numbers, and Pγn = αk ϕk , (2.8) k∈γn

where αk are certain complex numbers. The quantity en (f )p = en (f )ϕ,p = inf f − Pγn p αk ,γn

(2.9)

is called the best n-term approximation of the element f ∈ Sϕp in the space Sϕp . Finally, let Uϕp = {f ∈ Sϕp : f p ≤ 1} (2.10) and let ψUϕp be the set of the ψ-integrals of all elements from Uϕp . Note that if ψk = 0 ∀k ∈ N, then ψUϕp

= {f ∈

Sϕp

∞ fˆ(k) p : | | ≤ 1}, ψk

(2.11)

(2.12)

k=1

i.e., the set ψUϕp is a p -ellipsoid in the space Sϕp with semi-axes equal to |ψk |.

3.

Best Approximations and Widths of p-Ellipsoids 3.1. Assume that E(ψUϕp )ψ,p = sup En (f )ψ,p p f ∈ψUϕ

(3.1)

748

Approximations in the Spaces S p

Chapter 11

and En (ψUϕp )ψ,p = sup En (f )ψ,p . p f ∈ψUϕ

(3.2)

The following assertion is true: Theorem 3.1. Let ψ = {ψk }, k = 1, 2, . . . , be a system of numbers for which conditions (2.3) and (2.11) are satisfied. Then, for any p ∈ [1, ∞) and n ∈ N, the following equality is true: En (ψUϕp )ψ,p = En (ψUϕp )ψ,p = εn ,

(3.3)

where εn is the nth term of the characteristic sequence ε(ψ). Proof. In view of equalities (1.2) and (2.2) and the definition of numbers εn and sets gn , denoting · ϕ,p = · , for any element f ∈ ψUϕp we get En (f )ψ,p = fˆϕ (k)ϕk = ψk fˆϕψ (k)ϕk k∈¯ gn−1

=(

k∈¯ gn−1

|ψk |p |fˆϕψ (k)|p )1/p ≤ εn f ψ = εn ,

(3.4)

k∈¯ gn−1

g¯n−1 = N \ gn−1 . Thus, we always have En (f )ψ,p ≤ εn

∀f ∈ ψUϕp .

(3.5)

ψ and let f∗ = ψk ϕk , Further, let k be an arbitrary point from γn = gnψ \ gn−1 ψ ψ so that f∗ = |ψk | = εn . Since f∗ = ϕk , we have f∗ = 1, and, therefore, f∗ ∈ ψUϕp . It is clear that

En (f∗ )ψ,p = f∗ = εn .

(3.6)

Hence, combining relations (3.5) and (3.6) and taking into account that we always have En (ψUϕp ) ≤ En (ψUϕp ), we arrive at equality (3.3). The statement below describes the values of the widths dn (ψUϕp ). Theorem 3.2. Let ψ = {ψk }, k = 1, 2, . . . , be a system of numbers satisfying conditions (2.3) and (2.11). Then, for any p ∈ [1, ∞) and n ∈ N, one has dδn−1 (ψUϕp ) = dδn−1 +1 (ψUϕp ) = . . . = dδn −1 (ψUϕp ) = Enψ (ψUϕp )ϕ = εn ,

(3.7)

Section 3

Best Approximations and Widths of p-Ellipsoids

749

where δs and εs , s = 1, 2, . . . , are elements of the characteristic sequences δ(ψ) and ε(ψ) of the system ψ, and δ0 = 0. (ψ)

Proof. First, let n > 1. The dimension of the subspace Φn−1 of the polynomials Φn−1 = ak ϕk (3.8) ψ k∈gn−1

is equal to δn−1 . Therefore, taking (3.3) into account, we get εn = En (ψUϕp ) ≥ dδn−1 (ψUϕp ) ≥ dδn−1 +1 (ψUϕp ) ≥ . . . ≥ dδn −1 (ψUϕp ). Hence, to prove equality (3.7), it remains to show that dδn −1 (ψUϕp ) ≥ εn , n = 1, 2, . . . .

(3.9)

For this purpose, we use the known theorem on the width of a ball. According to this theorem, if a set M of a linear normed space X with norm · X contains a ball γUν+1 of radius γ from a certain (ν + 1)-dimensional subspace Mν+1 of X, i.e., if M ⊃ γUν+1 = {y : y ∈ Mν+1 , yX ≤ γ}, then dν (M)X = inf

sup inf f − uX ≥ γ,

Fν ⊂Gν f ∈Mu∈Fν

where Gν is the set of all ν-dimensional subspaces of X. ψ Let εn Un,Φ be the intersection of a ball of radius εn in Sϕp with the space Φψ n (of dimension δn ) of polynomials of the form (3.8), i.e., ψ = {Φn ∈ Φ(ψ) εn Un,Φ n : Φn ≤ εn }.

(3.10)

ψ For the ψ-derivative Φψ n of an arbitrary element Φn ∈ εn Un,Φ , in view of (3.10) we have ak |ak |p 1 1/p ϕk = ( ) ≤ ( |ak |p )1/p ≤ 1. Φψ n = p ψ |ψ | ε n k k ψ ψ ψ k∈gn

k∈gn

k∈gn

ψ of the δn -dimensional subspace Therefore, Φn ∈ ψUϕp . Thus, the ball εn Un,Φ p p Φn (ψ) of Sϕ belongs to the class ψUϕ , which yields relation (3.9) by virtue of the theorem indicated above. Hence, the theorem is proved in the case n > 1. For n = 1, the proof remains the same if we assume that Φ0 (ψ) consists of the zero element θ and its dimension is equal to zero.

750

4.

Approximations in the Spaces S p

Chapter 11

Approximations of Individual Elements from the Sets ψSϕp 4.1. The following two theorems are the main ones in this section:

Theorem 4.1. Suppose that f ∈ ψSϕp , p > 0, and the sequence ψ = {ψk }∞ k=1 satisfies condition (2.3). Then the series ∞

(εpk − εpk−1 )Ekp (f ψ )ψ,p

k=1

converges and, for any n ∈ N, the following equality is true: Enp (f )ψ,p = εpn Enp (f ψ )ψ,p +

∞

(εpk − εpk−1 )Ekp (f ψ )ψ,p ,

(4.1)

k=n+1

where the quantities En (x)ψ,p are determined by equality (2.7) and εk , k = 1, 2, . . . , are elements of the characteristic sequence ε(ψ). Theorem 4.2. Suppose that f ∈ Sϕp , p > 0, and the sequence ψ = {ψk }∞ k=1 satisfies condition (2.3). Further, let lim ε−1 Ek (f )ψ,p k→∞ k

= 0.

(4.2)

Then, in order that the inclusion f ∈ ψSϕp

(4.3)

be true, it is necessary and sufficient that the series ∞

−p p (ε−p k − εk−1 )Ek (f )ψ,p

(4.4)

k=1

converge. If this series is convergent, then, for any n ∈ N, the following equality is true: p Enp (f ψ )ψ,p = ε−p n En (f )ψ,p +

∞

−p p (ε−p k − εk−1 )Ek (f )ψ,p ,

(4.5)

k=n+1

where the quantities En (x)ψ,p and εk have the same sense as in Theorem 4.1.

Section 4

Approximations of Individual Elements from the Sets ψSϕp

751

These theorems are analogs of the corresponding statements from Sections 6.2 and 6.3 for the spaces Sϕp . Theorem 4.1 establishes a relationship between the best approximation of an element f and the best approximations of its derivatives. In approximation theory, such statements (as already noted in Section 6.2) are called direct theorems. In this sense, Theorem 4.2 is an inverse theorem, namely, here, using the properties of the best approximation of an element f, one concludes about the existence of its derivatives and obtains information about the best approximations of these derivatives. 4.2. The quantities εn are strictly decreasing. Hence, relation (4.1) yields Enp (f )ψ,p ≤ εpn Enp (f ψ )ψ,p ∀f ∈ ψSϕp , ∀n ∈ N.

(4.6)

Note that, by virtue of (2.5) and Proposition 1.2, we always have lim Enp (f ψ )ψ,p = 0.

n→∞

On important subsets N of ψSϕp . relation (4.6) gives an exact result. Let us indicate one of such cases. According to the Jensen inequality, for any nonnegative sequence a = {ak }∞ k=1 , ak ≥ 0, we have (

∞ k=1

∞ apk )1/p ≤ ( aqk )1/q , 0 < q ≤ p.

(4.7)

k=1

Therefore, the following inclusions hold: Sϕq ⊂ Sϕp , 0 < q ≤ p,

(4.8)

ψUϕq ⊂ ψUϕp , 0 < q ≤ p.

(4.9)

and Taking this into account, we take the set ψUϕq for 0 < q ≤ p then f ψ ∈ Uϕq and, all the more, f ψ ∈ Uϕp . Hence, f ψ pp Enp (f ψ )ψ,p ≤ 1. Therefore,

as N. If f ∈ ψUϕq , ≤ 1. Consequently,

Enp (f )ψ,p ≤ εpn ∀f ∈ ψUϕq , 0 < q ≤ p.

(4.10)

ψ On the other hand, let k be an arbitrary point from the set gnψ \ gn−1 and let ψ ψ f∗ = ψk ϕk (ψk = 0). Since f∗ = ϕk , we have f∗ q = 1 for any q > 0. Consequently, f∗ ∈ ψUϕq for any q > 0. However, it is clear that

En (f∗ )ψ,p = f∗ ϕ,p = ψk = εn .

(4.11)

Approximations in the Spaces S p

752

Chapter 11

Thus, combining relations (3.10) and (3.11) and setting En (ψUϕq )ψ,p = sup En (f )ψ,p , En (ψUϕq )ψ,p = sup En (f )ψ,p , q f ∈ψUϕ

q f ∈ψUϕ

we arrive at the following statement, which generalizes equality (3.3): Theorem 4.3. Let ψ = {ψk }∞ k=1 be a system of numbers satisfying conditions (3.3) and (2.11). Then, for any n ∈ N and 0 < q ≤ p < ∞, the following equalities hold: En (ψUϕq )ψ,p = En (ψUϕq )ψ,p = εn , (4.12) where εn is the nth term of the characteristic sequence ε(ψ). 4.3. Proof of Theorem 4.1. Let ik , k = 1, 2, . . . , be natural numbers such that |ψik | = ψ¯k , where ψ¯k = εn for k ∈ (δn−1 , δn ], n ∈ N.

(4.13)

If f ∈ ψSϕp , then S[f ] =

∞

∞

fˆ(i)ϕi =

i=1

ψik fˆψ (ik )ϕik

k=1

and

δn−1

Sgψ (f ) = n−1

fˆ(ik )ϕik .

k=1

Therefore, ∞

Enp (f )ψ,p = Epn (f )ψ,p =

ψ¯kp |fˆψ (ik )|p

k=δn−1

=

∞ δ ν −1

ψ¯kp |fˆψ (ik )|p =

ν=n k=δν−1

∞

εpν Δν ,

(4.14)

ν=n

where Δν =

δ ν −1

|f ψ (ik )|p = (

k=δν−1

=

p Eν−1 (f ψ )ψ,p

Further, we use Lemma 6.2.1.

∞

−

k=δν−1

− Eν (f )ψ,p . ψ

∞

)|f ψ (ik )|p

k=δν

(4.15)

Section 4

Approximations of Individual Elements from the Sets ψSϕp

753

In the case considered, we set ck = Δk and λk = εpk . Since f ψ p ≤ 1, by virtue of (4.15) we have ∞

ck =

k=1

∞

Δk =

k=1

∞

|f ψ (k)|p = f ψ pp ≤ 1,

k=1

and since εn → 0 for n → ∞, we get lim εn An = 0, An =

n→∞

∞

Δk ,

k=n

i.e., all conditions of the lemma are satisfied. The inclusion ψSϕp ⊂ Sϕp yields the estimate En (f )ψ,p ≤ f ϕ,p , which guarantees the convergence of all series in (4.14). Therefore, according to the lemma, the series ∞

(εpk − εpk−1 )Ekp (f ψ )ψ,p

k=1

converges and equality (4.1) is true. Proof of Theorem 4.2. First, assume that inclusion (4.3) holds. Then the element f has the ψ-derivative f ψ , which belongs to Sϕp , and, moreover, according to (2.2), fˆ(k) fˆψ (k) = , k ∈ N. ψk In this case, choosing the numbers ik according to (4.13), we get S[f ψ ] =

∞

fˆψ (i)ϕi =

i=1

∞

(ψik )−1 fˆ(ik )ϕik

(4.16)

k=1

and

δn−1 ψ

Sgψ (f ) = n−1

(ψik )−1 fˆ(ik )ϕik .

k=1

Therefore, Enp (f ψ )ψ,p

=

Epn (f ψ )ψ,p

=

∞ k=1

ψ¯k−p |fˆ(ik )|p =

∞ ν=n

ε−p ν Δν ,

(4.17)

Approximations in the Spaces S p

754 where

δ ν −1

Δν =

Chapter 11

p |fˆ(ji )|p = Eν−1 (f )ψ,p − Eνp (f )ψ,p .

i=δν−1

Setting ck = Δk and λk = ε−p k , we obtain ∞

ck =

k=1

and λn An+1 = λn

∞

|fˆ(k)|p < ∞

k=1 ∞

p ck = ε−p n En (f )ψ,p .

(4.18)

k=n+1

Taking (4.2) into account, we conclude that all conditions of Lemma 6.2.1 are satisfied. The series in (4.17) is convergent. Hence, according to the lemma, series (4.4) also converges, and equality (4.5) is true. The necessity of the conditions of Theorem 4.2 is established. Let us show that they are sufficient. Assume that conditions (4.3) and (4.4) are satisfied and, as above, ck = Δk and λk = ε−p k . Since f ∈ Sϕp , by virtue of (4.17) we get ∞

ck =

k=1

∞ δ ν −1

|fˆ(ji )|p = f pp < ∞,

k=1 i=δk−1

i.e., series (6.2.4) converges; by virtue of (4.18), condition (6.2.4 ) is guaranteed by (4.2). By condition (4.4), the series on the right-hand side of (6.2.6) converges. Consequently, according to the lemma, the series on the left-hand side of (6.2.6) also converges, and equality (6.2.6) holds. However, in the case considered, by virtue of (4.17) we have ∞ k=n

λk ck =

∞

p ψ ε−p ν Δν = En (f )ψ,p , n = 1, 2, . . . ,

ν=n

where f ψ is an element whose Fourier series has the form (4.16). For n = 1, we have ∞ p ψ ψ p E1 (f ) = f p = λk ck < ∞, k=1

i.e., f ψ ∈ Sϕp , which proves the theorem.

Best n-Term Approximations

Section 5

5.

755

Best n-Term Approximations

5.1. In this section, we consider the quantities en (f )p , p > 0, determined by equality (2.9) in the case where the approximated element belongs to ψUϕq , 0 < q ≤ p. Namely, we consider the quantities en (ψUϕq )p = en (ψUϕq ; Sϕp ) = sup en (f )p q f ∈ψUϕ

= sup

inf f − Pγn p , 0 < q ≤ p.

q αk ,γn f ∈ψUϕ

(5.1)

As before, we assume that the systems of numbers ψ satisfy conditions (2.3). In this case, as already noted, ψUϕp ⊂ Sϕp . Therefore, if 0 < q ≤ p, then, by virtue of (4.8) and (4.9), we have the natural inclusion ψUϕq ⊂ ψUϕp ⊂ Sϕp and, consequently, quantity (5.1) is meaningful. The following statement is true: Theorem 5.1. Let ψ = {ψk }∞ k=1 be a system of numbers satisfying conditions (2.3) and (2.11) and let p and q be arbitrary numbers such that 0 < q ≤ p. Then, for any n ∈ N, the following equality holds: ∗

epn (ψUϕq )p where ψ¯ =

l l −p −p ψ¯k−q ) q = (l∗ − n)( ψ¯−q ) q , = sup(l − n)(

{ψ¯k }∞ k=1

l>n

k=1

k=1

is the sequence determined by the relations

ψ¯k = εn for k ∈ (δn−1 , δn ], n = 1, 2, . . . ,

(5.2)

where εn and δn are the terms of the characteristic sequences ε(ψ) and δ(ψ), and l∗ is a certain natural number. Proof. According to (1.2), if f ∈ Sϕp , then f − Pγn pp = |fˆ(k)|p + |fˆ(k) − αk |p ≥ |fˆ(k)|p ¯ γn k∈

= f pp −

¯ γn k∈

k∈γn

|fˆ(k)|p .

k∈γn

Thus, we conclude that epn (f ) = f pp − sup γn

k∈γn

|fˆ(k)|p .

Approximations in the Spaces S p

756

Chapter 11

Let ik , k = 1, 2, . . . , be arbitrary numbers determined by (4.13). Hence, by virtue of (5.2) and (2.2), for any element f ∈ ψUϕq we have epn (f )p

=

∞

ψ¯kp |fˆψ (ik )|p − sup γn

k=1

ψ¯kp |fˆψ (ik )|p

(5.3)

k∈γn

and, consequently, epn (ψUϕq )p

∞ p ψ¯kp |fˆψ (ik )|p − sup ψ¯k |fˆψ (ik )|p ) = sup ( q f ∈ψUϕ k=1

γn

k∈γn

∞ p p ψ¯kp mrk − sup ψ¯k mrk ), r = . ≤ sup ( q γn |m|≤1 k=1

(5.4)

k∈γn

5.2. In order to find the values of the right-hand side in (5.4) we use the following lemma: Lemma 5.1. Let α = {αk }, k = 1, 2, . . . , be a nonincreasing sequence of positive numbers (αk > 0 for all k ∈ N ) such that lim αk = 0,

(5.5)

k→∞

and let m = {mk }, k = 1, 2, . . . , be a sequence of nonnegative numbers (mk ≥ 0 for all k ∈ N ) such that |m| =

∞

mk ≤ 1

(5.6)

k=1

(in this case, we write α ∈ A and m ∈ M, respectively). Further, assume that r is an arbitrary number, r ≥ 1, and S (r) (m) =

∞

αk mrk ,

Sγ(r) (m) = n

k=1

αk mrk ,

k∈γn

where γn is an arbitrary collection of n natural numbers, (m) = S (r) (m) − sup Sγ(r) (m), En (m) = E(α,r) n n γn

and En = E(α,r) = sup E(α,r) (m). n n |m|≤1

(5.7)

Best n-Term Approximations

Section 5

757

Then, for any natural n, there exists a number l∗ > n such that ∗

l −1 En = (l − n)( αk r )−r . ∗

(5.8)

k=1

The number l∗ is determined by the equality ∗

l l − r1 −r −1 ∗ sup(l − n)( αk ) = (l − n)( αk r )−r . l>n

k=1

k=1

Moreover, for the sequence m = {mk }∞ k=1 , where ⎧ l∗ ⎪ ⎨ − r1 − r1 −r αk ( αi ) , k = 1, 2, . . . , l∗ , mk = i=1 ⎪ ⎩ 0, k > l∗ , the following equality holds: ∗

l 1 En (m ) = (l − n)( αk − r )−r .

∗

k=1

5.3. Assume that the lemma is proved. Setting ψ¯kp = αk , k ∈ N, we conclude that, according to (2.3) and (5.2), the numbers αk thus chosen satisfy the conditions of the lemma. Therefore, by virtue of (5.4) and (5.8), we have epn (ψUϕq )p

≤

En(α,r)

∗

∗

k=1

k=1

l l −1 −p ψ¯k−q ) q , = (l − n)( αk r )−r = (l∗ − n)( ∗

and, to complete the proof, it remains to show that the set ψUϕq contains an element f∗ such that l∗ −p p ∗ en (f∗ )p = (l − n)( ψ¯k−q ) q . (5.9) k=1

To this end, we set h=

∞ k=1

cik ϕik ,

758

Approximations in the Spaces S p

where the numbers ik are chosen according to (4.13) and ⎧ l∗ ⎪ ⎨ ¯−q ¯−q −1 ψk ( ψi ) , k = 1, 2, . . . , l∗ , cqik = i=1 ⎪ ⎩ 0, k > l∗ .

Chapter 11

(5.10)

The element h is a linear combination of finitely many elements ϕj , and, hence, it belongs to the spaces Sϕp for any p > 0. Since hqϕ,q

=

∞

cqik = 1,

k=1

we have h ∈ Uϕq . Therefore, setting f∗ = J ψ h, we conclude that f∗ ∈ ψUϕq and f∗ψ = h. By virtue of (5.10), ⎧ l∗ ⎪ ⎨ ¯−p ¯−q − pq ψk ( ψi ) , k = 1, 2, . . . , l∗ , cpik = k=1 ⎪ ⎩ 0, k > l∗ , and, consequently,

ψ¯kp cpik

⎧ l∗ ⎪ ⎨ ¯−q − pq ( ψi ) , k = 1, 2, . . . , l∗ , = ⎪ ⎩ k=1 0, k > l.

Therefore, in view of (5.3), equality (5.9) holds for epn (f∗ ), which completes the proof of the theorem. 5.4. Proof of Lemma 5.1. By virtue of (5.6), the series in (5.7) converges for any r ≥ 1. Hence, we always have αk mrk → 0 as k → ∞. Therefore, one can always find at least one set γn∗ = γn∗ (m, r) satisfying the condition (r) (m) = Sγn∗ (m) = αk mrk . sup Sγ(r) n γn

∗ k∈γn

Further, let μ = μn (m, r) = min∗ αk mrk . k∈γn

Let us prove the following statement:

Best n-Term Approximations

Section 5

759

Proposition 5.1. If α ∈ A, then, for any sequence m ∈ M, one can find a sequence ν ∈ M and number l > n such that |ν| = |m|, ⎧ μ, k = 1, 2, . . . , l, ⎪ ⎪ ⎨ αk νkr = λμ, k = l + 1, ⎪ ⎪ ⎩ 0, k > l + 1, where λ ∈ [0, 1), and the following inequality is true: E(α,r) (m) ≤ E(α,r) (ν). n n

(5.11)

The idea of the proof of Proposition 5.1 is based on the following three facts: Fact 1. If α ∈ A, m ∈ M, and r ≥ 1, then, for any natural k and s, s > k ≥ 1, the following inequality is true: αk (mk + ms )r ≥ αk mrk + αs mrs .

(5.12)

Indeed, relation (5.12) follows from the inequality ar + br ≤ (a + b)r , which is true for all a ≥ 0, b ≥ 0, and r ≥ 1 in view of the fact that αk ≥ αs . Fact 2. Let α ∈ A, m ∈ M, r ≥ 1, and s > k ≥ 1. Moreover, let αk mrk < αs mrs =

and ms = m ¯ s + ms , where the value of m ¯ s is defined by the condition αk mrk = αs m ¯ rs .

(5.13)

Then the following inequality holds: =

αk (mk + ms )r + αs m ¯ rs ≥ αk mrk + αs mrs . Indeed, according to (5.13) and (5.14), we must show that =

αk (mk + ms )r ≥ αs mrs or

1/r

=

=

αk (mk + ms ) ≥ αs1/r (m ¯ s + ms ).

(5.14)

Approximations in the Spaces S p

760

Chapter 11 1/r =

In view of (5.13), the last inequality reduces to the inequality αk which is true because α ∈ A.

=

ms ≥ αs1/r ms ,

Fact 3. Let α ∈ A, m ∈ M, r ≥ 1, s > k ≥ 1, and αk mrk > αs mrs .

(5.15)

αk (mk + x)r + αs (ms − x)r > αk mrk + αs mrs .

(5.16)

Then, for any x ∈ (0, ms ),

Proof. The derivative f (x) of the function f (x) = αk (mk + x)r + αs (ms − x)r − αk mrk − αs mrs , which is equal to αk r(mk + x)r−1 − αs r(ms − x)r−1 , is nonnegative for all x ∈ [0, ms ]. Indeed, according to (5.15), in view of the inequality αk ≥ αs , we have ms αk 1 αk 1 < ( ) r < ( ) r−1 mk αs αs or 1 1 αsr−1 ms < αkr−1 mk . Hence,

1

1

1

1

αsr−1 (ms − x) ≤ αsr−1 ms < αkr−1 mk ≤ αkr−1 (mk + x) and, consequently, αk (mk + x)r−1 − αs (ms − x)r−1 > 0. Thus, indeed, f (x) > 0 for all x ∈ (0, ms ), and, since f (0) = 0, we have f (x) > 0 for all x ∈ (0, ms ], which yields (5.16). Facts 1–3 imply that the sequence ν = {νk }, k ∈ N, can be constructed, e.g., as described below. The first step is as follows: If α1 mr1 < μ, then we denote by s1 the largest natural number (greater than 1) such that s 1 −1 A1 = α1 ( mi )r ≤ μ. i=1

In this case, ms1 > 0 and

s1 mi )r > μ. α1 ( i=1

(5.17)

Best n-Term Approximations

Section 5

761

Consider the sequence m(1) = {mk }∞ k=1 , where (1)

⎧s −1 1 ⎪ ⎪ ⎪ mi , k = 1, ⎪ ⎨

(1)

mk =

i=1

⎪ 0, ⎪ ⎪ ⎪ ⎩ mk ,

1 < k < s1 , k ≥ s1 .

Since α1 mr1 < μ, the numbers 1, 2, . . . , s1 − 1 are not included into γn∗ , and, thus, the terms αi mri with such numbers are included into the sum that determines the quantity En (m) : En (m) =

αk mrk .

∗ ¯ γn k∈

Therefore, in view of Fact 1 (see (5.12)), we conclude that En (m) ≤ En (m(1) ). There are two cases: A1 < μ

(5.18)

A1 = μ.

(5.19)

and If (5.18) holds, then we have the following two possibilities: A1 < αs1 mrs1

(5.20)

A1 ≥ αs1 mrs1 .

(5.21)

and Assume that condition (5.20) is satisfied. Then we represent the quantity ms1 in = ¯ s1 + ms1 , where m ¯ s1 is determined by the condition the form ms1 = m ¯ rs1 . A1 = αs1 m In this case, according to Fact 2 (see (5.14)), we have s 1 −1

α1 (

i=1

=

mi + ms1 )r + αs1 m ¯ rs1 ≥ A1 + αs1 mrs1 .

762

Approximations in the Spaces S p

Chapter 11

(1a) ∞ }k=1 ,

Hence, setting m(1a) = {mk

(1a)

mk

where ⎧ = (1) ⎪ m1 + ms , k = 1, ⎪ ⎪ ⎨ = m(1) , k= s1 , k ⎪ ⎪ ⎪ ⎩m ¯ s1 , k = s1 ,

we get En (m) ≤ En (m(1) ) ≤ En (m(1a) ).

(5.22)

Here, also two cases are possible: (1a) r

) αs1 m ¯ rs1 and using Fact 3 (see (5.16)),

En (m) ≤ En (m(1) ) ≤ En (m(1a) ) ≤ En (m(1b) ).

(5.25)

If, instead of (5.23), we have (5.24), then we set m(1b) = m(1a) . Now assume that condition (5.21) is satisfied together with (5.18). Then, in view of Fact 3, we find a number x , x ∈ (0, ms1 ), such that (1a)

α1 (m1 (1b ) ∞ }k=1

and define a sequence {mk

(1b )

mk

+ x )r = μ,

by setting

⎧ (1a) ⎪ + x , k = 1, m ⎪ ⎨ 1 = m1k , k = s1 , ⎪ ⎪ ⎩ ms1 − x , k = s1 .

Best n-Term Approximations

Section 5

763

It is clear that, in this case, the following inequality is also true:

En (m) ≤ En (m(1) ) ≤ En (m(1a) ) ≤ En (m(1b ) ).

(5.26)

If, instead of (5.18), we have (5.19), then we set m(1b ) = m(1) . Finally, if, from the beginning, we have α1 mr1 ≥ μ, then we set m(1b ) = m. These constructions show that, for any sequence m ∈ M, one can indicate a (1) sequence ν (1) = {νk }∞ k=1 from M such that

(1)

νk

⎧ (1) (1) ⎪ ν1 , α1 (ν1 )r ≥ μ, ⎪ ⎪ ⎪ ⎪ ⎨0, = ⎪ ⎪ ms 1 − y 1 , ⎪ ⎪ ⎪ ⎩ mk ,

k = 1, 1 < k < s1 , k = s1 , k > s1 ,

where y1 is a certain number from the interval [0, ms1 ] that depends on the combination of relations (5.18)—(5.21), (5.23), and (5.24) that is realized. This completes the first step of the construction of the sequence ν. The second step is to construct a sequence ν (2) ∈ M on the basis of the sequence ν (1) so that ⎧ (1) ⎪ k = 1, ⎪ ⎪ν1 , ⎪ ⎪ ⎪ ⎪ ⎪ν2(2) , α2 (ν2(2) )r ≥ μ, k = 2, ⎪ ⎨ (2) νk = 0, 2 < k < s2 , ⎪ ⎪ ⎪ ⎪ (1) ⎪ νs 2 − y 2 , k = s2 , ⎪ ⎪ ⎪ ⎪ ⎩ mk , k > s2 , where s2 > s1 , y2 is a certain number from the interval [0, ms2 ], and, moreover, En (m) ≤ En (ν (1) ) ≤ En (ν (2) ).

(5.27) (1)

It is clear that this can be done if, for the system of numbers νk , k ≥ 2, we repeat the reasoning used at the first step for the sequence m.

Approximations in the Spaces S p

764

Chapter 11

Continuing this procedure, at a certain step (say, at the jth step) we construct a sequence ν (j) = {νkj }∞ k=1 from M such that ⎧ (j−1) ⎪ νk , k = 1, 2, . . . , j − 1, ⎪ ⎪ ⎪ ⎪ ⎪ (j) (j) ⎪ ⎪ ν , αj (νj )r ≥ μ, k = j, ⎪ ⎨ j (j) νk = 0, j < k < sj , ⎪ ⎪ ⎪ ⎪ (j−1) ⎪ νsj − yj , k = sj , ⎪ ⎪ ⎪ ⎪ ⎩ mk , k > sj , where yj is certain number from the interval [0, mj ]. For this sequence, we have En (m) ≤ En (ν (1) ) ≤ . . . ≤ En (ν (j) )

(5.28)

and, moreover, αj ( ν (j) )r = αj (νs(j−1) − yj + mk )r < μ. j k≥j

k>j (j+1) ∞ }k=1 ,

At the next step, we set ν (j+1) = {νk

(j+1)

νk

where

⎧ (j) k = 1, 2, . . . , j, νk , ⎪ ⎪ ⎪ ⎨ (j) νk , k = j + 1, = ⎪ ⎪ ⎪ ⎩ k>j 0, k > j + 1.

Taking into account relations (5.22) and (5.25)—(5.28) and Fact 1, we conclude that |ν (j+1) | = |m|, i.e., ν (j+1) ∈ M, En (m) ≤ En (ν (j+1) ),

(5.29)

and, moreover, (j+1) r

αk (νk

(j)

(j+1)

) = αk (νk )r ≥ μ, k = 1, 2, . . . , j, αj+1 (νj+1 )r < μ.

It is also clear that the number j satisfies the condition j ≥ n. We now represent the quantity j μ (j+1) β= ((νkj+1 )r − ) + νj+1 αk k=1

Best n-Term Approximations

Section 5

765

in the form β = βj+1 + βj+2 + . . . + βj+l , βj+i ≥ 0, i = 1, l, where the numbers βs and l are chosen so that r = μ, i = 1, l − 1, αj+i βj+i r < μ. aj+l βj+l

We also set ν = {νk }∞ k=1 , where ⎧ (μ/αk )1/r , k = 1, 2, . . . , j + l − 1, ⎪ ⎪ ⎨ νk = βj+l , k = j + l, ⎪ ⎪ ⎩ 0, k > j + l. The sequence ν is the required one. To verify this, it suffices to set l = j + l − 1 r and λ = aj+l βj+l and note that, in view of relation (5.29), inequality (5.11) holds and |ν| = |m|. 5.5. Proposition 5.1 is proved. Let us proceed with the proof of the lemma. For a given natural n, we denote by Mn the subset of sequences m from M such that, for a certain natural l, l > n, the following representation holds: ⎧ μ, k = 1, 2, . . . , l, ⎪ ⎪ ⎨ r (5.30) αk mk = λμ, k = l + 1, λ ∈ [0, 1), ⎪ ⎪ ⎩ 0, k > l + 1, where μ is some positive number. Since the sequence ν constructed earlier belongs to Mn , it follows from (5.11) that E(α,r) = sup E(α,r) (m) = sup E(α,r) (m), (5.31) n n n m∈M

m∈Mn

(α,r)

which means that, in order to find the value of the quantity En consider sequences from Mn . If m ∈ Mn , then, according to (5.30), En(α,r) (m)

=

l k=n+1

μ + λμ = (l − n + λ)μ.

, it suffices to

(5.32)

Approximations in the Spaces S p

766

Chapter 11

Moreover, |m| =

l+1 k=1

Hence,

mk =

l l μ λμ 1/r λ 1/r −1/r ( )1/r + ( ) = μ1/r ( αk +( ) ). αk αl+1 αl+1 k=1

k=1

l λ 1/r −r −1 μ = |m|r ( αk r + ( ) ) . αl+1 k=1

Therefore, by virtue of (5.32), l λ 1 −r − r1 r E(α,r) (m) = (l − n + λ)|m| ( α )r ) . n k + (α l+1

(5.33)

k=1

For fixed r ≥ 1 and n ∈ N and natural l > n, we consider the functions l −1 αk r )−r f (l) = (l − n)( k=1

and

l λ 1 −r −1 f1 (l, λ) = (l − n + λ)( αk r + ( ) r ) , λ ∈ [0, 1]. αl+1 k=1

We see that f1 (l, 0) = f (l),

f1 (l, 1) = f (l + 1).

We have 1 l l ∂f1 (l, λ) λ 1/r −r−1 − r1 (l − n)λ r −1 − r1 αk + ( ) ) ( αk − ). =( 1/r ∂λ αl+1 αl+1 k=1 k=1

This implies that if r = 1, then this derivative preserves its sign for λ ∈ [0, 1]. Consequently, the function f1 (l, λ) either increases or decreases on this interval. In this case, f1 (l, λ) ≤ max(f (l), f (l + 1)) ∀λ ∈ (0, 1). (5.34) If r > 1, then the function f2 (l, λ) =

l k=1

−1 αk r

1

−

(l − n)λ r −1 1/r

αl+1

Best n-Term Approximations

Section 5

767

has at most one zero λ0 on the interval (0, 1). Moreover, lim f2 (l, λ) = −∞.

λ→0+0

Therefore, if λ0 ∈ (0, 1), then λ0 is a minimum point of the function f1 (l, λ). Consequently, inequality (5.34) is also true in this case. Relations (5.33) and (5.34) yield E(α,r) (m) ≤ |m| max(f (l), f (l + 1)) ∀m ∈ Mn . n Thus, in view of (5.31), we get E(α,r) n

l −1 ≤ sup f (l) = sup(l − n)( αk r )−r . l>n

l>n

(5.35)

k=1

For any r ≥ 1 and λ ∈ (0, 1), l > n, we have f2 (l, λ) =

n

−1 αk r

+

k=1

≤

n

l

(

k=n+1 −1 αk r

l

+

k=1

1

1

−

1/r

αk

− r1

(αk

λ r −1 1/r

)

αl+1 −1

r − αl+1 ).

(5.36)

k=n+1

The quantity l

− r1

(αk

−1

r − αl+1 )

k=n+1

is negative,and, by virtue of (5.5), its absolute value does not decrease and tends to infinity. Thus, relation (5.36) implies that there exists a value l0 such that, beginning with it, we have f2 (l, λ) < 0 ∀λ ∈ (0, 1].

(5.37)

Consequently, for l > l0 , the function f1 (l, λ) is strictly decreasing on the interval (0, 1). This implies that, for l > l0 , the function f (l) is also strictly decreasing. Hence, on the interval (n, l0 ], there exists a point l∗ such that ∗

l −1 sup f (l) = max f (l) = f (l ) = (l − n)( αk r )r . ∗

l>n

l∈(n,l0 ]

∗

k=1

Approximations in the Spaces S p

768 Therefore,

Chapter 11

∗

E(α,r) n

l −1 ≤ (l − n)( αk r )−r . ∗

(5.38)

k=1

To complete the proof of the lemma, it suffices to show that the strict inequality is impossible in this relation. To this end, we consider the sequence m = {mk }∞ k=1 , where ⎧ l∗ ⎪ ⎨ − r1 − r1 −1 αk ( αi ) , k = 1, 2, . . . , l∗ , mk = i=1 ⎪ ⎩ 0, k > l∗ . It is clear that |m | = 1 and m ∈ M. According to (5.32), for n < l∗ we get ∗

E(α,r) (m ) n

l 1/r = (l − n)( αk )−r . ∗

k=1

Combining this relation with (5.38), we complete the proof of all assertions of the lemma. 5.6. Remark 5.1. In the proof of Lemma 5.1, condition (5.5) is used only for obtaining relation (5.37). Therefore, all arguments of this proof up to obtaining relation (5.35) remain valid without assumption (5.5). In particular, inequality (5.35) is true if the sequence α takes arbitrary constant values c, c > 0. For c = 1 and r > 1, by virtue of (5.35) we have l l−n E(1,r) ≤ sup (l − n)( 1)−r = sup r , n l>n l>n l

(5.39)

k=1

(1,r)

(α,r)

where Ek = En The function

for αk ≡ 1.

x−n , x ≥ n, r > 1, xr attains its maximum value at the point ϕ(x) =

x0 =

rn . r−1

The number x0 is not necessarily an integer, and, thus, the value sup l>n

l−n lr

(5.40)

Best n-Term Approximations

Section 5

is obtained for the natural l∗ determined by the relation ⎧ [x0 ] − n [x0 ] + 1 − n ⎪ ⎪ [x0 ] if > , ⎪ ⎪ r ⎨ [x0 ] ([x0 ] + 1)r ∗ l = ⎪ ⎪ [x0 ] − n [x0 ] + 1 − n ⎪ ⎪ < , ⎩[x0 ] + 1 if [x0 ]r ([x0 ] + 1)r

769

(5.41)

Therefore, sup l>n

l−n l∗ − n [x0 ] − n [x0 ] + 1 − n = = max{ ; }. lr l∗ r [x0 ]r ([x0 ] + 1)r

(5.42)

Here, [d], is, as usual, the integer part of the number d. In this case, the extremal sequence mk takes the form ⎧1 ⎪ ⎨ ∗ , k = 1, 2, . . . , l∗ , l mk = ⎪ ⎩ 0, k > l∗ . According to (5.32) (for αk ≡ 1), for this sequence we get (m ) = E(1,r) n

l∗ − n , l∗ r

where l∗ is determined by (5.41), i.e., relation (5.39) is, in fact, an equality. Therefore, the following statement is true: Lemma 5.1. Suppose that m ∈ M, r is an arbitrary number, r > 1, En(1,r) (m)

=

∞

mrk − sup γn

k=1

mrk ,

k∈γn

where γn is an arbitrary collection of n natural numbers, and (m). En(1,r) = sup E(1,r) n m∈M

Then, for any natural n, the following equality holds: E(1,r) = n where l∗ is determined by (5.41).

l∗ − n , l∗ r

(5.43)

770

Approximations in the Spaces S p

Chapter 11

On the basis of this lemma, one can obtain the following statement: Theorem 5.1. Let p and q be arbitrary numbers such that 0 < q < p. Then, for any n ∈ N, the following equality is true: epn (Uϕq )p =

l∗ − n p , r= , l∗ r q

(5.44)

where en (Uϕq )p is the quantity defined by relation (5.1), i.e., the quantity of the best n-term approximation of the unit ball Uϕq in the space Sϕq in the metric of the space Sϕp , and l∗ is defined by (5.41). Proof. According to (5.1), (5.3), (2.10), and (5.43), we get epn (Uϕq )p

∞ = sup ( |fˆ(k)|p − sup |fˆ(k)|p ) q f ∈Uϕ k=1

γn

k∈γn

∞ mrk − sup mrk ) = E(1,r) , ≤ sup ( n γn

m∈M k=1

i.e., we always have

l∗ − n . l∗ r

epn (Uϕq )p ≤ On the other hand, let f∗ =

k∈γn

∞

ck ϕk ,

k=1

where the numbers ck are such that ⎧1 ⎪ ⎨ ∗ , k = 1, 2, . . . , l∗ , l q ck = ⎪ ⎩ 0, k > l∗ . It is clear that f∗ ∈ Uϕq and, by virtue of (5.3), epn (f∗ ) = which proves equality (5.44).

l∗ − n , l∗ r

Best n-Term Approximations (q > p)

Section 6

771

Remark 5.2. If the system ψ = {ψk }∞ k=1 is such that, for a certain given ψ ψ n ∈ N, the set gn \ gn−1 contains more than one point, then, by virtue of (3.7), dν (ψUϕp ) = εn ∀ν ∈ [δn−1 , δn − 1]. The quantities eν (ψUϕp )p always strictly decrease as the number ν increases, namely, q−ν−1 q−ν < sup q ≤ epν (ψUϕp ), q q>ν+1 −p q>ν+1 −p ψ¯ ψ¯

epν+1 (ψUϕp )p = sup

k

k

k=1

k=1

i.e., we always have eν+1 (ψUϕp ) < eν (ψUϕp ) and, moreover, eδn−1 (ψUϕp ) < εn = dδn−1 (ψUϕp ). Indeed, according to Theorem 5.1, q − δn−1

epδn−1 (ψUϕp )p = sup

q>δn−1

δn−1

ψ¯k−p +

q>δn−1

ψ¯k−p

k=δn−1 +1

k=1

< sup

q

q − δn−1 q − δn−1 ≤ sup q ¯−p q>δn−1 (q − δn−1 )ψ δn−1 +1 ψ¯k−p k=δn−1 +1

= ψ¯δpn−1 +1 = εpn . Thus, the inequality eν (ψUϕp )p < dν (ψUϕp ) is always true.

6.

Best n-Term Approximations (q > p)

6.1. In this section, we consider the quantities en (f )p , p > 0, defined by (2.9) in the case where the approximated element belongs to the set ψUϕq for q > p. Namely, we consider the quantities en (ψUϕq )p = en (ψUϕq ; Sϕp ) = sup en (f )p q f ∈ψUϕ

= sup

inf f − Pγn p , q > p > 0.

q αk ,γn f ∈ψUϕ

(6.1)

Approximations in the Spaces S p

772

Chapter 11

As before, we assume that the systems of numbers ψ satisfy condition (2.3) and, moreover, ∞ pq q−p ψl pq = ( |ψk | q−p ) pq < ∞. (6.2) q−p

k=1

This condition guarantees the inclusion ψUϕq ⊂ Sϕp .

(6.3)

Indeed, if f ∈ ψUϕq , then f ψ ∈ Uϕq and, according to (2.10), f ψ qq =

∞

|fˆkψ |q ≤ 1.

(6.4)

k=1

Consequently, by virtue of the Hölder inequality, f pp =

∞

∞ ∞ pq q−p p |ψk |p |fˆψ (k)|p ≤ ( |ψk | q−p ) q ( |fˆψ (k)|q ) q < ∞.

k=1

k=1

k=1

In what follows, we show that condition (6.2) is also necessary for the validity of (6.3). The following statement is true: Theorem 6.1. Let ψ = {ψk }∞ k=1 be a system of numbers satisfying conditions (2.3) and (2.11) and let p and q be arbitrary numbers such that q > p > 0. Then, for any n ∈ N, the following equality is true: −p

q

p

epn (ψUϕq )p = σ ¯1 q [(s − n) q−p + σ ¯1q−p σ ¯2 ] where ¯1 (s) = σ ¯1 = σ

s

ψ¯k−q , σ ¯2 = σ ¯2 (s) =

k=1

q−p q

∞

,

(6.5) pq

ψ¯kq−p ,

(6.6)

k=s+1

¯ ψ¯ = {ψ¯k }∞ k=1 is a sequence for which ψk = εn for k ∈ (δn−1 , δn ], εn and δn are terms of the characteristic sequences ε(ψ) and δ(ψ), and the number s is chosen from the condition 1 ¯−q −q ψk < ψ¯s+1 . s−n s

ψ¯s−q ≤

k=1

Such a number s always exists and is unique.

(6.7)

Best n-Term Approximations (q > p)

Section 6

773

6.2. Proof. This proof coincides in the form with the proof of Theorem 5.1. The difference lies in the formulation of the corresponding lemmas for numerical series (Lemmas 5.1 and 6.1). This can be seen from the following reasoning: According to (1.2), if f ∈ Sϕp and Pγn is a polynomial of the form (2.8), then f − Pγn pp =

|fˆ(k)|p +

k∈γn

k∈γn

≥

|fˆ(k) − αk |p

|fˆ(k)|p = f pp −

|fˆ(k)|p .

k∈γn

k∈γn

Hence, epn (f )p = f pp − sup γn

|fˆ(k)|p .

(6.8)

k∈γn

Let ik , k = 1, 2, . . . , be natural numbers such that |ψik | = ψ¯k , where ψ¯k = εn for k ∈ (δn−1 , δn ], n ∈ N.

(6.9)

Then, in view of (6.9) and (2.2), for any element f ∈ Uϕq we have epn (f )p =

∞

ψ¯kp |fˆψ (ik )|p − sup γn

k=1

ψ¯kp |fˆψ (ik )|p

k∈γn

and, consequently, ∞ p ψ¯kp |fˆψ (ik )|p − sup ψ¯k |fˆψ (ik )|p ) epn (f )p = sup ( q f ∈ψUϕ k=1

γn

k∈γn

∞ p p ψ¯kp mrk − sup ψ¯k mrk ), r = , ≤ sup ( q γn |m|≤1 k=1

(6.10)

k∈γn

where m = {mk }∞ k=1 , mk ≥ 0, and |m| =

∞

mk .

(6.11)

k=1

In order to find the value of the right-hand side of (6.10), we use the following lemma:

774

Approximations in the Spaces S p

Chapter 11

Lemma 6.1. Let α = {αk }∞ k=1 be a nonincreasing sequence of positive numbers (αk > 0 ∀k ∈ N ) such that, for a given r ∈ (0, 1), ∞

1

αk1−r < ∞

(6.12)

k=1

and let m = {mk }∞ k=1 be a sequence of nonnegative numbers (mk ≥ 0 ∀k ∈ N ) such that ∞ |m| = mk ≤ 1 (6.13) k=1

(in this case, we write α ∈ Ar and m ∈ M, respectively). Also assume that ∞ αk mrk , Sγ(r) (m) = αk mrk , (6.14) S (r) (m) = n k=1

k∈γn

where γn is a certain collection of n natural numbers, En (m) = E(α,r) (m) = S (r) (m) − sup Sγ(r) (m), n n γn

and En = E(α,r) = sup E(α,r) (m). n n

(6.15)

|m|≤1

Then, for any natural n, the following relation is true: 1

r

En = En (α, r) = σ1−r (s)[(s − n) 1−r + σ11−r (s)σ2 (s)]1−r ,

(6.16)

where σ1 (s) = σ1 (α; s) =

s

−1

αk r ,

σ2 (s) = σ2 (α; s) =

k=1

∞

1

αk1−r ,

k=s+1

and the number s is chosen from the condition − r1

σ1 (s) − r1 , s > n. (6.17) < αs+1 s−n Such a number always exists and is unique. The upper bound in (6.15) is realized by the sequence m = {mk }∞ k=1 , where ⎧ ts 1/r ⎪ ⎪ ) , k = 1, 2, . . . , s, ( ⎪ ⎪ ⎪ αk ⎪ ⎪ ⎨ 1/r 1 mk = 1 − ts σ1 (s) 1−r α ⎪ k , k = s + 1, s + 2, . . . , ⎪ ⎪ σ2 (s) ⎪ ⎪ ⎪ 1 σ1 (s) 1−r ⎪ ⎩ ts = (σ1 (s) + ( ) σ2 (s))−r . s−n αs

≤

Best n-Term Approximations (q > p)

Section 6

775

6.3. Assume that the lemma is proved. Setting ψ¯kp = αk , k ∈ N, we conclude that, by virtue of (2.3) and (6.2), the numbers αk thus chosen satisfy the conditions of the lemma. Therefore, according to (6.10) and (6.16), we have r

1

epn (ψUϕq )p ≤ En (α, r) = σ1−r (s)[(s − n) 1−r + σ11−r (s)σ2 (s)]1−r ,

(6.18)

where r = p/q, σ1 = σ1 (s) =

s

− r1

αk

=

k=1

and σ2 = σ2 (s) =

∞

s

ψ¯k−q = σ ¯1 (s),

(6.19)

k=1

1 1−r

αk

=

k=s+1

∞

pq

ψ¯kq−p = σ ¯2 (s).

(6.20)

k=s+1

The number s is chosen from the condition 1 ¯−q −q ψk < ψ¯s+1 . s−n s

ψ¯s−q ≤

(6.21)

k=1

Such a number s always exists and is unique. Comparing relations (6.18)–(6.21) and (6.5)–(6.7), we conclude that, to complete the proof of the theorem, it remains to show that the strict inequality in (6.18) is impossible. To this end, we verify that the set ψUϕq contains an element fε such that −p/q

epn (fε ) > σ ¯1

q

p

(s)[(s − n) q−p + σ ¯1q−p (s)¯ σ2 (s)]

for any ε > 0. Assume that h=

∞

q−p q

−ε

(6.22)

cik ϕik ,

k=1

where the numbers ik are chosen according to (6.9) and ⎧ ts 1/r ⎪ ⎪ ( ) , k = 1, 2, . . . , s, ⎪ ⎪ αk ⎪ ⎪ ⎨ 1/r 1 1 − ts σ1 (s) 1−r cqik = k = s + 1, s + 2, . . . , αk , ⎪ σ (s) 2 ⎪ ⎪ ⎪ 1 σ1 (s) 1−r p ⎪ ⎪ ) σ2 (s))−r , r = . ⎩ts = (σ1 (s) + ( s−n q

(6.23)

Approximations in the Spaces S p

776 Then

hqϕ,q =

∞

Chapter 11

cqik = 1.

(6.24)

k=1

Further, let ε be an arbitrary positive number and let Nε be a sufficiently large natural number such that ∞ 1/r 1 (1 − ts σ1 (s))r 1−r α < ε. (6.25) k σ2r (s) k=Nε +1

In this case, the element hε =

Nε

cik ϕik

k=1

is a linear combination of finitely many elements from ϕ. Hence, it belongs to all spaces Sϕp for any p > 0, and, since, hε qϕ,q ≤ 1, we have hε ∈ Uϕq . Therefore, setting fε = J ψ hε , we conclude that fε ∈ ψUϕq and fεψ = hε . According to (6.23), we have ⎧ ts ⎪ ⎪ , k = 1, 2, . . . , s, ⎪ ⎪ ⎨ αk cpik = 1/r ⎪ r ⎪ (1 − ts σ1 (s))r 1−r ⎪ ⎪ α ⎩ k , k = s + 1, s + 2, . . . . σ2r (s) Consequently,

ψ¯kp cpik =

⎧ ⎪ ts , ⎪ ⎪ ⎨

k = 1, 2, . . . , s,

1/r 1 ⎪ (1 − ts σ1 (s))r 1−r ⎪ ⎪ α ⎩ k , k = s + 1, s + 2, . . . . σ2r (s)

It follows from the results presented below (see, e.g., the proof of the Proposition 7.11) that the numbers ψ¯k cik , k = 1, 2, . . . , do not increase. Thus, by virtue of (6.8), we obtain epn (fε )p

Nε 1/r 1 (1 − ts σ1 (s))r 1−r = (s − n)ts + α k σ2r (s) k=s+1

r 1−r = (s − n)ts + (1 − t1/r s σ1 (s)) σ2 (s)

−

1/r

(1 − ts σ1 (s))r σ2r (s)

∞ k=Nε +1

1

αk1−r .

Section 7

Proof of Lemma 6.1

777

Hence, in view of (6.24) and (6.25), we get r 1−r epn (fε )p ≥ (s − n)ts + (1 − t1/r s σ1 (s)) σ2 (s) − ε.

Substituting the value of ts from (6.23) into the right-hand side and using equalities (6.19) and (6.20), we obtain relation (6.22), whence, in view of the arbitrariness of ε, the required result follows.

7.

Proof of Lemma 6.1

7.1. First, note that condition (6.12) is sufficient (and necessary, as shown in what follows) for series in (6.14) to be convergent for any m ∈ M. Thus, we always have (7.1) lim αk mrk = 0 k→∞

and, consequently, one can always find at least one set γn∗ = γn∗ (m) = γn∗ (m, r) such that (r) sup Sγ(r) (m) = S (m) = αk mrk . (7.2) ∗ γ n n γn

∗ k∈γn

Our next purpose is to restrict the range of sequences m ∈ M for which we must (α,r) examine the upper bounds of the quantities En (m) in order to find En . First, let M be the subset of sequences m ∈ M for which ∞

mk = 1.

(7.3)

k=1

Then (m). En = sup E(α,r) n

(7.4)

m∈M

Indeed, let m ∈ M. Then En (m) =

∞

αk mrk −

αk mrk .

∗ (m) k∈γn

k=1

Consider a sequence m = {mk }∞ k=1 such that mk

=

mk ,

k ∈ γn∗ (m),

∗ m+ k , k∈γn (m),

Approximations in the Spaces S p

778

Chapter 11

+ where m+ k are such that mk ≥ mk and ∞

mk = 1.

k=1

It is clear that, in this case, we have En (m) ≤ En (m ), and, since m ∈ M, relation (7.4) is indeed true. 7.2. The following statement is not as trivial: Proposition 7.1. Let Mα be a subset of M such that the numbers αk mrk do not increase. Then En = sup En (m). (7.5) m∈Mα

Proof. Let m be an arbitrary sequence from M and let s be a natural number for which sup αk mrk = αs mrs . (7.6) k≥1

If α1 mr1 < αs mrs , −

(7.7)

−

=

then we set ms =ms + ms , where ms is such that − r

α1 mr1 = αs ms .

(7.8)

Consider a sequence ν1 = {ν1,k }∞ k=1 for which ⎧ = ⎪ m1 + ms , k = 1, ⎪ ⎪ ⎨ − ν1,k = m k = s, s, ⎪ ⎪ ⎪ ⎩ mk , k = 1, k = s.

(7.9)

Then r + αs m ¯ rs ≥ α1 mr1 + αs mrs . α1 ν1,1

Indeed, according to (7.7) and (7.9), we must show that =

α1 (m1 + ms )r ≥ αs mrs , or

1/r

=

−

=

α1 (m1 + ms ) ≥ αs1/r (ms + ms ).

(7.10)

Section 7

Proof of Lemma 6.1

779 1/r =

In view of (7.8), the last inequality reduces to the inequality α1 which is true for any α ∈ Ar . With regard for (7.7), relation (7.9) yields r α1 ν1,1 ≥ αs mrs .

=

ms ≥ αs1/r ms ,

(7.11)

It follows from (7.6) that s ∈ γn∗ (m). Therefore, if we also have 1 ∈ γn∗ (m), then, by virtue of (7.8) and (7.11), we get 1 ∈ γn∗ (ν1 ), s ∈ γn∗ (ν1 ), and, consequently, En (m) = En (ν1 ). (7.12) If 1∈γn∗ (m), then, by virtue of (7.11), we have 1 ∈ γn∗ (ν1 ), and, by virtue of (7.8), s∈γn∗ (ν1 ). Hence, relation (7.12) remains valid. If, instead of (7.7), we have α1 mr1 = αs mrs , then we set ν1 = m. Thus, for an arbitrary sequence m ∈ M , one can construct a sequence ν1 ∈ M for which (7.12) is true and r r sup αk ν1,k = α1 ν1,1 . (7.13) k≥1

If n ≥ 2, then, on the basis of the sequence ν1 , acting by analogy with the construction of ν1 , we construct a sequence ν2 ∈ M for which ν2,1 = ν1,1 , r r r ≥ α2 ν2,2 = sup αk ν2,k , α1 ν2,1 k≥2

and En (ν2 ) = En (ν1 ) = En (m).

(7.14)

Continuing this procedure, at the nth step we obtain a sequence νn ∈ M for which νn,i = νi,i , i = 1, 2, . . . , n − 1, r r r r ≥ α2 νn,2 ≥ . . . ≥ αn νn,n = sup αk νn,k , αn νn,1

(7.15)

En (νn ) = En (νn−1 ) = . . . = En (ν1 ) = En (m).

(7.16)

k≥n

and, moreover,

The next (n + 1)th and the subsequent steps are carried out by analogy with the previous ones with regard for the fact that the numbers s chosen according to an analog of equality (7.6), namely r r = αs νn+i−1,s , i = 1, 2, . . . , sup αk νn+i−1,k

k≥n+i

Approximations in the Spaces S p

780

Chapter 11

no longer belong to the set γn∗ (νn+i−1 ) and, hence, belong to the sums that determine the quantities ∞

En (νn+i−1 ) =

r αk νn+i−1,k .

k=n+1 r As a result, we obtain a sequence ν = {νk }∞ k=1 for which the numbers αk νk do α not increase and ν ∈ M , i.e., ν ∈ M . At the same time,

En (m) ≤ En (ν), which proves Proposition 7.1. 7.3. Proposition 7.2. Let Mαn be a subset from Mα such that the quantities αk mrk for k = 1, 2, . . . , n are equivalent: αk mrk = a, k = 1, 2, . . . , n,

(7.17)

where a is certain (positive) number. Then En = sup En (m). m∈Mα n

(7.18)

Proof. Let m be an arbitrary sequence from Mα . We set a = αn mrn

(7.19)

and define numbers βk , k = 1, 2, . . . , n − 1, by the equalities αk (mk − βk )r = a, k = 1, 2, . . . , n − 1, i.e.,

a 1/r ) , k = 1, 2, . . . , n − 1. αk Further, we represent the quantity βk = mk − (

B=

n−1

βk

k=1

in the form B=

∞ k=n+1

ξk ,

(7.20)

Section 7

Proof of Lemma 6.1

781

where the quantities ξk are chosen so that the numbers αk (mk + ξk )r , k = n + 1, n + 2, . . . , decrease and the relation αn+1 (mn+1 + ξn+1 )r ≤ a holds. It is clear that such numbers ξk always exist. Then we set m = {mk }∞ k=1 , where ⎧ ⎨mk − βk , k = 1, 2, . . . , n − 1, mk = mn , k = n, ⎩ mk + ξk , k > n. By construction, m ∈ Mαn and En (m) ≤ En (m ). By virtue of (7.5), this yields equality (7.18). Note that, for m ∈ Mαn , the greatest value a of the quantities a is determined by equalities (7.19) under condition (7.3), which implies that n −1 a=( αk r )−r = σ1−r (n). k=1

Therefore, for any m ∈ Mαn , the admissible values of a satisfy the condition 0 < a ≤ σ1−r (n).

(7.21)

Further, consider the extremal problem fn (x) =

∞

αk xrk −→ sup

(7.22)

k=n+1

under the following conditions: α ∈ Ar , 0 < r < 1, xk ≥ 0,

∞

xk = β,

(7.23) (7.24)

k=n+1

αk xrk

≤ a, a > 0, k = n + 1, n + 2, . . . .

(7.25)

Approximations in the Spaces S p

782

Chapter 11

A solution of this problem (problem (7.22)–(7.25)) is understood as a sequence x ¯=x ¯(a) = x ¯n+1 , x ¯n+2 , . . .

(7.26)

x), sup{fn (x) : x ∈ D} = fn (¯

(7.27)

for which where D is the set of sequences satisfying conditions (7.24) and (7.25). On the basis of Proposition 7.2 and relation (7.21), for every fixed a ∈ (0, σ1−r (n)] we denote by Mαn (a) the set of sequences m ∈ Mαn satisfying condition (7.17). Then Mαn =

∪

a∈(0,σ1−r (n)]

Mαn (a),

(7.28)

and, consequently, in view of (7.18), En ≤

sup

sup

a∈(0,σ1−r (n)] m∈Mα n (a)

En (m).

(7.29)

Proposition 7.3. The following equality holds: sup

m∈Mα n (a)

En (m) = En (m), ¯

(7.30)

where the sequence m ¯ = m(a) ¯ = {m ¯ k }∞ k=1 is such that αk m ¯ rk = a, k = 1, 2, . . . , n, a ∈ (0, σ1−r (n)],

(7.31)

and the elements m ¯ k for k ≥ n + 1 are solutions of problem (7.22)–(7.25) for the given value a and β = 1 − a1/r σ1 (n). (7.32) Indeed, if m ∈ Mαn (a), then, according to the definition of a solution of problem (7.22)–(7.25), we have En (m) =

∞

αk mrk ≤ fn (¯ x) = En (m). ¯

(7.33)

k=n+1

Hence, relation (7.30) is a consequence of estimate (7.33) and the inclusion m ¯ ∈ α Mn (a). Moreover, this implies that (7.29) is, in fact, an equality. Consequently, En =

sup a∈(0,σ1−r (n)]

En (m(a)). ¯

(7.34)

Section 7

Proof of Lemma 6.1

783

Further, we need the explicit form of the solution of problem (7.22)–(7.25) for the value β defined by equality (7.32). To this end, we first consider this problem for n = 0 and β = c, where c is an arbitrary positive number, i.e., we consider the problem ∞ f0 (x) = f (x) = αk xrk −→ sup (7.35) k=1

under the conditions α ∈ Ar , 0 < r < 1, xk ≥ 0,

∞

(7.36)

xk = c, c > 0,

(7.37)

k=1

αk xrk ≤ a, a > 0, k = 1, 2, . . . .

(7.38)

7.4. To obtain a solution of problem (7.35)–(7.38), we first establish several auxiliary facts. With any number q ∈ N, we associate a number aq that is a solution of the equation 1

βq−1 (aq )αqr(1−r) = a1/r q , σ2 (q − 1) βs (a) = c − a

1/r

(7.39)

σ1 (s), β0 (a) ≡ c, σ1 (s) =

s

−1/r

αk

,

k=1

σ2 (s) =

∞

1

αk1−r , 0 < r < 1,

k=s+1

where αk are elements of the sequence α ∈ Ar , r > 0. In this case, 1

a1 = cr α11−r σ2−r (0), σ2 (0) =

∞

1

αk1−r ,

(7.40)

, q = 2, 3, . . . ,

(7.40 )

k=1 1 − r(1−r) −r

aq = cr (σ1 (q − 1) + σ2 (q − 1)αq

)

and, in particular, r

a2 = c

−1 (α1 r

+

− 1 α2 r(1−r)

∞ k=2

1

αk1−r )−r .

(7.41)

Approximations in the Spaces S p

784

Chapter 11

Proposition 7.4. Let q ∈ N. Then aq = aq+1 if

αq = αq+1 ,

(7.42)

aq > aq+1 if

αq > αq+1 .

(7.43)

Moreover, lim aq = 0.

(7.44)

q→∞

Proof. Since

− r1

σ1 (q − 1) = σ1 (q) − αq and

(7.45)

1

σ2 (q − 1) = σ2 (q) + αq1−r ,

(7.46)

we have σ1 (q − 1) + σ2 (q − 1)αq−γ = σ1 (q) + σ2 (q)αq−γ , γ =

1 . r(1 − r)

(7.47)

Consequently, if q ≥ 2, then, according to (7.40 ), −γ −1 −γ −1 − c(σ1 (q) + σ2 (q)αq+1 ) . a1/r q − aq+1 = c(σ1 (q) + σ2 (q)αq ) 1/r

This yields relations (7.42) and (7.43) for all q ≥ 2. If q = 1, then these relations follow from (7.40) and (7.41). Since lim σ1 (q) = ∞, (7.48) q→∞

equality (7.44) follows from (7.40 ). Also note that, in view of (7.40 ) and (7.47), aq = cr (σ1 (q) + σ2 (q)αq−γ )−r , γ =

1 , q ≥ 2. r(r − 1)

(7.49)

7.5. Let Jq = {a : aq < a ≤ aq+1 }. If aq = aq+1 , then Jq is an empty set. If aq > aq+1 , then Jq is the interval (aq , aq+1 ]. Proposition 7.4 yields ∪ Jq =

q≥1

∪

aq =aq+1

Jq = (0, a1 ).

Section 7

Proof of Lemma 6.1

785

In what follows, a ∈ (aq , aq+1 ] means that a belongs to the set Jq if aq > aq+1 and, consequently, αq > αq+1 . Proposition 7.5. If, for certain q ∈ N and r ∈ (0, 1), a ∈ (aq+1 , aq ],

(7.50)

then both following inequalities are true: βq−1 (a)αqγ ≥ a1/r σ2 (q − 1) and

γ βq (a)αq+1

σ2 (q)

< a1/r , γ =

1 , β0 (a) ≡ c. r(1 − r)

(7.51)

(7.52)

Proof. If a ∈ (aq+1 , aq ] and q ≥ 2, then, according to (7.40 ), −γ −1 c(σ1 (q) + σ2 (q)αq+1 ) < a1/r ≤ c(σ1 (q − 1) + σ2 (q − 1)αq−γ )−1 .

(7.53)

Therefore, c−

cσ1 (q − 1) ≤ βq−1 (a) = c − a1/r σ1 (q − 1) σ1 (q − 1) + σ2 (q − 1)αq−γ 0, relations (7.51) and (7.52) are true simultaneously. Since a increases, the values βs (a) do not increase and inequalities (7.51) and (7.52) remain valid up to the value of a determined by the equality βq−1 (a)αqγ = a1/r , σ2 (q − 1) i.e., up to aq inclusive. Since a decreases, the values βs (a) do not decrease and inequalities (7.51) and (7.52) remain valid up to the value of a equal to aq+1 . This means that a ∈ (aq+1 , aq ]. Therefore, the following statement is true: Proposition 7.5. Let α ∈ Ar , r > 0, and q ∈ N. In order that inclusion (7.50) be valid, it is necessary and sufficient that inequalities (7.51) and (7.52) be true. 7.6. In the notation accepted, we prove the following statement:

Section 7

Proof of Lemma 6.1

787

Proposition 7.6. Suppose that, for given α ∈ Ar and a > 0 and some q ∈ N, q ≥ 2, the following inequality is true: βq−1 (a)αqγ ≤ a1/r . σ2 (q − 1) Then

γ βq (a)αq+1

σ2 (q)

≤ a1/r .

(7.55)

(7.56)

Proof. In view of (7.39), we have βq (a) = βq−1 (a) − (

a 1/r ) αq

(7.57)

and, by virtue of (7.55), βq−1 (a) ≤ a1/r σ2 (q − 1)αq−γ . Hence, βq (a) ≤ (

− 1 a 1/r ) (σ2 (q − 1)αq 1−r − 1) αq 1

= a1/r αq−γ (σ2 (q − 1) − αq1−r ) = a1/r αq−γ σ2 (q). Consequently, since α ∈ Ar , r ∈ (0, 1), we get γ βq (a)αq+1

σ2 (q)

≤ a1/r (

αq+1 γ ) ≤ a1/r . αq

The statement proved yields the following result: Corollary 7.1. If, for certain q ∈ N, q ≥ 1, the inequality γ βq (a)αq+1

σ2 (q)

≥ a1/r

(7.58)

holds, then, for such q and a, the following relation is true: βq−1 (a)αqγ ≥ a1/r . σ2 (q − 1)

(7.59)

Indeed, if relation (7.59) is not true, then (7.55) holds and, as proved above, we obtain (7.56) instead of (7.58).

Approximations in the Spaces S p

788

Chapter 11

Proposition 7.6 and Corollary 7.1 yield the following result: Corollary 7.2. If, for certain natural q = q0 , q0 ≥ 1, relation (7.55) holds, then it also holds for all q > q0 ; if, for some q1 ∈ N, we have (7.58), then this inequality is also true for all q ∈ [1, q1 ]. Remark 7.1. Suppose that, for given α ∈ Ar , the number a1 is determined by (7.40) and a ∈ (0, a1 ). Denote by qa the least natural number that satisfies (7.56): βqa (a)αqγa +1 (7.60) < a1/r . σ2 (qa ) Since lim βq (a) = −∞ ∀a > 0,

q→∞

(7.61)

such a number exists. According to the choice of qa , βqa −1 (a)αqγa ≥ a1/r . σ2 (qa − 1)

(7.62)

By virtue of Corollary 7.2, in this case we have γ βq (a)αq+1

σ2 (q) and

< a1/r ∀q ≥ qa

βq−1 (a)αqγ ≥ a1/r ∀q = 1, 2, . . . , qa . σ2 (q − 1)

Thus, qa can be uniquely determined as a natural number that satisfies both inequalities (7.60) and (7.62). Therefore, a ∈ (aqa +1 , aqa ]. 7.7. Let us return to problem (7.35)–(7.38). First, consider its particular case where conditions (7.38) are absent, i.e., consider problem (7.35)–(7.37). Problem (7.35)–(7.37) is a problem of convex programming, to which we can apply the well-known Kuhn–Tucker theorem in the subdifferential form. As a result, we obtain its solution 1

cα 1−r x ¯ = {¯ xk }∞ ¯k = k , k = 1, 2, . . . . k=1 , x σ2 (0)

(7.63)

Section 7

Proof of Lemma 6.1

789

However, the theorem indicated is rather general and, to verify its conditions and obtain solution (7.63), one needs a series of definitions and auxiliary statements. For this reason, here we find solution (7.63) of problem (7.35)–(7.37) in a simpler way. Consider a finite-dimensional analog of problem (7.35)–(7.37), i.e., fn (x) =

n

αk xrk −→ sup, n ∈ N,

(7.64)

k=1

with the conditions α ∈ Ar , 0 < r < 1, xk ≥ 0,

n

(7.65)

xk = c, c > 0.

(7.66)

k=1

The Lagrange function of this problem has the form n L(x) = fn (x) + λ( xk − c) k=1

and, since ∂L = rαk xr−1 + λ, k = 1, 2, . . . , n, k ∂xk the corresponding critical points are determined from the system ⎧ r−1 + λ = 0, ⎪ k xk ⎨rα n k = 1, 2, . . . , n. ⎪ ⎩ xk = c, k=1

Thus, 1 λ 1 xk = (− ) r−1 αk1−r , k = 1, 2, . . . , n, r

and

n 1 λ 1 1−r (− ) r−1 αk = c. r k=1

Therefore, the solution of problem (7.64)–(7.66) has the form 1

(n) xk

cα 1−r = k , k = 1, 2, . . . , n, σ3 (n)

σ3 (n) =

n k=1

1

αk1−r .

(7.67)

Approximations in the Spaces S p

790

Chapter 11

Hence, under conditions (7.37), the nth-order partial sum Sn (x) of the series in (7.35) satisfies the estimate Sn (x) =

n

αk xrk

≤

k=1

n

1 cr 1−r = r αk = cr σ31−r (n). (7.68) σ3 (n)

n

(n) αk (xk )r

k=1

k=1

By virtue of conditions (7.36), the series in (7.35) and the series ∞

1

αk1−r

k=1

converge. Hence, according to (7.68), we always have f (x) ≤ cr σ21−r (0), σ2 (0) =

∞

1

αk1−r .

(7.69)

k=1

However, f (¯ x) =

∞

αk x ¯rk = cr σ21−r (0).

(7.70)

k=1

Comparing (7.69) and (7.70), we conclude that the sequence x ¯ is, in fact, a solution of problem (7.35)–(7.37). Note that the intermediate equality in (7.68) yields the necessity of condition (6.12) for the series in (6.14) to be convergent for any m ∈ M. This implies that condition (6.2) is necessary for the validity of (6.3). In view of (7.63), for any k ∈ N we have 1

1

¯rk = cr αk1−r σ2−1 (0) ≤ cr α11−r σ2−1 (0) = a1 , αk x

(7.71)

where the value a1 is determined by equality (7.40). Therefore, condition (7.38) for a ≥ a1 , in fact, does not impose any restrictions. Hence, the solution of problem (7.35)–(7.38) is determined by (7.63). The case a ∈ (0, a1 ) is more complicated. For definiteness, let a ∈ (aq+1 , aq ], where q is some natural number. Denote by D the set of sequences x that satisfy conditions (7.37) and (7.38) for α ∈ Ar . If x ∈ D, then f (x) =

∞ k=1

αk xrk ≤

q k=1

αk xrk + sup

∞

x∈Dk=q+1

αk xrk ≤ F (x1 , . . . , xq ),

(7.72)

Section 7

Proof of Lemma 6.1

791

where F (x1 , . . . , xq ) =

q

αk xrk + sup{

k=1

∞

αk xrk :

k=q+1

∞

xk = c −

k=q+1

q

xk }. (7.73)

k=1

Therefore, sup f (x) ≤ sup F (x1 , . . . , xq )

x∈D

x∈D

≤ sup{F (x1 , . . . , xq ) : 0 ≤ αk xrk ≤ a, k = 1, q}.

(7.74)

Let us find the value of Sq (x1 , . . . , xq ) = sup{

∞

αk xrk :

k=q+1

∞

xk = c −

k=q+1

q

xk }.

(7.75)

k=1

To this end, we set yi = xq+i , δi = αq+i , B = c −

q

xk .

(7.76)

yi = B}.

(7.77)

k=1

Then ∞ δi yir : Sq (x1 , . . . , xq ) = sup{

yi ≥ 0,

i=1

The problem

∞ i=1

∞

δi yir −→ sup

(7.78)

i=1

with the conditions δ ∈ Ar , 0 < r < 1, δ =

{δi }∞ i=1 ,

yi ≥ 0,

∞

yi = B

(7.79)

i=1

coincides with problem (7.35)–(7.37) for αi = δi , xi = yi , and c = B. Hence, its solution is given by formula (7.63), according to which 1

∞

1 Bδi1−r y¯ = {¯ yi }∞ ¯i = δi1−r . , σ ˜2 (0) = i=1 , y σ ˜2 (0) i=0

(7.80)

792

Approximations in the Spaces S p

Chapter 11

Consequently, in view of (7.76) and (7.80), an extremum in relation (7.75) is attained for (c −

q

1

xi )αk1−r

i=1

xk = x ¯k =

, k = q + 1, q + 2, . . . .

σ2 (q)

(7.81)

Hence, (c −

∞

Sq (x1 , . . . , xq ) =

αk x ¯rk q

xi )r

i=1 r σ2 (q)

=

k=q+1

= (c −

q

∞

1

αk1−r

k=q+1

xi )r σ21−r (q).

i=1

Substituting this value into (7.73), we get F (x1 , . . . , xq ) =

q

αk xrk + (c −

k=1

q

xk )r σ21−r (q).

(7.82)

k=1

Let us prove that, in the rectangle 0 < xi < (

a 1/r ) , i = 1, 2, . . . , q, αi

(7.83)

the following inequalities are true: ∂F ≥ 0, i = 1, 2, . . . , q. ∂xi By virtue of (7.82), we obtain ∂F = rαi xr−1 − r(c − xk )r−1 σ21−r (q). i ∂xi q

k=1

Thus, inequality (7.84) is equivalent to the inequality αi xr−1 i

≥ (c −

q k=1

xk )r−1 σ21−r (q),

(7.84)

Section 7

Proof of Lemma 6.1

or 1

αi1−r (c −

q

xk )

k =i

xi ≤

df

= Ai ,

1

σ2 (q) + αi1−r

q

xk =

k =i

q

793

xk − xi .

(7.85)

k=1

If relation (7.83) is true, then 1

αi1−r (c − a1/r

q

−1

αk r )

k =i

Ai ≥

df

= A¯i .

1 1−r

(7.86)

σ2 (q) + αi

Consequently, the required statement will be proved if we show that ( Since (

a 1/r ) ≤ A¯i , i = 1, q. αi

(7.87)

αq a 1/r a ) = ( )1/r ( )1/r , αi αq αi

according (7.86) and (7.87) the required inequality takes the form (

αq 1/r a 1/r ) ( ) ≤ A¯i . αi αq

(7.88)

In the case considered, we have a ∈ (aq+1 , aq ], and, hence, by virtue of Proposition 7.5, inequality (7.51) is valid. Thus, it suffices to establish the relation 1

αq βq−1 (a)αq1−r ( )1/r ≤ 1 αi 1−r σ2 (q) + αq

(c −

q 1 a 1/r ( ) )αi1−r αk k =i

1 1−r

(7.89)

σ2 (q) + αi

or the relation αqγ βq−1 (a) σ2 (q) + αqs

≤

a 1/r a ) − ( )1/r ) αi αq 1 , s= , s σ2 (q) + αi 1−r

αiγ (βq−1 (a) + (

(7.90)

which can be rewritten in the form αiγ αiγ a 1/r a 1/r αqγ (( ) −( ) ) ≤ ( − )βq−1 (a), (7.91) σ2 (q) + αis αq αi σ2 (q) + αis σ2 (q) + αqs

Approximations in the Spaces S p

794

Chapter 11

or (

a 1/r s 1/r ) αi (αi − αq1/r ) αq 1/r

≤

σ2 (q)(αiγ − αqγ ) + αis αqs (αi σ2 (q) + αqs

1/r

− αq )

βq−1 (a). (7.92)

If αi = αq , then (7.91) is obvious. Therefore, in what follows we assume that αi > αq . Relation (7.51) yields a 1/r σ2 (q − 1) a 1/r σ2 (q) + αqs =( ) . βq−1 (a) ≥ ( ) αq αqs αq αqs

(7.93)

Hence, relation (7.90) follows from the inequality 1/r

αis (αi

− αq1/r ) ≤ αq−s [σ2 (q)(αiγ − αqγ ) + αis αqs (αi

1/r

− αq1/r )],

which can be rewritten in the form αis Δi,q ≤ αq−s (σ2 (q) + αis αqs Δi,q ),

(7.94)

where 1/r

Δi,q = (αi or in the form

− αq1/r )/(αiγ − αqγ ),

αis αqs Δi,q ≤ 1. σ2 (q) + αis αqs Δi,q

(7.95)

(7.96)

Since αi > αq , we have Δi,q > 0. Therefore, relation (7.96) holds for any i = 1, q − 1. Consequently, relation (7.83), in fact, yields (7.84). Thus, in rectangle (7.83), the function F (x1 , . . . , xq ) does not decrease in each of its variables and, consequently, it attains there its maximum value for xk = ( αak )1/r , k = 1, q. According to (7.82), this maximum value is equal to q a ( )1/r )r σ21−r (q) = qa + (c − a1/r σ1 (q))r σ21−r (q). qa + (c − αk

(7.97)

k=1

Substituting it into (7.74), we get sup f (x) ≤ qa + (c − a1/r σ1 (q))r σ21−r (q).

x∈D

(7.98)

Section 7

Proof of Lemma 6.1

795

The sequence x∗ = {x∗k }∞ k=1 , where ⎧ a 1/r ⎪ k = 1, 2, . . . , q, ⎪ ⎨( αk ) , 1 x∗k = (c − a1/r σ1 (q))αk1−r ⎪ ⎪ ⎩ , k ≥ q + 1, σ2 (q)

(7.99)

belongs to D, which is determined by conditions (7.36)–(7.38). Indeed, we have x∗k > 0, αk xrk = a for k ≤ q, and αk x∗k r =

(c − a1/r σ1 (q))r αks σ2r (q)

≤ αq+1 x∗q+1 r =

s βqr (a)αq+1 (c − a1/r σ1 (q))r s = α q because a ∈ (aq+1 , aq ] and estimate (7.52) is true; finally, ∞

x∗k = a1/r σ1 (q) +

k=1

(c − a1/r σ1 (q))σ2 (q) = c. σ2 (q)

For this sequence, the value of f (x∗ ) coincides with the value of the right-hand side of (7.98). Therefore, relation (7.98) is, in fact, an equality. Thus, we have proved the following statement: Proposition 7.7. If a ≥ a1 , then the solution of problem (7.35)–(7.38) is determined by the formulas 1

cα 1−r x ¯k = k , k = 1, 2, . . . , σ2 (0)

σ2 (0) =

∞

1

αk1−r ,

(7.100)

k=1

and, thus, f (¯ x) = cr σ21−r (0).

(7.101)

If a ∈ (aq+1 , aq ), then the solution of this problem is determined by (7.99) and, furthermore, f (¯ x) = qa + (c − a1/r σ1 (q))r σ21−r (q). (7.102) The numbers aq are determined by (7.40) and (7.40 ). For a given value a > 0, the number q = qa is such that a ∈ (aq+1 , aq ] can be determined as the least natural number that satisfies inequality (7.60).

Approximations in the Spaces S p

796

Chapter 11

Consider problem (7.22)–(7.25) under condition (7.32). We set yi = xn+i , δi = αn+i .

(7.103)

Then this problem coincides with the problem fˆ(y) =

∞

δi yir −→ sup

(7.104)

i=1

under the conditions δ ∈ Ar , 0 < r < 1, yi ≥ 0,

∞

yi = β = 1 − a1/r σ1 (n),

(7.105)

i=1

δi yir ≤ a, a > 0, i = 1, 2, . . . .

(7.106)

The difference between this problem and problem (7.35)–(7.38) is that here the sum of the values of all variables depends on the parameter a. Hence, its solution does not automatically follow from Proposition 7.7, but the scheme of the proof of this statement enables us to solve problem (7.104)–(7.106). 7.8. First, for any l ∈ N, we determine the quantities a ˆl from analogs of equalities (7.39), namely, βˆl−1 (ˆ al )δlγ 1/r =a ˆl , l = 1, 2, . . . , σ ˆ2 (l − 1)

(7.107)

where βˆs (a) = 1 − a1/r (σ1 (α; n) + σ1 (δ; s)) = 1 − a1/r σ1 (n + s), ∞

1 1−r

∞

(7.108)

1

αk1−r = σ2 (n + s),

(7.109)

γ (1 − a ˆlr σ1 (n + l − 1))αn+l 1/r =a ˆl , l = 1, 2, . . . . σ2 (n + l − 1)

(7.110)

σ ˆ2 (s) =

δi

=

i=s+1

k=n+s+1

i.e., from the equalities 1

Setting n + l = q, q = n + 1, n + 2, . . . , we rewrite this equality in the form 1

r σ1 (q − 1))αqγ (1 − a ˆq−n 1/r =a ˆq−n . σ2 (q − 1)

(7.111)

Section 7

Proof of Lemma 6.1

Thus,

797

a ˆq−n = (σ1 (q − 1) + σ2 (q − 1)αq−γ )−r = aq ,

(7.112)

where aq are the numbers defined by (7.40 ) for c = 1. Therefore, a ˆl coincides with aj , where j − l = n. In particular, −γ −r ) = an+1 . a ˆ1 = (σ1 (n) + σ2 (n)αn+1

(7.113)

Further, following the solution of problem (7.35)–(7.38), we consider problem (7.104)–(7.106). We see that, for a fixed value of a, this problem coincides with problem (7.35)–(7.38) for c = β. Consequently, in view of (7.63), its solution has the form 1

1

1−r (1 − a1/r σ1 (n))αn+i βδi1−r y¯ = {¯ yi }∞ , y ¯ = = , i = 1, 2, . . . . (7.114) i i=1 ∞ 1 σ2 (n) 1−r δj

j=1

This yields 1

1

1−r 1−r (1 − a1/r σ1 (n))r αn+i (1 − a1/r σ1 (n))r αn+1 δi y¯ir = ≤ , σ2r (n) σ2r (n)

(7.115)

and if a ≥ a ˆ1 , then, by virtue of (7.111) and (7.113), 1

1/r

1−r (1 − a1 σ1 (n))r αn+1 δi y¯ir ≤ =a ˆ1 = an+1 . σ2r (n)

(7.116)

Thus, if the value of the parameter a in problem (7.104)–(7.106) is not less than a ˆ1 , then condition (7.106) is automatically satisfied, and, hence, in this case, the solution of this problem is given by (7.114). Now assume that a ∈ (ˆ al+1 , a ˆl ], where l is a certain natural number. Deˆ the set of sequences y = {yi }∞ that satisfy conditions (7.105) and note by D i=1 ˆ then (7.106) for this a. If y ∈ D, fˆ(y) =

∞

δi yir ≤

i=1

l

δi yil + sup

k=1

∞

ˆ i=l+1 y∈D

δi yir ≤ Fˆ (y1 , . . . , yl ),

(7.117)

where Fˆ (y1 , . . . , yl ) =

l k=1

δi yir

+ sup{

∞ i=l+1

δi yir

:

∞ i=l+1

yi = β −

l i=1

yi }. (7.118)

Approximations in the Spaces S p

798

Chapter 11

Consequently, sup fˆ(y) ≤ sup Fˆ (y1 , . . . , yl )

ˆ y∈D

ˆ y∈D

≤ sup{Fˆ (y1 , . . . , yl ) : 0 ≤ δi yir ≤ a, i = 1, l}.

(7.119)

Let us find the value ∞

Sˆl (y1 , . . . , yl ) = sup{

δi yir

i=l+1

∞

:

yi = β −

l

yi }.

(7.120)

i=1

i=l+1

Setting zν = yl+ν ; ξν = δl+ν , B = β −

l

yi ,

(7.121)

i=1

we conclude that the problem of finding the value ∞ ∞ Sˆl (y1 , . . . , yl ) = sup{ ξν zνr : zν = B} ν=1

(7.122)

ν=1

is equivalent to the problem ∞

ξν zνr −→ sup

(7.123)

ν=1

under the conditions ξ ∈ Ar , 0 < r < 1, zν > 0,

∞

zν = B.

(7.124)

ν=1

We obtain its solution by using relations (7.63), namely, 1

Bξν1−r z¯ν = , ν = 1, 2, . . . . σ2 (ξ, 0)

(7.125)

Hence, Sˆl (y1 , . . . , yl ) =

∞

∞

ξν z¯νr =

ν=1

= (β −

ν=1

l i=1

= (β −

1 B r 1−r ξ = B r σ21−r (ξ; 0) ν σ2r (ξ, 0)

l i=1

∞ 1 yi )r ( ξν1−r )1−r ν=1

yi )r σ21−r (δ; l).

(7.126)

Section 7

Proof of Lemma 6.1

799

Consequently, Fˆ (y1 , . . . , yl ) =

l

l

δi yir + (β −

i=1

yi )r σ21−r (δ; l).

(7.127)

i=1

Thus, in view of (7.103), we get Fˆ (y1 , . . . , yl ) = Fˆ (xn+1 , . . . , xn+l ) =

l+n

αk xrk + (1 − a1/r σ1 (n)

k=n+1 l+n

−

r

xk ) (

k=n+1

=

q

αk xrk

k=n+1

= F ((

∞

1

αk1−r )1−r

k=n+l+1

q n a r + (1 − ( ( ) + xk )r σ21−r (q) αk k=1

k=n+1

a a ) , . . . , ( ) , xn+1 , . . . , xq ) α1 αn 1 r

−

1 r

n k=1

αk [(

a 1 r )r ] , αk

(7.128)

where l + n = q and F (·) is the function defined by (7.82) for c = 1. ˆ then If y ∈ D, 0 ≤ xk ≤ (

a r ) , k = n + 1, . . . , q. αk

(7.129)

Therefore, taking into account the properties of the function F, we can conclude that, according to (7.97), sup Fˆ (y) = F (( ˆ y∈D

a 1 a 1 ) r , . . . , ) r ) − na α1 αq

= (q − n)a + (1 − a1/r σ1 (q))r σ21−r (q), q = l + n.

(7.130)

Hence, by virtue of (7.119), we get sup fˆ(y) ≤ (q − n)a + (1 − a1/r σ1 (q))r σ21−r (q).

ˆ y∈D

(7.131)

Approximations in the Spaces S p

800

Chapter 11

7.9. Consider the sequence y¯ = {yi }∞ i=1 , where

y¯i =

⎧ a 1/r ⎪ ⎪ ⎨( δi ) ,

i = 1, 2, . . . , l, 1

(1 − a1/r σ1 (n + l))δi1−r ⎪ ⎪ ⎩ , i = l + 1, l + 2, . . . . σ2 (δ; l)

(7.132)

We have fˆ(¯ y) =

l i=1

∞ 1 a (1 − a1/r σ1 (n + l))r 1−r δi (( )1/r )r + δi δi σ2 (δ; l) i=l+1

= la + (1 − a1/r σ1 (n +

l))r σ21−r (δ; l)

= (q − n)a + (1 − a1/r σ1 (q))r σ21−r (q), q = l + n.

(7.133)

At the same time, for i > l, in view of (7.115) and (7.110), we get 1

1 (1 − a1/r σ1 (n + l))r δi1−r (1 − a1/r σ1 (n + l))r 1−r δi y¯ir = ≤ δ l+1 σ2r (δ; l) σ2r (δ; l) 1

1/r

1−r (1 − a ˆl+1 σ1 (n + l))r αn+l+1 < =a ˆl+1 < a σ2r (n + l)

because a ∈ (ˆ al+1 , a ˆl ]. Moreover, ∞

y¯i = a1/r

i=1

l

− r1

δi

i=1

= a1/r

l

+

∞ 1 1 − a1/r σ1 (n + l) 1−r δi σ2 (δ; l) i=l+1

−1

r αn+i + 1 − a1/r σ1 (n + l) = 1 − a1/r σ1 (n).

i=1

ˆ Consequently, (7.131) is, in fact, an equality. This means that y¯ ∈ D. Therefore, problem (7.104)–(7.106) has the following solution: If a > a ˆ1 , then the solution is determined by (7.114) and ∞ 1 (1 − a1/r σ1 (n))r 1−r 1/r ˆ sup f (y) = α σ1 (n))r σ21−r (n). (7.134) n+i = (1 − a r (n) σ ˆ 2 y∈D i=1

Section 7

Proof of Lemma 6.1

801

If a ∈ (ˆ al+1 , a ˆl ], l ∈ N, then the solution of this problem is determined by (7.132) and sup fˆ(y) = (q − n)a + (1 − a1/r σ1 (q))r σ21−r (q), q = l + n.

(7.135)

ˆ y∈D

In view of (7.103) and (7.112), this yields the following statement: Proposition 7.8. If a ≥ an+1 , then the solution of problem (7.22)–(7.25) under conditions (7.32) is determined by the formulas 1

(1 − a1/r σ1 (n))αk1−r x ¯k = , k = n + 1, n + 2, . . . , σ2 (n) and fn (¯ x) =

∞

αk x ¯rk = (1 − a1/r σ1 (n))r σ21−r (n).

(7.136)

(7.137)

k=n+1

If a ∈ (aq+1 , aq ], q ≥ n + 1, then the solution of this problem is determined by the formulas ⎧ a 1/r ⎪ k = n + 1, . . . , q, ⎪ ⎨( αk ) , 1 x ¯k = (7.138) 1/r σ (q))α 1−r (1 − a ⎪ 1 k ⎪ ⎩ , k = q + 1, q + 2, . . . , σ2 (q) and fn (¯ x) = (q − n)a + (1 − a1/r σ1 (q))r σ21−r (q).

(7.139)

Combining Propositions 7.3 and 7.8, we obtain the following statement: Proposition 7.9. The sequence m∗ in (7.30) has the following form: (1) if a ≥ an+1 , then ⎧ a 1/r ⎪ k = 1, 2, . . . , n, ⎪ ⎨( αk ) , 1 ∗ m = (1 − a1/r σ1 (n))αk1−r ⎪ ⎪ ⎩ , k = n + 1, . . . , σ2 (n) and

En (m∗ ) = (1 − a1/r σ1 (n))r σ21−r ;

(7.140)

(7.141)

Approximations in the Spaces S p

802

Chapter 11

(2) if a ∈ (aq+1 , aq ], q ≥ n + 1, then ⎧ a 1/r ⎪ k = 1, 2, . . . , q, ⎪ ⎨( αk ) , 1 ∗ mk = (1 − a1/r σ1 (q))αk1−r ⎪ ⎪ ⎩ , k = q + 1, . . . , σ2 (q) and

En (m∗ ) = (q − n)a + (1 − a1/r σ1 (q))r σ21−r (q).

(7.142)

(7.143)

The numbers aq are determined by equality (7.40 ) for c = 1. Further, according to relation (7.29) and Proposition 7.9, we consider the quantity sup a∈(0,σ1−r (n))

En (m∗ ) = max{

En (m∗ ),

sup a∈(0,an+1 )

sup a∈[an+1 ,σ1−r (n)]

En (m∗ )}. (7.144)

By virtue of (7.113), we have an+1 < σ1−r (n). Therefore, according to (7.141), sup a∈[an+1 ,σ1−r (n)]

En (m∗ ) = (1 − an+1 σ1 (n))r σ21−r (n) 1/r

γ = σ2 (n)(σ2 (n) + σ1 (n)αn+1 )−r ,

γ=

(7.145)

1 . r(1 − r)

Taking (7.143) into account, we obtain En (m∗ )

sup a∈(0,an+1 )

= max

sup

q≥n+1 a∈(aq+1 ,aq ]

((q − n)a + (1 − a1/r σ1 (q))r σ21−r (q)). (7.146)

For fixed values of n, r ∈ (0, 1), and q ≥ n + 1, we set fq (t) = (q − n)t + (1 − t1/r σ1 (q))r σ21−r (q), t > 0. The critical point tq of this function determined from the equation fq (t) = q − n − t

1−r r

(1 − t1/r σ1 (q))r−1 σ1 (q)σ21−r (q) = 0

(7.147)

Section 7

Proof of Lemma 6.1

has the form tq = (σ1 (q) + (

1 σ1 (q) 1−r ) σ2 (q))−r . q−n

803

(7.148)

Assume that tq ∈ (aq+1 , aq ]. Then, according to (7.40 ), (7.49) (for c = 1), and (7.148), we get −γ −r ) < (σ1 (q) + ( (σ1 (q) + σ2 (q)αq+1

1 σ1 (q) 1−r ) σ2 (q))−r q−n

≤ (σ1 (q) + σ2 (q)αq−γ )−r , which is possible only if − r1

αq

≤

σ1 (q) − r1 . < αq+1 q−n

(7.149)

Thus, condition (7.149) is necessary and, obviously, sufficient for the inclusion tq ∈ (aq+1 , aq ]. 7.10. Let us prove the following statement: Proposition 7.10. For every α ∈ Ar , 0 < r < 1, and n ∈ N, one can always find q ∗ ∈ N, q ∗ > n, for which −1 αq∗r

q∗

1 − r1 −1 ≤ ∗ αk < αq∗r+1 , q −n

(7.150)

k=1

and such q ∗ is unique. Proof. Let l1 (t) and l2 (t) be the polygonal lines that connect the points σ1 (q) −1 (q, αq r ) and (q, ), q = n + 1, n + 2, . . . , respectively. If q = n + 1, q−n then n+1 −1 − r1 αk r > αn+1 = l1 (n + 1). l2 (n + 1) = k=1

On the other hand, q n 1 − r1 1 −1 αk + αk r l2 (q) = q−n q−n k=1

− r1

= αq

+

k=n+1

q n 1 − r1 −1 −1 αk + (αk r − αq r )). ( q−n k=1

k=n+1

Approximations in the Spaces S p

804

Chapter 11

The second term on the right-hand side of this equality is negative for all sufficiently large values of q. This means that, for such q, − r1

l2 (q) < αq

= l1 (q).

Consequently, the polygonal lines l1 (t) and l2 (t) necessarily intersect at a certain point t¯q∗ . Thus, the required q ∗ exists. Let us verify that it is unique. Assume that, together with (7.150), relation (7.149) holds for some q = q ∗ + p. It is clear that we can restrict ourselves to the case p ∈ N. Then, in view of (7.150), we get ∗

∗

(q + p −

−1/r n)αq∗ +p

≤

q k=1

or

−1

−1 αk r

+

∗ +p q

−1 αk r

∗

< (q −

k=q ∗ +1 −1

−1 n)αq∗r+1

−1

+

∗ +p q

−1

αq r ,

k=q ∗ +1

−1

(q ∗ − n)αq∗r+p + pαq∗r+p < (q ∗ − n)αq∗r+1 + pαq∗r+p , or

−1

−1

(q ∗ − n)(αq∗r+p − αq∗r+1 ) < 0, − r1

which is impossible because q ∗ > n and the numbers αi Proposition 7.10 is proved.

do not decrease.

Thus, relation (7.149) always holds for q > n, and the corresponding q = q ∗ is unique. Exactly for this value fq∗ (t) has its (unique) critical point tq∗ on the interval (aq∗ +1 , aq∗ ]. Moreover, it follows from the results proved above that

and

σ1 (q) −1 > αq r , q = n + 1, . . . , q ∗ − 1, q−n

(7.151)

σ1 (q) −1 < αq r , q = q ∗ + 1, q ∗ + 2, . . . . q−n

(7.152)

Taking into account (7.151) and (7.148), we get tq < (σ1 (q) + αq−γ σ2 (q))−r = aq , q = n + 1, . . . , q ∗ − 1.

(7.153)

By analogy, in view of (7.152), we obtain tq > (σ1 (q) + αq−γ σ2 (q))−r = aq , q = q ∗ + 1, q ∗ + 2, . . . .

(7.154)

Section 7

Proof of Lemma 6.1

805

If q ≤ q ∗ − 1, then the point tq cannot lie on the interval (aq+1 , aq ]. Consequently, relation (7.153), in fact, means that tq ≤ aq+1 , q = n + 1, . . . , q ∗ − 1.

(7.155)

Let us establish several more properties of the functions fq (t). We have fq (t) = −σ1 (q)σ21−r (q)[

1 1 − r 1 −2 t r (1 − t r σ1 (q))r−1 r

1 1 1 1 + t r −1 (r − 1)(1 − t r σ1 (q))r−2 (− σ1 (q)t r −1 )] < 0. r

Hence, every function fq (t) is convex upwards, and, consequently, tq are their maximum points. Let us show that fq (aq ) = fq−1 (aq ), q = n + 1, . . . ,

(7.156)

where aq is determined by equalities (7.40 ) or (7.49) for c = 1, namely, aq = (σ1 (q − 1) + σ2 (q − 1)αq−γ )−r = (σ1 (q) + σ2 (q)αq−γ )−r .

(7.157)

We have fq (aq ) − fq−1 (aq ) r 1−r 1/r r 1−r = aq + (1 − a1/r q σ1 (q)) σ2 (q) − (1 − aq σ1 (q − 1)) σ2 (q − 1).

By virtue of (7.157), r 1−r (1 − a1/r q σ1 (q)) σ2 (q) = (1 −

σ1 (q) r 1−r −γ ) σ2 (q) σ1 (q) + σ2 (q)αq

1 − 1−r

= aq αq

σ2 (q)

and r 1−r (1 − a1/r q σ1 (q − 1)) σ2 (q − 1)

= (1 −

1 σ1 (q − 1) r 1−r (q − 1) = aq α− 1−r σ2 (q − 1). −γ ) σ σ1 (q − 1) + σ2 (q − 1)αq

Approximations in the Spaces S p

806

Chapter 11

Thus, 1 − 1−r

fq (aq ) − fq−1 (aq ) = aq (1 + αq

(σ2 (q) − σ2 (q − 1))) = 0,

which proves equality (7.156). We set Φn (t) = fq (t), t ∈ (aq+1 , aq ], q = n + 1, . . . .

(7.158)

Taking into account the obtained information about the functions fq (t), we conclude that Φn (t) is continuous on the interval (0, an+1 ), convex upwards on every interval (aq+1 , aq ], nonincreasing for q < q ∗ , and nondecreasing for q > q ∗ . Consequently, this function attains its maximum value Φn (t) on the interval (aq∗ +1 , aq∗ ] at the point tq∗ , and this value is determined by the formula Φn (tq∗ ) = fq∗ (tq∗ ) = (q ∗ − n)tq∗ + (1 − tq∗ σ1 (q ∗ ))r σ21−r (q ∗ ) 1/r

r σ1 (q ∗ ) 1−r σ2 (q ∗ ) ) q∗ − n = 1 σ1 (q ∗ ) 1−r (σ1 (q ∗ ) + ( ∗ ) σ2 (q ∗ ))r q −n

q∗ − n + (

r

1

= σ1−r (q ∗ )[(q ∗ − n) 1−r + σ11−r (q ∗ )σ2 (q ∗ )]1−r .

(7.159)

Hence, taking into account that the value of fq (t) for t = a coincides with the value of the quantity under the sign of the upper bound on the right-hand side of (7.146), we arrive at the following statement: Proposition 7.11. If α ∈ Ar , 0 < r < 1, and n ∈ N, then sup a∈(0,an+1 )

1

r

En (m∗ ) = σ1−r (q ∗ )[(q ∗ − n) 1−r + σ1r−1 (q ∗ )σ2 (q ∗ )]1−r ,

(7.160)

where q ∗ is determined by condition (7.150). Combining relations (7.144), (7.145), and (7.160), we get sup a∈(0,σ1−r (n))

γ En (m∗ ) = max{σ2 (n)(σ1 (n)αn+1 + σ2 (n))−r ; 1

r

σ1−r (q ∗ )((q ∗ − n) 1−r + σ11−r (q ∗ )σ2 (q ∗ ))1−r }. (7.161)

Section 7

Proof of Lemma 6.1

807

Computing the value Φn (an+1 ), we obtain γ Φn (an+1 ) = fn+1 (an+1 ) = σ2 (n)(σ1 (n)αn+1 + σ2 (n))−r .

Hence, in view of (7.161), we get sup a∈(0,σ1−r (n))

En (m∗ ) =

En (m∗ )

sup a∈(0,an+1 ]

r

1

= σ1−r (q ∗ )((q ∗ − n) 1−r + σ11−r (q ∗ )σ2 (q ∗ ))1−r . (7.162) Thus, by virtue of (7.29), (7.30), and (7.162), r

1

En ≤ σ1−r (q ∗ )((q ∗ − n) 1−r + σ11−r (q ∗ )σ2 (q ∗ ))1−r ,

(7.163)

where q ∗ is determined by condition (7.150). Let us verify that (7.163) cannot be a strict inequality. To this end, we set

m ¯ ∗k =

⎧ t∗ q 1/r ⎪ ⎪ ⎨( αk ) ,

k = 1, 2, . . . , q ∗ , 1

(1 − tq∗ σ1 (q ∗ ))αk1−r ⎪ ⎪ ⎩ , k = q ∗ + 1, q ∗ + 2, . . . . σ2 (q ∗ ) 1/r

(7.164)

¯ ∗k do The sequence m ¯ ∗ belongs to Mαn (for a = tq∗ ). Indeed, the numbers m not increase, condition (7.17) is satisfied, and ∞

m ¯ ∗k = 1.

k=1

At the same time, by virtue of (7.159), we get ¯ ∗ ) = (q ∗ − n)tq∗ + (1 − tq∗ σ1 (q ∗ ))r σ21−r (q ∗ ) En (m 1/r

1

r

= σ1−r (q ∗ )((q ∗ − n) 1−r + σ11−r (q ∗ )σ2 (q ∗ ))1−r . Therefore, relations (7.163) and (7.29) are, in fact, equalities. Lemma 6.1 is proved.

Approximations in the Spaces S p

808

8.

Chapter 11

Best Approximations by q-Ellipsoids in the Spaces Sϕp

8.1. The best approximations of elements f ∈ ψUϕq in the spaces Sϕp are quantities (2.7), i.e., the quantities En (f )ψ,p = inf f − Φgψ p , Φ

where Φg ψ

n−1

=

(8.1)

n−1

ψ gn−1

ck ϕk

(8.2)

ψ k∈gn−1

and ck are complex numbers. Here we establish a statement that complements the result of Theorem 4.3 in the case where q > p > 0. Theorem 8.1. Let ψ = {ψk }∞ k=1 be a system of numbers satisfying conditions (2.3) and (2.11) and let p and q be arbitrary numbers such that q > p > 0. Then, for any n ∈ N, the following equality holds: sup En (f )ψ,p = sup En (f )ψ,p = (

q f ∈ψUϕ

q f ∈ψUϕ

∞

pq

ψ¯kq−p )

q−p pq

,

(8.3)

k=δn−1 +1

where ψ¯ = {ψ¯k }∞ k=1 is a sequence for which ψ¯k = εk , δn−1 < k ≤ δn , and εn and δn are elements of the characteristic sequences ε(ψ) and δ(ψ). Proof. The first equality in (8.3) follows from Proposition 1.2, and, hence, it suffices to verify the second one. Let f be an arbitrary element of ψUϕq . Then, in view of (6.3), we have Epn (f )ψ,p = f − Sn−1 (f )ϕ,ψ p = |fˆ(k)|p

=

ψ ¯ gn−1 k∈

ψ ¯ gn−1 k∈

|ψk |p |fˆψ (k)|p .

(8.4)

Section 8

Best Approximations by q-Ellipsoids in the Spaces Sϕp

809

Now let ik , k = 1, 2, . . . , be natural numbers chosen according to condition (6.9). In this case, relation (8.4) can be rewritten in the form ∞

Epn (f )ψ,p =

ψ¯kp |fˆψ (ik )|p .

k=δn−1 +1

Consequently, setting αk = ψ¯kp , we get ∞

Epn (ψUϕq )p = sup Epn (f )ψ,p ≤ sup q f ∈ψUϕ

where m =

{mk }∞ k=1 ,

|m|=1 k=δ

n−1 +1

p αk mrk , r = , q

(8.5)

mk ≥ 0, and the quantity |m| is determined by (6.11).

8.2. Therefore, the problem of finding the upper bound of Epn (ψUϕq ) reduces to the solution of the extremal problem ∞

αk xrk −→ sup

(8.6)

k=δn−1 +1

under the conditions α ∈ Ar , 0 < r < 1, ∞ xk = 1. xk ≥ 0,

(8.7) (8.8)

k=δn−1 +1

By the substitution yi = xδn−1 +i , γi = αδn−1 +i ,

(8.9)

we reduce it to the problem ∞

γi yir −→ sup

(8.10)

i=1

under the conditions γ ∈ Ar , yi ≥ 0,

∞

yi = 1,

(8.11)

i=1

which coincides with problem (7.35)–(7.37) for c = 1. Its solutions y¯i are given by (7.63), namely, 1

γ 1−r y¯i = ∞ i , i = 1, 2, . . . . 1 1−r γj j=1

(8.12)

810

Approximations in the Spaces S p

Chapter 11

Hence, by virtue of (8.9), solutions of problem (8.6)–(8.8) have the form 1

x ¯k = αk1−r (

∞

1

1

αi1−r )−1 = αk1−r σ2−1 (δn−1 ),

(8.13)

i=δn−1 +1

k = δn−1 + 1, δn+1 + 2, . . . , and, thus, ∞

αk x ¯rk = (

k=δn−1 +1

∞

1

αk1−r )1−r = σ21−r (δn−1 ).

(8.14)

k=δn−1 +1

Therefore, according to (8.14) and (8.5), Epn (ψUϕq )p ≤ (

∞

1

αk1−r )1−r = σ21−r (δn−1 ),

(8.15)

k=δn−1 +1

and, to complete the proof of the theorem, it remains to show that this relation cannot be a strict inequality. We proceed by analogy with the end of the proof of Theorem 6.1. Namely, let ∞ cik ϕik , h= k=1

where the numbers ik are chosen according to (6.9) and the numbers cik are such that ⎧ k = 1, 2, . . . , δn−1 , ⎪ ⎨0, q (8.16) cik = 1 ⎪ ⎩ 1−r −1 αk σ2 (δn−1 ), k = δn−1 + 1, δn−1 + 2, . . . . In this case, hqϕ,q

=

∞

cqik = 1.

(8.17)

k=1

Further, let ε be an arbitrary positive number and let Nε be so large that ∞

σ2−r (δn−1 )

1

αk1−r < ε.

k=Nε +1

The element hε =

Nε k=1

cik ϕik

(8.18)

Section 9

Application of Obtained Results

811

belongs to all spaces Sϕp for any p > 0 and, by virtue of (8.17), we have hε qϕ,q ≤ 1, i.e., hε ∈ Uϕq . Setting fε = J ψ hε , we conclude that fε ∈ ψUϕq and fεψ = hε . Since, according to (8.16), ⎧ k = 1, 2, . . . , δn−1 , ⎪ ⎨0, p p ψ¯k cik = 1 ⎪ ⎩ 1−r αk σ2−r (δn−1 ), k = δn−1 + 1, δn−1 + 2, . . . , by virtue of (8.18) we get Epn (fε )ψ,p =

Nε

ψ¯kp |fˆεψ (k)|p

k=δn−1 +1

=

Nε

ψ¯kp cpik = σ2−r (δn−1 )

k=δn−1 +1

Nε

1

αk1−r

k=δn−1 +1

= σ21−r (δn−1 ) − σ2−r (δn−1 ) ×

∞

1

αk1−r > σ21−r (δn−1 ) − ε.

(8.19)

k=Nε +1

Comparing relations (8.15) and (8.19) and taking into account that ε is arbitrary, we obtain equality (8.3).

9.

Application of Obtained Results to Problems of Approximation of Periodic Functions of Many Variables

9.1. As in Section 1, we assume that L = L(Rm ) is the set of all functions f (x) = f (x1 , . . . , xm ) 2π-periodic in each variable and summable on the cube of periods Qm , m ≥ 1, and series (1.7) is the Fourier series of a function f ∈ L in system (1.8). We assume that equivalent functions are indistinguishable. Further, let S p be the space generated by the set L, system (1.8), and a certain number p, p ∈ (0, ∞), with the scalar product (1.9) and the norm ·p = · S p defined by (1.2): f p = ( |fˆ(k)|p )1/p . (9.1) k∈Zm

812

Approximations in the Spaces S p

Chapter 11

Now let ψ = {ψ(k)}k∈Z m be an arbitrary system of complex numbers (a multiple sequence). For functions f ∈ L, parallel with (1.7), we consider the series (2π)−m/2 ψ(k)fˆ(k)eikx . (9.2) k∈Z m

If, for a given function f and a system ψ, this series is the Fourier series of a certain function F from L, then we say that F is the ψ-integral of the function f and write F (x) = J ψ (f ; x). In this case, it is sometimes convenient to call the function f the ψ-derivative of the function F and write f (x) = Dψ (F ; x) = F ψ (x). The set of ψ-integrals of all functions f ∈ L is denoted by Lψ . If N is a certain subset of L, then Lψ N denotes the set of ψ-integrals of all functions from N. It is clear that if f ∈ Lψ , then the Fourier coefficients of the functions f and f ψ satisfy the following relation: fˆ(k) = ψ(k)fˆψ (k), k ∈ Z m .

(9.3)

As N, we consider the unit ball U p in the space S p , namely, U p = {f : f ∈ S p , |f p ≤ 1}.

(9.4)

ψ m In this case, we set Lψ U p = Lψ p = Lp (R ). We assume that the system ψ satisfies the condition (9.5) lim ψ(k) = 0. |k|→∞

Note that if f ∈ Lψ S p and |ψ(k)| ≤ C, k ∈ Z m , C = const, then f ∈ S p , p i.e., condition (9.5) always guarantees the inclusion Lψ p ⊂S . We define characteristic sequences ε(ψ), g(ψ), and δ(ψ) in the following way: ε(ψ) = ε1 , ε2 , . . . is the set of values of |ψ(k)|, k ∈ Z m , arranged in decreasing order; g(ψ) = {gn }∞ n=1 , where gn = gnψ = {k ∈ Z m : |ψ(k)| ≥ εn }; ψ m δ(ψ) = {δn }∞ n=1 , where δn = δn = |gn | is the number of vectors k ∈ Z belonging to the set gn . In the case considered, by virtue of condition (9.5), the sequences ε(ψ) and g(ψ) are defined by equalities (2.4) with regard for the fact that k ∈ Z m . As above, we assume that g0 = g0ψ is an empty set and δ0 = δ0ψ = 0.

Section 9

Application of Obtained Results

813

As approximating aggregates for functions f ∈ Lψ , we consider the trigonometric polynomials Sn (f ; x) = Sgψ (f ; x) = (2π)−m/2 fˆ(k)eikx , (9.6) n

ψ k∈gn

n ∈ N, S0 (f ; x) = 0, where gnψ are the elements of the sequence g(ψ). Let p and q be arbitrary numbers, p, q > 0, and let En (f )ψ,p = f (x) − Sn−1 (f ; x)S p ,

(9.7)

En (Lψ q )p = sup En (f )ψ,p , n = 1, 2, . . . .

(9.8)

f ∈Lψ q

En (f )ψ,p = inf f (x) − (2π)−m/2 ak

ak eikx S p

(9.9)

ψ k∈gn−1

and En (Lψ q )p = sup En (f )ψ,p , n = 1, 2, . . . .

(9.10)

f ∈Lψ q

Further, let dn (Lψ p )p = inf

Fn ∈Gn

sup inf f − uS p ,

f ∈Lψ p

u∈Fn

n ∈ N,

df

d0 (Lψ p )p = sup f S p ,

(9.11)

f ∈Lψ p

be the Kolmogorov widths of the classes Lψ p (Gn is the set of all n-dimensional p subspaces in S ) and let −m/2 en (Lψ q )p = sup inf f (x) − (2π) f ∈Lψ q

ak ,γn

ak eikx S p ,

(9.12)

k∈γn

p be the quantity of the best n-term approximation of the class Lψ q in the space S m (γn is an arbitrary collection of n vectors k ∈ Z ). In this notation, the following statements (analogs (in fact, special cases) of the theorems considered above) are true:

Approximations in the Spaces S p

814

Chapter 11

Theorem 9.1. Let ψ = {ψ(k)}k∈Zm be a system of numbers satisfying condition (9.5) and such that ψ(k) = 0

∀k ∈ Zm .

(9.13)

Then, for any n ∈ N and 0 < q ≤ p < ∞, we have ψ En (Lψ q )p = En (Lq )p = εn ,

(9.14) ∗

epn (Lψ q )p

l l −p −p ψ¯k−q ) q = (l∗ − n)( ψ¯k−q ) q , = sup(l − n)( l>n

k=1

(9.15)

k=1

where ψ¯ = {ψ¯k }∞ k=1 is the sequence defined by the relations ψ¯k = εn

for

k ∈ (δn−1 , δn ], n = 1, 2, . . . ,

(9.16)

εn and δn are the elements of the characteristic sequences of the system ψ, and l∗ is a certain natural number, which always exists under the conditions of the theorem. Theorem 9.2. Let f ∈ Lψ p , p > 0, and let ψ = {ψ(k)}k∈Zm be a system of numbers satisfying condition (9.5). Then the series ∞

(εpk − εpk−1 )Ekp (f ψ )ψ,p

k=1

converges and, for any n ∈ N, the following equality is true: Enp (f )ψ,p = εpn Enp (f ψ )ψ,p +

∞

(εpk − εpk−1 )Ekp (f ψ )ψ,p ,

k=n+1

where the quantities Eν (·)ψ,p are defined by equality (9.9) and εk are the elements of the characteristic sequence ε(ψ) of the system ψ. Theorem 9.3. Suppose that f ∈ S p , p > 0, the system ψ = {ψ(k)}k∈Zm satisfies condition (9.5), and lim ε−1 Ek (f )ϕ,p k→∞ k Then, in order that the inclusion f ∈ Lψ p

= 0.

Section 9

Application of Obtained Results

815

be true, it is necessary and sufficient that the series ∞

−p p (ε−p k − εk−1 )Ek (f )ψ,p

k=1

be convergent. If this series is convergent, then, for any n ∈ N, the following equality is true: Enp (f ψ )ψ,p

=

p ε−p n En (f )ψ,p

∞

+

p p (ε−p k − εk−1 )Ek (f )ψ,p ,

k=n+1

where the quantities En (·)ψ,p and εk have the same meaning as in Theorem 9.2. Let ψ = {ψ(k)}k∈Zm be an arbitrary system of numbers satisfying condition (9.5). Let us renumber all vectors k ∈ Zm by natural numbers s in a certain fixed order. We say that the system ψ belongs to the set Ap,q for certain values of p and q, q > p > 0, if ∞

pq

|ψ(ks )| q−p < ∞.

(9.17)

s=1

It is clear that the sets Ap,q are independent of the enumeration of k ∈ Zm and are completely determined by the quantities |ψ(k)| and the numbers p and q. Theorem 9.4. Suppose that the system ψ = {ψ(k)}k∈Zm belongs to the set Ap,q for given p and q, q > p > 0. Then ψ En (Lψ q )p = En (Lq )p = (

∞

pq

ψ¯ q−p )

q−p pq

, n = 1, 2, . . . ,

(9.18)

k=δn−1 +1

and

−p

q

p

¯1 q (s)[(s − n) q−p + σ1q−p (s)¯ σ2 (s)] epn (Lψ q )p = σ where σ ¯1 (s) =

s

ψ¯k−q ,

σ ¯2 (s) =

k=1

∞

q−p q

,

(9.19)

pq

ψkq−p ,

k=s+1

ψ¯ = {ψk }∞ k=1 is a sequence such that ψ¯k = εk

for k ∈ (δn−1 , δn ],

n = 1, 2, . . . ,

(9.20)

Approximations in the Spaces S p

816

Chapter 11

εn and δn are the elements of the characteristic sequences ε(ψ) and δ(ψ), respectively, and the number s in (9.19) is chosen according to the condition 1 ¯−q −q ψk < ψs+1 . s−n s

ψ¯s−q ≤

(9.21)

k=1

Such a number s always exists and is unique. Theorems 9.1–9.4 are proved on the basis of the corresponding theorems in the previous sections as follows: For the given systems ψ appearing in these statements, we renumber all vecψ tors k ∈ Zm so that all vectors k from the sets gnψ \ gn−1 are enumerated by the numbers s, s ∈ (δn−1 , δn ], in a certain fixed order. Then we define a sequence ψ = {ψs }∞ s=1 by setting ψs = ψ(ks ), s = 1, 2, . . . .

(9.22)

Since −m/2

S[f ] = (2π)

fˆ(k)eikx = (2π)−m/2

k∈Zm

∞

fˆ(ks )eiks x ,

(9.23)

s=1

relation (9.22) yields J

ψ

−m/2

(f ; x) = (2π)

∞

ψ(ks )fˆ(ks )eiks x = J (f ; x)∀ f ∈ L.

(9.24)

s=1

p p Therefore, Lψ = Lψ . However, we have Lψ p = ψ U , where the set ψ U is defined according to equality (2.12), namely

ψ U p = {f ∈ L : f ψ ∈ U p }, p −m/2 eiks x }∞ . Furthermore, the sequences where U p = U ϕ and ϕ = {2π s=1 ψ E(ψ ) and E as well as δ(ψ ) and δ(ψ) coincide, and the following equalities are true:

Sgψ (f ) = Sgψ (f ; x), En (f )ψ ,p n

n

= En (f )ψ,p , En (ψ U q )p = En (Lψ q )p , En (f )ψ ,p = En (f )ψ,p

Section 10 and

Remarks

817

En (ψ U q )p = En (Lψ q )p ,

where the left-hand sides are defined by equalities (2.6)–(2.9), (3.1), (3.2) and the q right-hand sides by (9.6)–(9.10). It is also clear that en (Lψ q )p = en (ψ U ) and ¯ This implies that equality (9.14) follows from (4.12), equality (9.15) ψ¯ = ψ. follows from Theorem 5.1, Theorems 4.1 and 4.2 yield Theorems 9.2 and 9.3, and Theorems 6.1 and 8.1 yield Theorem 9.4. For completeness, we reformulate Theorem 3.2 for the Kolmogorov widths of the sets Lψ p. Theorem 9.5. Let ψ = {ψ(k)}k∈Z m be a system of numbers satisfying conditions (9.5) and (9.13) and let p ∈ [1, ∞). Then, for any n ∈ N, the following relation is true: p ψ p dδn−1 (Lψ p ; S ) = dδn−1+1 (Lp ; S ) p ψ = . . . = dδn −1 (Lψ p ; S ) = En (Lp )p = εn ,

(9.25)

where εn and δn are the elements of the characteristic sequences ε(ψ) and δ(ψ), respectively.

10.

Remarks

10.1. On Sequences ψ. In all previous constructions, the key role is played by the sequences ψ : they define approximated sets, the approximation apparatus is constructed on their basis, and the approximation characteristics are expressed in terms of these sequences. We do not impose any restrictions on the sequences ψ except conditions of the form (9.5) and (9.17), without which our considerations would become almost meaningless. Therefore, generally speaking, the systems ψ and their characteristic sequences ε(ψ), g(ψ), and δ(ψ) can be fairly complex. In the multidimensional case, apparently the simplest and most natural are the systems ψ in which the numbers ψ(k) are represented by the products of the values of one-dimensional sequences ψj = {ψj (kj )}∞ kj =1 : ψ(k) = ψ(k1 , . . . , km ) =

m $

ψj (kj ), kj ∈ Z 1 , j = 1, m.

j=1

If, in addition, ψ(−kj ) = ψj (kj ), j = 1, m

(10.1)

Approximations in the Spaces S p

818

Chapter 11

(by z¯ we denote the number conjugate to z), then the sets gnψ are symmetric with respect to all coordinate planes and, as can easily be verified,

ψ(k)eikt =

2m−q(k)

m k∈Z+

k∈Z m

m $

|ψj (kj )| cos(kj tj −

j=1

βkj π ), 2

(10.2)

m = {k ∈ Z m , k ≥ 0, i = 1, m}, q(k) is the number of the coordiwhere Z+ i nates of the vector k that are equal to zero, and the numbers βkj are defined by the equalities

cos

βkj π βk π Re ψj (kj ) Im ψj (kj ) = , sin j = . 2 |ψj (kj )| 2 |ψj (kj )|

In this case, the set Lψ of ψ-integrals of real functions ϕ from L(Rm ) consists of real functions f. Moreover, if the series in (10.2) is the Fourier series of a certain summable function Dψ (t), then a necessary and sufficient condition for the inclusion f ∈ Lψ N is the representability of f by a convolution of the form −m ϕ(x − t)Dψ (t)dt, f (x) = (2π) Qm

where ϕ ∈ N and ϕ(x) = f ψ (x) almost everywhere. In particular, this means that the classes Lψ N include the classes of functions representable by convolutions with fixed summable kernels. 10.2. On a Relationship between the Spaces S p and Lp . Let Lp = Lp (Rm ), p ∈ [1, ∞), be the space of functions f ∈ L with finite norm · Lp , namely, f Lp = (

|f (t)|p dt)1/p .

(10.3)

Qm

The relationship between the sets Lp and S p is established by the known Hausdorff–Young theorem, which asserts the following: I. If f ∈ Lp , p ∈ (1, 2], and fˆ(k) are the Fourier coefficients of the function f defined according to the formula −m/2 ˆ f (k) = (2π) f (t)e−ikt dt, (10.4) Qm

Section 10

Remarks

then (

m( 12 − p1 )

|fˆ(k)|p )1/p ≤ (2π)

819

f Lp ,

k∈Z m

1 1 + = 1. p p

II. Let {ck }k∈Z m be a sequence of complex numbers for which |ck |p < ∞, p ∈ (1, 2]. k∈Z m

In this case, there exists a function f ∈ Lp for which fˆ(k) = ck and f Lp ≤ (2π)

m( 12 − p1 )

(

|ck |p )1/p ,

k∈Z m

1 1 + = 1. p p

It follows from this theorem that if p ∈ (1, 2], then m( 12 − p1 )

Lp ⊂ S p

and f S p ≤ (2π)

S p ⊂ Lp

and f Lp ≤ (2π)

m( 12 − p1 )

f Lp ,

(10.5)

f S p .

(10.6)

In particular, for p = p = 2, the following equalities are true: L2 = S 2

and

· L2 = · S 2 .

(10.7)

Relations (10.5) and (10.6) imply that the theorems proved for the spaces S p also contain information for the spaces Lp , which, by virtue of (10.7), is most complete in the case p = 2. Since this case is especially important, we give here the exact formulations of the corresponding statements. As above, let ψ = {ψ(k)}k∈Z m be an arbitrary system of complex numbers and let Lψ N be the set of ψ-integrals of all functions f ∈ N, where N is a certain subset of L = L(Rm ), m ≥ 1. As N, we take the unit ball UL2 in the space L2 : UL2 = {f : f ∈ L2 , f L2 ≤ 1}. (10.8) Here, the norm · L2 is defined by equality (10.3) for p = 2. In this case, we set Lψ UL2 = ULψ2 . Assuming that condition (9.5) is satisfied, we define the characteristic sequences ε(ψ), g(ψ), and δ(ψ); we also define the polynomials Sn (f ; x) according to (9.6), and, for f ∈ ULψ2 , we set ψ ψ Eψ n (f )L2 = f (x) − Sn−1 (f ; x)L2 , En (UL2 )L2 = sup En (f )L2 , ψ f ∈UL

2

Approximations in the Spaces S p

820

Enψ (f )L2 = inf f (x) − (2π)−m/2 ak

Chapter 11 ak eikx L2 ,

ψ k∈gn−1

and En (ULψ2 )L2 = sup Enψ (f )L2 . ψ f ∈UL

2

We also set dn (ULψ2 )L2 = inf

Fn ∈Gn

sup inf f − uL2 ,

ψ f ∈UL p

n ∈ N,

u∈Fn

d0 (ULψ2 ) = sup f L2 , ψ f ∈UL

2

where Gn is the set of all n-dimensional subspaces of L2 and ak eikx L2 en (ULψ2 )L2 = sup inf f (x) − (2π)−m/2 ψ f ∈UL

ak,γn

k∈γn

2

is the quantity of the best n-term approximation of the class ULψ2 in the space L2 (γn is an arbitrary collection of n vectors k ∈ Z m ). The following statement is true: Theorem 10.1. Let ψ = ψ(k)k∈Z m be a system of numbers satisfying conditions (9.5) and (9.13). Then, for any n ∈ N, we have En (ULψ2 )L2 = En (ULψ2 )L2 = εn ,

(10.9)

dδn−1 (ULψ2 )L2 = dδn−1 +1 (ULψ2 ) = . . . = dδn −1 (ULψ2 )L2 = En (ULψ2 )L2 = εn , e2n (ULψ2 )L2

= sup(q − n)/ l>n

l s=1

(10.10)

∗

ψ¯s−2

∗

= (l − n)/

l

ψ¯s−2 .

(10.11)

s=1

Here, εs and δs are the elements of the characteristic sequences ε(ψ) and δ(ψ), respectively, δ0 = 0, l∗ is a certain natural number, and ψ¯s = εn , δn−1 < s ≤ δn , n = 1, 2, . . . .

Section 10

ULψ2

Remarks

821

Proof. It follows from relations (10.7) and (10.8) that UL2 = U 2 and, hence, = Lψ 2 . Therefore, we have

ψ ψ ψ ψ ψ ψ Eψ n (UL2 )2 = En (L2 )2 , En (UL2 )L2 = En (L2 )2 , dn (UL2 )L2 = dn (L2 )2

and en (ULψ2 )2 = en (Lψ 2 )2 , n = 1, 2, . . . . This implies that equalities (10.8)–(10.11) follow from relations (9.14), (9.25), and (9.15). Note that, in the one-dimensional case, i.e., for m = 1, equalities (10.9) and (10.10), in a somewhat different terminology, were obtained as early as 1936 by Kolmogorov, who initiated the investigation of widths of various functional classes. Also note that, as follows from equalities (10.9) and (10.10), the values of the widths of the sets ULψ2 in the space L2 realize the approximations by sums (9.6), i.e., by polynomials that are the best in the sense of the widths in the spaces S p for all p ∈ [1, ∞) for the classes Lψ p . This fact allows one to assume that exactly sums (9.6) are the best apparatus for approximation (again, in the sense of Kolmogorov widths) in the spaces Lp for all p ≥ 1 for the corresponding sets ULψp : ULψp = Lψ ULp , ULp = {f : f ∈ Lp , f Lp ≤ 1}. They are a direct generalization of the known Sobolev spaces, which can be obtained from ULψp if we take ψ(k) in the form (10.1) for ψj (kj ) =

1,

kj = 0,

(ikj )rj , kj = 0, j = 1, m,

(10.12)

where rj are some real numbers. Let m = 2 and let the sequences ψ1 (k1 ) and ψ2 (k2 ) be given by (10.12) under the condition r1 = r2 = r > 0. The classes ULψ2 defined by such sequences were first considered by Babenko in [2] and [3] from the viewpoint of the problem of finding their widths. In fact, Babenko obtained relation (10.10) in this case. In the case under consideration, the characteristic sequence ε(ψ) consists of the elements εn = n−r , n ∈ N, and the sets gnψ are the sets of vectors k = (k1 , k2 ) ∈ Z 2 satisfying the condition k1 k2 ≤ n,

Approximations in the Spaces S p

822 where

kj =

1,

Chapter 11

kj = 0,

|kj |, kj = 0, j = 1, 2.

Relations of this type first appeared in the cited works of Babenko, and it is now customary to call them hyperbolic crosses.

11.

Theorems of Jackson and Bernstein in the Spaces S p

11.1. In this section, let S p , 1 ≤ p < ∞, be the space of 2π-periodic summable functions f (f ∈ L) defined on the real axis and satisfying the relation f S p = ( |fˆ(ν)|p )1/p < ∞, (11.1) ν∈Z

where fˆ(ν) = (2π)−1/2

π

f (x)e−iνx dx

(11.2)

−π

are the Fourier coefficients of a function f in the trigonometric system (2π)−1/2 eiνx , ν ∈ Z. According to Subsection 1.1, among all sums of the form tn−1 = αν eiνx , |ν|≤n−1

where αν are arbitrary complex numbers, the least deviation from a function f from S p for a given natural number n (n ∈ N ) is exhibited by its partial sum Sn−1 (f, x) =

eiνx fˆ(ν) √ . 2π |ν|≤n−1

(11.3)

In other words, if En (f )S p = inf f − tn−1 S p tn−1

is the best approximation of a function f ∈ S p by trigonometric polynomials tn−1 of degree n − 1 in the metric of the space S p , then the following equalities are true: |fˆ(ν)|p . (11.4) Enp (f )S p = f − Sn−1 (f )pS p = |ν|≥n

Section 11

Theorems of Jackson and Bernstein in the Spaces S p

823

Here, we continue the investigation of the approximation properties of the spaces S p . We introduce the notion of the kth modulus of continuity and establish statements analogous to the well-known theorems of Jackson and Bernstein. Let k be an arbitrary natural number (k ∈ N ) and let f ∈ S p . The modulus of continuity of order k of a function f is defined as the function ωk (t)S p = ωk (f, t)S p = sup Δku f (·)S p , |u|≤t

where Δku f (x)

=

k

(−1)k−i (

i=0

k )f (x + iu), i

(11.5)

is the finite difference of the function f of the kth order with step u. By using standard arguments, one can show that the functions ωk (t)S p possess all main properties of ordinary moduli of continuity. In particular, they possess the following properties: (a) ωk (0)S p = 0; (b) ωk (t)S p does not decrease on the interval [0, ∞); (c) ωk (t)S p is continuous on [0, ∞); (d) ωk (mt)S p ≤ mk ωk (t)S p , m ∈ N. In addition, the function ω1 (t) is semiadditive for k = 1: (e) ω1 (t1 + t2 )S P ≤ ω1 (t1 )S p + ω1 (t2 )S p , t1 > 0, t2 > 0. In this section, we prove direct and inverse theorems on approximation in the space S p in terms of the best approximations and moduli of continuity, establish Jackson inequalities of the form τ En (f )S p ≤ K(τ )ωk (f, )S p , τ > 0, n and study the problem of the least constant in these inequalities for fixed values of the parameters n, k, τ, and p, i.e., En (f )S p p Kn,k (τ )S = sup : f ∈ S p , f ≡ const . (11.6) ωk (f, nτ )S p

Approximations in the Spaces S p

824

Chapter 11

We also consider questions related to the constructive characteristics of functional classes by the majorants of the moduli of continuity of their elements. 11.2. Direct Theorems. The main results of this section are formulated in the statements presented below. Let M (τ ) be a set of bounded nondecreasing functions μ that differ from a constant on the interval [0, τ ]. Theorem 11.1. Suppose that f ∈ S p , 1 ≤ p < ∞, and f ≡ const. Then, for any τ > 0, the following inequality is true: τ En (f )S p ≤ Cn,k (τ )S p ωk (f, )S p , k, n ∈ N, n where df

Cn,k (τ )S p = ( inf

μ∈M (τ )

and

μ(τ ) − μ(0) kp 2

(11.8)

2 In (τ, μ) τ

In (τ, μ) = In,k,p (τ, μ) = inf

)1/p

(11.7)

ν≥n ν∈N 0

ν kp (1 − cos t) 2 dμ(t). n

(11.9)

In this case, there exists a function μ∗ ∈ M (τ ) that realizes the greatest lower bound in (11.8). Inequality (11.7) is unimprovable on the set S p in the sense that, for any k and n ∈ N, the following equality is true: Cn,k (τ )S p = Kn,k (τ )S p .

(11.10)

Theorem 11.2. For any function f ∈ S p , 1 ≤ p < ∞, the following inequalities are true: Enp (f )S p

≤

1 kp 2

2 In ( kp 2 )

π

t ωkp (f, )S p sin tdt, n, k ∈ N, n

(11.11)

0

where df

π

In (λ) = inf

ν≥n ν∈N 0

ν (1 − cos t)λ sin tdt, λ > 0, n ∈ N. n

(11.12)

Theorems of Jackson and Bernstein in the Spaces S p

Section 11 If, in addition,

kp 2

825

∈ N, then kp

kp 2 2 +1 , In ( ) = kp 2 2 +1

(11.13)

and inequality (11.11) cannot be improved for any n ∈ N. In the theorems below, we obtain upper bounds independent of n for the constants Kn,k (τ )S p for τ = π, which are unimprovable in several important cases. Theorem 11.3. For any function f ∈ S p , 1 ≤ p < ∞, and any k and n ∈ N, the following inequalities are true: p Kn,k (π)S p ≤

1 2

kp −1 2

In ( kp 2 )

≤

2kp +

kp 2 +1 kp 2 2 −1 ( kp 2 +

1)σ( kp 2 )

,

(11.14)

where the quantities In (λ) are defined by (11.12), df

σ(λ) = −

∞ m=[ λ ]+1 2

1 − (−1)[λ] 2m 1 ( λ ) 2m−1 ( ( ) 2m 2 m 2 −

m−1

(

j=0

2 2m ) ), λ > 0, (11.15) j 2(m − j)2 − 1

and [λ] is the integer part of the number λ. If equal to zero and p Kn,k (π)S p ≤

kp 2

+1

2kp

,

kp 2

∈ N, then the value σ( kp 2 ) is

kp ∈ N, n ∈ N. 2

(11.14 )

The statement below establishes the uniform boundedness of the constants Kn,k (π)S p with respect to all parameters under consideration (k, n ∈ N, 1 ≤ p < ∞). Theorem 11.4. For any function f ∈ S p , 1 ≤ p < ∞, different from a constant, the following inequalities are true: √ 4−2 2 π ω (f, (11.16) )S p , n, k = 1, 2, . . . . En (f )S p < k n 2k/2

Approximations in the Spaces S p

826

Chapter 11

In the case p = 2, the space S 2 coincides with the space L2 of functions f from L with the finite norm π f L2 = f 2 = (

|f (t)|2 dt)1/2 .

−π

In view of the importance of this case, we present the corresponding statements, which follow from Theorems 11.2 and 11.3. Theorem 11.2. For any function f ∈ L2 , the following inequality is true: En2 (f )2

k+1 ≤ 2k+1 2

π

t ωk2 (f, )2 sin tdt, n, k = 1, 2, . . . . n

(11.11 )

0

This inequality cannot be improved for any k and n ∈ N. Theorem 11.3. For any function f ∈ L2 different from a constant, the following inequalities are true: √ En (f )2
0, λ+1

(11.57)

0

which can be proved as follows: Expanding the function (1 − x)λ in a binomial series and taking into account the uniform convergence of the latter on the interval [−1, 1] and relations (11.42), (11.45), (11.46), and (11.48)–(11.51), for any θ > 0 and λ > 0 we can write df

π (1 − cos θt)λ sin tdt − (

Δ(θ, λ) =

2λ+1 + σ(λ)) λ+1

0

π ((1 − cos θt)λ − (1 − cos t)λ ) sin tdt − σ(λ)

= 0

=

∞ m=1

−

m−1 1 2m − 1 1 + cos(2m − 2j − 1)θπ λ ( ) ( ) 2m − 1 22m−2 j ((2m − 2j − 1)θ)2 − 1

∞ m=1

+

∞ m=1

j=0

m−1 1 2m 1 + cos(2(m − j)θπ) λ ( ( ) ) 2m 22m−1 j (2(m − j)θ)2 − 1 j=0

(

m−1 1 2m 2 λ ) 2m−1 ( ) − σ(λ). (11.58) 2m 2 j (2(m − j))2 − 1 j=0

First, assume that [λ] is even. In this case, we have

Theorems of Jackson and Bernstein in the Spaces S p

Section 11

λ ) > 0, 2m − 1

( (

λ ) ≥ 0, 2m − 1

( (

m = 1, 2, . . . , [λ/2], (11.59)

m = [λ/2] + 1, [λ/2] + 2, . . . ,

λ ) > 0, 2m

λ ) ≤ 0, 2m

837

m = 0, 1, 2, . . . , [λ/2], (11.59 )

m = [λ/2] + 1, [λ/2] + 2, . . .

and, according to (11.58), Δ(θ, λ) =

∞

m−1 1 2m − 1 1 + cos((2m − 2j − 1)θπ) λ ) 2m−2 ( ) 2m − 1 2 j ((2m − 2j − 1)θ)2 − 1

(

m=1

[λ/2]

+

m=1

+ (−

(

j=0

m−1 1 2m 2 1 + cos(2(m − j)θπ) λ ) 2m−1 ( )( − ) 2 2m 2 j (2(m − j)) − 1 (2(m − j)θ)2 − 1 j=0

∞ m=[λ/2]+1

m−1 1 2m 1 + cos(2(m − j)θπ) λ ( ( ). ) ) 2m 22m−1 j (2(m − j)θ)2 − 1

(11.60)

j=0

Hence, by using relations (11.59) and (11.59 ), we establish that each term on the right-hand side of equality (11.60) is nonnegative. Therefore, relation (11.57) is true. Now let [λ] be odd. Then ( λ ) > 0, m = 0, 1, . . . , [λ/2], 2m (11.61) λ ( ) ≥ 0, m = [λ/2] + 1, [λ/2] + 2, . . . , 2m

(

λ ) > 0, m = 0, 1, . . . , [λ/2], 2m + 1

λ ) ≤ 0, m = [λ/2] + 1, [λ/2] + 2, . . . , ( 2m + 1 and

(11.61 )

Approximations in the Spaces S p

838

[λ/2]

Δ(θ, λ) =

1 λ ) 2m−1 2m 2

(

m=1

×

m−1

(

2 1 + cos(2(m − j)θπ 2m )( − ) j (2(m − j))2 − 1 (2(m − j)θ)2 − 1

(

1 λ ) 2m + 1 22m

(

2m + 1 1 + cos((2m − 2j + 1)θπ) ) j ((2m − 2j + 1)θ)2 − 1

j=0

[λ/2]

+

m=0

×

m−1 j=0

∞

+{

(

m=[λ/2]+1

−

1 22m−1

m−1 j=0

∞

+

(

m=[λ/2]+1

×

Chapter 11

(

1 λ 2m ( )( ) 2m 22m−1 m 2m 1 + cos(2(m − j)θπ ) ) j (2(m − j)θ)2 − 1

1 λ ) 2m + 1 22m

m 1 + cos((2m − 2j + 1)θπ) ( 2m + 1 ) }. j ((2m − 2j + 1)θ)2 − 1

(11.62)

j=0

Taking into account (11.61) and (11.61 ), we conclude that the first two terms on the right-hand side of equality (11.62) are nonnegative. To verify the nonnegativity of the term in braces, note that this term is equal to ∞ m=[λ/2]+1

λ (( ) 2m

π cos

2m

λ θt sin tdt − ( ) 2m + 1

0

π cos2m+1 θt sin tdt). 0

This quantity is definitely nonnegative. This proves inequality (11.57) for θ > 1. Since σ(λ) ≤ 0 for any λ > 0, taking inequality (11.44) into account we also establish the validity of (11.57) in the case θ = 1. 11.7. Proof of Theorem 11.4. First, we show that In (λ) ≥ 2 ∀λ ≥ 1.

(11.63)

Section 11

Theorems of Jackson and Bernstein in the Spaces S p

839

If [λ] is even, then, using inequalities (11.44 ) and (11.59) for any θ ≥ 1, we get π (1 − cos θt) sin tdt = λ

∞ m=0

0

≥

∞ m=0

π =

(−1) ( λ ) m

π

m

cosm θt sin tdt 0

( λ ) 2m

π cos2m θt sin tdt 0

(1 + cos θt)λ + (1 − cos θt)λ sin tdt. 2

(11.64)

0

For λ ≥ 1, the function uλ is convex downwards. Therefore, (1 + cos θt)λ + (1 − cos θt)λ ≥ 1, 2 whence π

(1 + cos θt)λ + (1 − cos θt)λ sin tdt ≥ 2

0

π sin tdt = 2.

(11.65)

0

Comparing formulas (11.64) and (11.65) and taking into account the arbitrariness of the choice of the parameter θ ≥ 1, we get (11.63). If [λ] is odd, then, using inequality (11.57) and taking into account that, in this case, σ(λ) = −

∞

λ ( ) 2m

m=[λ/2]+1

π cos2m θt sin tdt

([λ] is odd),

0

we have [λ/2] λ 2λ+1 ( ) cos2m θt sin tdt ≥ 2. + σ(λ) = In (λ) ≥ 2m λ+1 π

m=0

(11.66)

0

Inequality (11.63) is proved. Let us show that In (λ) ≥ 1 + 2λ−1 , λ ∈ (0, 1).

(11.67)

Since, in this case, we have [λ] = 0, it follows from (11.57) and (11.15) that

Approximations in the Spaces S p

840

Chapter 11

∞ m−1 2λ+1 2 1 2m λ ( ( ) ) 2m−1 In (λ) ≥ + 2m j λ+1 2 (2(m − j))2 − 1 m=1

= 2(1 +

∞

(

m=1

=2+

∞ m=1

j=0

1 λ 2m ( ) )) 2m 22m m

∞ 1 λ ( ( λ )( 2m−1 ( 2m ) − 1) )+ 2m 2m 2 m m=1

= 1 + 2λ−1 +

∞

(

m=1

1 λ 2m )( 2m−1 ( ) − 1). 2m 2 m

(11.68)

2m ) ≤ 22m−1 , m ∈ N, m we can conclude that the sum on the right-hand side of (11.68) is nonnegative. Therefore, relation (11.67) is true. Combining (11.63) and (11.67) for λ = kp 2 , k ∈ N, p ∈ [1, ∞), we get

By virtue of relation (11.59 ) and the inequality (

In (

kp 1 )≥1+ √ . 2 2

Therefore, Kn,k (π)S p

√ 2 2 1/p √ ) ≤ k( 22 1 + 2 1

(11.69)

(11.70)

by virtue of (11.14) and (11.69). This yields (11.16). 11.8. Inverse Theorems. The main result in this direction is the following theorem: Theorem 11.6. If f ∈ S p , p ≥ 1, then the following inequality holds for any natural k and n : n π π k kp ωk (f, )S p ≤ k ( (ν − (ν − 1)kp )Eνp (f )S p )1/p . n n ν=1

(11.71)

Section 11

Theorems of Jackson and Bernstein in the Spaces S p

841

Proof. Let f ∈ S p and u ∈ [0, π/n]. Then, taking (11.19) into account, we obtain kp kp |fˆ(ν)|p (1 − cos νu) 2 Δku f (·)pS p = 2 2 ν∈Z

= 2kp

|fˆ(ν)|p | sin

|ν|≤n−1

+ 2kp

νu kp | 2

|fˆ(ν)|p | sin

|ν|≥n

νu kp | . 2

(11.72)

It is clear that the second term on the right-hand side of (11.72) does not exceed the value 2kp |fˆ(ν)|p = 2kp Enp (f )S p |ν|≥n

and

kp

2

|ν|≤n−1

νu kp |fˆ(ν)|p | sin | = 2kp 2

n−1 ν=1

νu (|fˆ(ν)|p + |fˆ(−ν)|p ) sinkp 2

π ν kp (|fˆ(ν)|p + |fˆ(−ν)|p ). ≤ ( )kp n n−1 ν=1

Therefore, π ν kp (|fˆ(ν)|p + |fˆ(−ν)|p ). Δku f (·)pS p ≤ 2kp Enp (f )S p + ( )kp n n−1

(11.73)

ν=1

The statement below can be verified directly and is necessary for what follows.

Lemma 11.1. Suppose that the numerical series ∞

cν

ν=1

is convergent. Then, for any sequence αν , ν ∈ N, the following equality holds for all natural m and M, m ≤ M : M ν=m

αν cν = αm

∞ ν=m

cν +

M ν=m+1

(αν − αν−1 )

∞ i=ν

ci − αM

∞ ν=M +1

cν . (11.74)

Approximations in the Spaces S p

842

Chapter 11

Setting αν = ν kp , cν = |fˆ(ν)|p + |fˆ(−ν)|p , m = 1, and M = n − 1 in (11.74), we get π ( )kp ν kp (|fˆ(ν)|p + |fˆ(−ν)|p ) n n−1 ν=1

∞ π = ( )kp ( (|fˆ(ν)|p + |fˆ(−ν)|p ) n ν=1

+

n−1

(ν kp − (ν − 1)kp )

ν=2

∞

(|fˆ(i)|p + |fˆ(−i)|p )

i=ν

− (n − 1)kp

∞

(|fˆ(ν)|p + |f (−ν)|p ))

ν=n

π = ( )kp ( (ν kp − (ν − 1)kp )Eνp (f )S p − (n − 1)kp Enp (f )S p ). (11.75) n n−1 ν=1

Hence, Δku f (·)pS p π ≤ ( )kp ( (ν kp − (ν − 1)kp )Eνp (f )S p n n−1 ν=1

− (n − 1)kp Enp (f )S p ) + 2kp Enp (f )S p π ≤ ( )kp (ν kp − (ν − 1)kp )Eνp (f )S p , n n

(11.76)

ν=1

which yields (11.71). For every fixed k ∈ N, inequality (11.71) cannot be improved in order on the entire space S p , p ∈ [1, ∞). Indeed, for any f ∈ S p and k, n ∈ N, we have π νπ kp ωkp (f, )S p ≥ Δkπ f (·)pS p = 2kp |fˆ(ν)|p | sin | n n 2n ν∈Z νπ kp ≥ 2kp |fˆ(ν)|p | sin | 2n |ν|≤n

2 ≥ ( )kp ν kp (|fˆ(ν)|p + |fˆ(−ν)|p ). n n

ν=1

(11.77)

Theorems of Jackson and Bernstein in the Spaces S p

Section 11

843

Applying to the last sum equality (11.74), where αν = ν kp , cν = |fˆ(ν)|p + |fˆ(−ν)|p , m = 1, and M = n, we get ωkp (f,

n π 2kp kp p (f )S p ). (11.78) )S p ≥ kp ( (ν − (ν − 1)kp )Eνp (f )S p − nkp En+1 n n ν=1

If f is a trigonometric polynomial Tn of at most nth degree, then En+1 (Tn )S p = 0. Therefore, by virtue of (11.71) and (11.78), we have n 2k kp ( (ν − (ν − 1)kp )Eνp (Tn )S p )1/p nk ν=1

n π πk ≤ ωk (Tn , )S p ≤ k ( (ν kp − (ν − 1)kp )Eνp (Tn )S p )1/p . n n ν=1

Since ν kp − (ν − 1)kp ≤ kpν kp−1 , it follows from inequality (11.71) that n π π k (kp)1/p kp−1 p ωk (f, )S p ≤ ( ν Eν (f )S p )1/p . n nk

(11.71 )

ν=1

This, in particular, yields the following statement: Corollary 11.1. Suppose that the sequence of the best approximations En (f )S p of a function f from S p , 1 ≤ p < ∞, satisfies the following relation for some α > 0 : 1 En (f )S p = O( α ). n Then, for any k ∈ N, one has ⎧ for α < k, O(tα ) ⎪ ⎪ ⎨ ωk (f, t)S p = O(tk | ln t|1/p ) for α = k, ⎪ ⎪ ⎩ O(tk ) for α > k. 11.9. The assertions obtained enable us to make a conclusion about constructive characteristics of the classes of functions defined by the kth moduli of continuity in the space S p . We say that a function ϕ(t), 0 ≤ t ≤ π, belongs to the class Φ and write ϕ ∈ Φ if the following conditions are satisfied: (i) ϕ(t) is continuous on [0, π]; (ii) ϕ(t) monotonically increases; (iii) ϕ(0) = 0.

844

Approximations in the Spaces S p

Chapter 11

ω , k ∈ N, ω ∈ Φ, denote the class of all functions f ∈ S p , Let Hk,S p 1 ≤ p < ∞, for which ωk (f, t)S p ≤ Cω(t), (11.79)

where C is a positive constant, generally speaking, different for different functions. We say that a function ϕ ∈ Φ satisfies the Bari condition (Br ), r ≥ 1, if n ν=1

1 1 ν r−1 ϕ( ) = O(nr ϕ( )). ν n

Theorem 11.7. Let ω(t) ∈ Φ be such that ω p (t), p ≥ 1, satisfies condition ω , it is (Bkp ). Then, in order that a function f ∈ S p belong to the class Hk,S p necessary and sufficient that 1 En (f )S p = O(ω( )). n ω . By virtue of Theorem 11.4, we get Proof. Let f ∈ Hk,S p

1 En (f )S p = O(ωk (f, )S p ), f ∈ S p . n

(11.80)

Therefore, relation (11.79) yields (11.80). On the other hand, according to (11.71 ), we have n 1 Ak kp−1 p ν Eν (f )S p )1/p , f ∈ S p . ωk (f, )S p ≤ k ( n n ν=1

Hence, taking (11.79) into account, we obtain n 1 1 kp−1 p 1 1/p ν ω ( )) ). ωk (f, )S p = O( k ( n ν n

(11.81)

ν=1

Since ω p satisfies condition (Bkp ), we have n ν=1

1 1 ν kp−1 ω p ( ) = O(nkp ω p ( )). ν n

(11.82)

Comparing (11.81) and (11.82), we get relation (11.79). The function ϕ(t) = tα , α < r, satisfies condition (Br ). Hence, denoting α ω α by Hk,S p the class Hk,S p for ω(t) = t , 0 < α ≤ k, we establish the following statement:

Section 11

Theorems of Jackson and Bernstein in the Spaces S p

845

Corollary 11.2. Let k ∈ N, 0 < α < k. In order that a function f ∈ S p α , p ≥ 1, it is necessary and sufficient that belong to Hk,S p En (f )S p = O(

1 ). nα

11.10. Let us make several more remarks on the case p ∈ (0, 1). As in the case p ≥ 1, we denote by En (f )S p the best approximation of a function f ∈ S p , p ∈ (0, 1), by trigonometric polynomials tn−1 of degree n − 1 in the metric of S p , and by ωk (f, t)S p , p ∈ (0, 1), we denote the kth modulus of continuity of this function, i.e., En (f ) = inf f − tn−1 S p , tn−1

ωk (f, t)S p = sup Δku f (·)S p , |u|≤t

p ∈ (0, 1), k ∈ N. Analyzing the proofs of Theorems 11.1–11.3, one can conclude that the condition 1 ≤ p < ∞ is not essential and can be replaced by the condition 0 < p < ∞; in this case, the statements of Theorems 11.1–11.3 remain true. An analog of Theorem 11.4 in the case p ∈ (0, 1) is the following statement: Theorem 11.4. If f ∈ S p , p ∈ (0, 1), then En (f )S p ≤

1 4 1/p π ωk (f, )S p , n, k = 1, 2, . . . . k ( ) n 22 3

(11.83)

The proof of this theorem is based on inequalities (11.14), which are also valid, as noted above, in the case where 0 < p < 1, and on inequalities (11.63) and (11.67) for λ = kp 2 , k ∈ N, p ∈ (0, 1). Under these restrictions, we have In (

kp ) ≥ 1.5, 2

which yields (11.83) by virtue of (11.14). The analysis of the proof of Theorem 11.6 shows that, for any f ∈ S p , inequality (11.71) remains true for p ∈ (0, 1). Hence, inequality (11.71 ) is also true in this case.

Approximations in the Spaces S p

846

Chapter 11

11.11. Also note that the results obtained here admit a generalization to the case of moduli of continuity whose order is defined by an arbitrary positive number k (including fractional numbers). These moduli are defined as follows: ωk (f, t)

Sp

= sup 0≤|u|≤t

Δku f (·)S p

= sup 0≤|u|≤t

∞ i=0

(−1)i (

k )f (· + (k − i)u)S p . i

In particular, Theorems 11.1–11.3, 11.6, and 11.7 remain true in the case where k is an arbitrary positive number.

12. APPROXIMATIONS BY ZYGMUND AND ´ POUSSIN SUMS DE LA VALLEE 0.1. In this chapter, we consider the approximation properties of the linear methods of summation of Fourier series generated by triangular matrices Λ = (n) (n) λk whose elements λk are defined by the equalities (n)

(n)

λk = λk (s) = 1 − (k/n)s , k = 0, 1, . . . , n − 1, s > 0, and (n)

λk =

(0.1)

⎧ ⎨1,

k = 0, 1, . . . , n − p, k−n+p , k = n − p + 1, . . . , n − 1, 1 ≤ p ≤ n. ⎩1 − p

(0.2)

In the first case, the polynomials Un (f ; x; Λ) for a given function f ∈ L, de(s) noted by Zn (f ; x) and called Zygmund sums, have the form a0 = (1 − (k/n)s )Ak (f ; x), + 2 n−1

Zn(s) (f ; x)

(0.3)

k=1

where Ak (f ; x) = ak cos kx + bk sin kx,

(0.4)

and ak = ak (f ) and bk = bk (f ) are the Fourier coefficients of the function f (·). In the second case, they are denoted by Vn,p (f ; x) and have the form Vn,p (f ; x) =

Vpn (f ; x)

n−1 1 = Sk (f ; x), p

(0.5)

k=n−p

where Sk (f ; x) are the partial sums of order k of Fourier series of the function f (·); these polynomials are called de la Vallée Poussin sums. 847

848

Approximations by Zygmund and de la Vallée Poussin Sums

Chapter 12

Zygmund and de la Vallée Poussin sums are characterized by the simplicity of (n) determination of multipliers λk and good approximation properties. The combination of these qualities, apparently, stimulated the extensive investigation of these sums and their special cases (Fejér and Riesz sums) for many decades by many prominent experts in the theory of functions. At present, a large amount of factual material is accumulated in numerous publications, some of which are not widely available.One of the most important directions in this field is the investigation of approximation properties of these sums for various classes of functions in different metrics. A systematic exposition of the results obtained in this direction may be the subject of a monograph. However, it is obviously beyond the scope of the present monograph. The aim of this chapter is to give a survey of known results related to the approximation properties of Zygmund and de la Vallée Poussin sums and to present new facts established for these sums. We are mainly interested in asymptotic equalities for the quantities E(N; Un (f ; x)) = sup f − Un (f ; x)X f ∈N

that realize solutions of the corresponding Kolmogorov–Nikol’skii problems. Recall that we say that, for a given method Un (f ; Λ) on the class M in the space X, the Kolmogorov–Nikol’skii problem is solved if the function ϕ(n) = ϕ(n, Λ; M) is determined in explicit form and is such that E(M; Un (f ; Λ)) = sup f (x) − Un (f ; x; Λ)X = ϕ(n) + O(ϕ(n)) f ∈M

as n → ∞.

1.

Fejér Sums: Survey of Known Results (s)

1.1. For s = 1, the sums Zn (f ; x) coincide with the known Fejér sums 1 = σn (f ; x) = Sk (f ; x). n n−1

Zn(1) (f ; x)

k=0

The famous Fejér theorem (see Fejér [1]) states: For any function f ∈ C, lim (f (x) − σn (f ; x)) = 0

n→∞

uniformly in x.

(1.1)

Section 1

Fejér Sums: Survey of Known Results

849

As noted above, this theorem gained Fejér a well-deserved reputation because it renewed an interest in Fourier series once spoiled by investigations of du Bois-Reymond [1], who constructed an example of a continuous periodic function whose Fourier series is divergent at certain points. The rate of convergence of the sequence σn (f ; x) was, apparently, investigated for the first time by Bernstein [1], who showed that the following relation holds for any f ∈ H α and α ∈ [0, 1): |f (x) − σn (f ; x)| < Kn−α ,

(1.2)

where K is an absolute constant and |f (x) − σn (f ; x)| ≤ K ·

ln n , n

(1.3)

if α = 1. (Here and in what follows, we use the notation of Chapter 3.) According to Theorem 3.5.3, if f ∈ H α , 0 < α < 1, then f˜ ∈ H α . Thus, ˜α: by virtue of (1.2), the following inequality holds for any f ∈ H |f (x) − σn (f ; x)| ≤ Kn−α .

(1.4)

˜ 1 , then, as proved by Alexits [2], If f ∈ H |f (x) − σn (f ; x)| ≤ Kn−1 .

(1.5)

It was also noted in the indicated work that the relation f (x) − σn (f ; x)C = O(n−1 )

(1.6)

˜ 1. holds if and only if f ∈ H This, in particular, implies that Fejér sums cannot realize an approximation of order better than n−1 for functions different from constants. Note that, as Butzer and Nessel wrote in [1], Zygmund announced the following result in his letter of 1940 to Hill: If f (x) − σn (f ; x) = o(n−1 ),

(1.7)

then f (x) ≡ const. Zygmund published this fact in 1945 [1, 2]. These results of Alexits and Zygmund, as well as investigations of Zamanski [1] and Favard [3], led to the introduction of the notion of saturation of linear methods, which is considered in Chapter 2.

850

Approximations by Zygmund and de la Vallée Poussin Sums

Chapter 12

In the notation accepted in Chapter 2, relations (1.5) and (1.7) mean that the Fejér method is saturated, its saturation order is ϕΛ (n) = n−1 , and its saturation ˜1 = W ˜ 1. class is H The next significant step in the investigation of approximations by Fejér sums was made by Nikol’skii in 1945 [8]. He considered the quantities E(N; σn ) = sup f (x) − σn (f ; x)C f ∈N

(1.8)

r ˜ r , W r H α , and in the case where N is one of the classes Wrr , Wr+1 = W r r r α r α Wr+1 H = Wr H . He proved the following theorems:

Theorem 1.1. If numbers r and α satisfy the inequalities r ≥ 0 and 0 ≤ α ≤ 1, then ⎧ O(n−r ), 0 ≤ r < 1, ⎪ ⎪ ⎨ E(Wrr ; σn ) = O(n−1 ln n), r = 1, (1.9) ⎪ ⎪ ⎩ O(n−1 ), r > 1, ⎧ O(n−(r+α) ), 0 ≤ r + α < 1, ⎪ ⎪ ⎨ E(Wrr H α ; σn ) = O(n−1 ln n), r + α = 1, (1.10) ⎪ ⎪ ⎩ O(n−1 ), r + α > 1, as n → ∞. In relations (1.9) and (1.10) the symbol O cannot be replaced by o. In particular, if r ≥ 1, then the following exact formulas are true: ⎧ 2 ⎪ ln n + O(n−1 ), r = 1, ⎨ πn r (1.11) E(Wr ; σn ) = ⎪ ⎩ Cr n−1 + O(n−r ), r > 1, ⎧ ln n ⎪ ⎪ r = 1, α = 0, + O(n−1 ), ⎪ ⎪ πn ⎪ ⎪ ⎨ (1.12) E(Wrr H α ; σn ) = C n−1 + O(n−(1+α) ), r = 1, 0 < α ≤ 1, 1,α ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ Cr,α n−1 + O(n−(r+α) ), r > 1, 0 ≤ α ≤ 1, where C1,α

2α−1 = ( π

π/2 π t t α t cot dt + (π − t)α cot dt), 0 < α ≤ 1, 2 2 0

π/2

(1.13)

Section 1

Fejér Sums: Survey of Known Results

851

the coefficient Cr is the upper bound 1 Cr = sup | |ϕ|≤1 π

2π Fr (t)ϕ(t)dt|,

Fr (t) =

∞

k −r+1 cos(kt + rπ/2), (1.14)

k=1

0

over all 2π-periodic functions ϕ(·), |ϕ| ≤ 1, with mean value zero on the period, i.e., π ϕ(t)dt = 0, (1.15) −π

and the coefficients Cr,α for r > 1 are equal to the upper bound Cr,α

1 = sup | α π ϕ∈H0

2π Fr (t)ϕ(t)dt|

(1.16)

0

H0α

of 2π-periodic functions ϕ(·) for which ϕ(0) = 0. For over the class r = 2, 3, 4, . . . , one has Cr =

∞ 4 (−1)k(r−1) , π (2k + 1)r

(1.17)

k=0

and, for r = 3, 5, 7, . . . and 0 ≤ α ≤ 1, Cr,α

π/2 π ∞ 2α −r+1 α = k ( t sin ktdt + (π − t)α sin ktdt). π k=1

0

(1.18)

π/2

The following equality corresponding to the case r = 0 is also true: E(H α ; σn ) =

2Γ(α)n−α απ sin + O(n−1 ), 0 < α < 1. π(1 − α) 2

(1.19)

Theorem 1.2. If numbers r and α satisfy inequalities r ≥ 0 and 0 ≤ α ≤ 1, then O(n−r ), 0 < r < 1, r ˜ (1.20) E(Wr ; σn ) = O(n−1 ), r ≥ 1, ⎧ O(n−(r+α) ), 0 < r + α < 1, ⎪ ⎪ ⎨ r α (1.21) E(W O(n−1 ln n), r + α = 1, 0 < r < 1, r H ; σn ) = ⎪ ⎪ ⎩ O(n−1 ), r + α > 1, as r → ∞.

852

Approximations by Zygmund and de la Vallée Poussin Sums

Chapter 12

In particular, the following more exact formulas are true: ˜ r ; σn ) = C¯r n−1 + O(n−r ), r > 1, E(W r ⎧ πα ⎪ ⎪ + O(n−(r+α) ), r = 1, 0 < α ≤ 1, ⎨ (α + 1)n r α E(W r H ; σn ) = ⎪ ⎪ ⎩ ¯ Cr,α n−1 + O(n−(r+α) ), r > 1, 0 ≤ α ≤ 1,

(1.22)

(1.23)

where the constant C¯r is the upper bound 2π F¯r (t)ϕ(t)dt|, F¯r (t) =

C¯r = sup | |ϕ|≤1

∞

k −r+1 sin(kt + rπ/2),

(1.24)

k=1

0

over all 2π-periodic functions ϕ(·), |ϕ| ≤ 1, satisfying condition (1.15), and the constant C¯r,α is equal to the upper bound 1 = sup | α π ϕ∈H0

C¯r,α

2π F¯r (t)ϕ(t)dt|.

(1.25)

0

For r = 2, 3, 4, . . . , one has ∞ 4 (−1)kr C¯r = , π (2k + 1)r

(1.26)

k=0

and, for r = 2, 4, 6, . . . and 0 ≤ α ≤ 1, C¯r,α

π/2 π ∞ 2α ¯r+1 α k ( t sin ktdt + (π − t)α sin ktdt). = π k=1

0

(1.27)

π/2

˜ 1 ; σn ), the following asymptotic equality was found by Stechkin For E(W 1 (see Telyakovskii [2]): ˜ 1 ; σn ) = 2 E(W 1 πn

∞ ∞ sin t | dt|du + O(n−2 ). t2 0

(1.28)

u

Theorems 1.1 and 1.2 establish the exact orders of deviations of Fejér sums r α ) r and W on the classes Wrr and W r H α and on the conjugate classes W r rH

Section 1

Fejér Sums: Survey of Known Results

853

for any r ≥ 0 and 0 ≤ α ≤ 1, and, in a series of important cases (see equalities (1.11), (1.12), (1.22), and (1.23)), they completely solve the corresponding Kolmogorov–Nikol’skii problem in the uniform metric. One year later, investigating approximations of periodic functions by trigonometric polynomials in the integral metric, Nikol’skii [8], in particular, showed that relations (1.11) remain valid if E(Wrr , σn ) is replaced by r ; σn )1 = sup f (x) − σn (f ; x)1 , E(Wr,1 r f ∈Wr,1

i.e., the following equalities are true: ⎧ 2 ⎪ ln n + O(n−1 ), r = 1, ⎨ πn r E(Wr,1 ; σn )1 = ⎪ ⎩ Cr n−1 + O(n−r ), r > 1.

(1.29)

(1.30)

The constants Cr and C¯r defined by (1.17) and (1.26) were later called the ˜ r , respectively. They satisfy the Favard constants; they are denoted by Kr and K following equalities (see, e.g., Telyakovskii [4]): Kr = sup |f˜ (x)|, f ∈Wrr

¯ r = sup |f˜ (x)|, K

r = 2, 3, . . . .

(1.31)

¯r f ∈W r

Therefore, for r = 2, 3, 4, . . . , the second equality in (1.11) and equality (1.22) can be rewritten in the form 1 + O(n−r ) n

(1.31 )

1 + O(n−r ). n

(1.22 )

E(Wrr ; σn ) = sup |f˜ (x)| f ∈Wrr

and ˜ r ; σn ) = sup |f˜ (x)| E(W r ˜r f ∈W r

In this form, relations (1.11) and (1.22) were obtained by Nagy [7]; later, they were generalized by Telyakovskii [4] to the classes Wβr for any r > 1 and β ∈ R: E(Wβr ; σn ) = sup |f˜ (x)|n−1 + O(n−r ). (1.32) f ∈Wβr

The most complete results on the classes Wβr for r ≤ 1 also belong to Telyakovskii. Namely, he proved the following theorem [3]: Theorem 1.3. The following asymptotic equalities hold as n → ∞ :

854

Approximations by Zygmund and de la Vallée Poussin Sums

Chapter 12

(i) if r < 1, then E(Wβr ; σn ) = A(μr )n−r + O(n−1 ),

(1.33)

where ∞ ∞ | μr (u) cos(ut + βπ/2)du|dt, A(μr ) ≤ const, (1.34) A(μr ) = −∞

0

μr (u) =

1−r u , 0 ≤ u ≤ 1, u−r ,

u ≥ 1;

(1.35)

(ii) if r = 1, then E(Wβr ; σn ) =

2 βπ ln n | sin | + O(n−1 ); π 2 n

(1.36)

if, in addition, β = kπ, k ∈ Z, then E(Wβr ; σn ) = A(μr )n−1 + O(n−2 ).

(1.37)

Note that relations (1.32)–(1.37) give a complete solution of the Kolmogorov– Nikol’skii problem for Fejér sums on the classes Wβr for any r > 0 and β ∈ R1 . After the aforementioned results of Nikol’skii related to the approximation by Fejér sums on the classes Wβr Hω for ω(t) = tα and β = r, the most general results on these classes were obtained by Efimov [1–13]. Theorem 1.4. For any modulus of continuity ω = ω(t), the following asymptotic equalities hold as n → ∞ : (i) if r = 0, then E(Wβ0 Hω ; σn ) 1

2| cos βπ 2 | = nπ

2| sin βπ ω(t) 2 | dt + dn (ω) + O(ω(n−1 )), (1.38) 2 t π

1/n

where 1/n dn (ω) = sup{| 0

ϕ(t) dt| : ϕ ∈ Hω , ϕ(−t) = −ϕ(t)}; t

(1.39)

Section 1

Fejér Sums: Survey of Known Results

855

in the case where the function ω(t) is convex, we have 1 dn (ω) = 2

1/n ω(2t) dt; t 0

(ii) if 0 < r < 1, then, for any β ∈ R, 2| sin βπ 2 | = (k + 1)Δ2 (k 1−r )dn,k (ω)| | πn [n/2]

E(Wβr Hω ; σn )

k=0

[n/2] 1 −r + O( k ω(k −1 )) + O(n−r ω(n−1 )), (1.40) n k=1

where Δ2 (αk ) = αk − 2αk+1 + αk+2 , k = 0, 1, . . . , and 1/(k+1)

dn,k = sup{|

ϕ(t) dt| : ϕ ∈ Hω , ϕ(−t) = −ϕ(t)}, t

(1.41)

1/n

k = 0, . . . , n − 1; in the case where the function ω(t) is convex, we have 1 dn,k (ω) = 2

1/(k+1)

ω(2t) dt; t

1/n

(iii) if r = 1, then, for any β ∈ R, E(Wβ1 Hω ; σn ) =

2| sin(βπ/2)| dn,0 (ω) + O(n−1 ). πn

(1.42)

In view of the first equality in (1.20), for r = 0 and 0 < α < 1 we get E(W00 H α ; σn ) = O(n−α ).

(1.43)

Hence, in this case, relation (1.38) does not give a solution of the Kolmogorov– Nikol’skii problem because the remainder in (1.38) has the order O(n−α ).

856

Approximations by Zygmund and de la Vallée Poussin Sums

Chapter 12

At the same time, one can indicate moduli of continuity ω = ω(t) for which relation (1.38) gives a solution of this problem. For example, this is true for the modulus of continuity ω ∗ (t) = t ln 1t . Indeed, in this case, 1 n

1

ω ∗ (t) 1 2 1 1 dt = ln n = ω ∗ ( ) ln n, 2 t 2n 2 n

(1.44)

1/n

and equality (1.38) yields E(W00 Hω∗ ; σn ) =

1 ∗ −1 ω (n ) ln n + O(ω ∗ (n−1 )). π

(1.45)

It is clear that one can make the same remark for equality (1.40), comparing it with relation (1.20) for 0 < r + α < 1, and also for equality (1.42). If we restrict ourselves to integer values of the parameter r, then one can obtain final solutions of the Kolmogorov–Nikol’skii problem. In this connection, we present several statements established by Stepanets (see, e.g., [21, pp. 222– 251]). Theorem 1.5. For an arbitrary modulus of continuity ω(t), the following relations are true: 2 E(Hω ; σn ) = πn

π/2 sin2 nt dt, ω(2t) sin2 t

(1.46)

0

1 E(W 1 Hω ; σn ) ≤ nπ

π/2

ω(2t) dt + O(n−1 ω(n−1 )), sin t

(1.47)

1 ˜ Mr−1 (ω) + O(n−r ln nω(n−1 )), r ≥ 2, n

(1.48)

π/n

E(Wrr Hω ; σn ) = where

˜ r (ω) = {sup |f˜(0)| : f ∈ Wrr Hω , f ⊥1}. M

(1.49)

If ω(t) is a convex modulus of continuity, then relation (1.47) is an equality. Analogous statements are also true for the conjugate classes. Theorem 1.6. For an arbitrary modulus of continuity ω = ω(t), the following inequality holds:

Section 1

Fejér Sums: Survey of Known Results

857

˜ ω ; σn ) E(H 1 ≤ π

x1

∞ 2t sin t 2 ρi (t) − t sin t df ω( ) 2 dt + ω( )| |dt = En (ω), (1.50) n t π n t iπ

i=1 x

0

i

where xi , i = 1, 2, . . . , are the zeros of the function ∞ ˜ 1 (t) = Φ

sin τ dτ, τ2

t > 0,

(1.51)

t

enumerated in ascending order, and ρi (t) are the functions defined on [xi , iπ], i = 1, 2, . . . , by the equalities ˜ 1 (t) = Φ ˜ 1 (ρi (t)), Φ

xi ≤ t ≤ iπ ≤ ρi (t) ≤ xi+1 .

If ω(t) is a convex modulus of continuity and, in addition, the inequality 2i

(−1)k ω(

k=1

xi+1 − xi 1 2x2i+1 2x1 ) ≤ (ω( ) − ω( )) n 2 n n

(1.52)

holds for all i = 1, 2, . . . , [n/4] + 1, then ˜ ω ; σn ) = En (ω) + γn(0) , γn(0) ≤ 0, γn(0) = O(n−1 ω(n−1 )). E(H

(1.53)

Moreover, the quantity En (ω) admits the representation 1 En (ω) = π

x1

sin t 2 ω(2t/n) 2 dt + t πn

0

∞

˜ 1 ; t)ω (t/n)dt, Ψ1 (Φ

0

where ˜ 1 ; t) = Ψ1 (Φ

∞ ¯ ˜ 1,i (t), Φ i=1

¯˜ (t) is a decreasing rearrangement of the function and Φ 1,i ˜ 1 (t)|, t ∈ [xi , xi+1 ], |Φ ˜ 1,i (t) = Φ ¯ [xi , xi+1 ], i = 1, 2, . . . , 0, t∈ ˜ 1,i (y) > y). i.e., the function inverse to M (y) = mesE(Φ

(1.54)

858

Approximations by Zygmund and de la Vallée Poussin Sums

Chapter 12

It is clear that, under conditions that guarantee equality (1.53), this equality gives a solution of the corresponding Kolmogorov–Nikol’skii problem. It is easy to verify that such conditions are satisfied if ω(t) = tα , 0 < α ≤ 1. In the general case, relation (1.50) yields ˜ σn ) < 2 E(Hω; π

x1 /n

ω(t) 0

dt 1 ω(π/x1 ), + t πx1

2.15624 < x1 < 2.15625.

This, in particular, implies that if 1/n

ω(t) dt = O(ω(n−1 )), t

n → ∞,

(1.55)

0

then

˜ ω ; σn ) = O(ω(n−1 )). E(H

(1.56)

Theorem 1.7. For an arbitrary modulus of continuity ω = ω(t), the following relation is true: ⎧ ⎪ π/2 ⎪ ⎪ 2 ⎪ ⎪ ω(2t)dt + O(n−1 ω(n−1 )), r = 1, (1.57) ⎪ ⎨ πn r 0 E(W r Hω ; σ n ) = ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ Mr−1 (ω) + O( ln n ω(n−1 )), r = 2, 3, . . . , (1.58) n nr where Mr (ω) = {sup |f (0)| : f ∈ Wrr Hω , f ⊥1}.

(1.59)

Equality (1.46) is almost obvious, but it lacks information. In the case where ω(t) = tα , using this equality, Nikol’skii obtained formula (1.19) by fairly subtle reasoning. Further investigations of the integral π/2 sin2 nt tα dt, sin2 t

0 < α ≤ 1,

0

were carried out by Telyakovskii [12], who established the following statement: Theorem 1.8. Suppose that n → ∞, taking only even or only odd values. Then the following asymptotic expansions are true:

Section 1

Fejér Sums: Survey of Known Results

E(H 1 ; σn ) ≈

859

2 ln n 2 + (1 + c + ln 2) π n πn ∞ 1 −(2k+1) n (2 − 1/k)[1 + (−1)n (1 − 22k )]B2k , + π k=1

E(H α ; σn ) ≈

2Γ(α) sin(απ/2) π(1 − α)nα ∞

+

Aα sin(απ/2) −(2k+α) Γ(2k − 1 + α) B2k n + n π (2k − 2)! k k=1

+

∞ k=1

+

∞

(−1)n+k+1 2Γ(2k + 1 − α) { 2k+2−α n2k+1 π Γ(2 − α) (−1)i

i=1

Γ(2i − 1 + α) π 2i−2−2k+α B2i }, (2i − 2)! Γ(2i − 2k + α) i

where c is the Euler constant and B2k are Bernoulli numbers, i.e., B2k = (−1)k−1 2

∞ (2k)! −2k s . (2π)2k s=1

Nikol’skii [8] showed that, for certain linear methods of approximation, the upper bounds of deviations in the metric of C and the corresponding upper bounds in the metric of L over classes of differentiable functions coincide or are asymptotically equal. More exactly, Nikol’skii established that, for any λmethod, the following relation is true: r E(Wr,1 ; Un (f ; λ))1 ≤ E(Wrr ; Un (f ; λ))C , r ∈ N.

(1.60)

Later, Stechkin and Telyakovskii [1] showed that the same relation is also true r for all r > 0 and β ∈ R, namely for the classes Wβ,1 r ; Un (f ; Λ))1 ≤ E(Wβr ; Un (f ; Λ))C . E(Wβ,1

(1.61)

Motornyi [2] extended this statement to the classes Wβr Hω1 , where π Hω1 = {ϕ ∈ L,

|f (x + t) − f (x)|dx ≤ ω(t)} −π

(1.62)

860

Approximations by Zygmund and de la Vallée Poussin Sums

Chapter 12

and ω(t) is a given modulus of continuity, and showed that, for all r ≥ 0 and β ∈ R, the following relation is true: E(Wβr Hω1 ; Un (f ; Λ))1 ≤ E(Wβr Hω ; Un (f ; Λ))C .

2.

(1.63)

Riesz Sums: A Survey of Available Results

The sums Zns (f ; x) with s = 2 are called Riesz sums. However, in the mathematical literature, we most often encounter the so-called spherical Riesz sums of order δ introduced by Bochner in [1] in the multidimensional case. In the analyzed one-dimensional case, these sums take the form a0 k2 + = (1 − 2 )δ Ak (f ; x), δ > 0, 2 n n−1

Snδ (f ; x)

k=1

and, hence, coincide with Zn2 (f ; x) for δ = 1, For the sums Sn1 (f ; x), it is possible to establish several facts unknown for the Zygmund sums with s > 2 . We now present some of these results obtained by Stepanets in [21]. Theorem 2.1. Let ω(t) be an arbitrary modulus of continuity. Then E(Hω ; Sn1 ) = sup f (x) − Sn1 (f ; x)C f ∈Hω

x1 ∞ ci 4 ρi (t) − t ≤ ( ω(t/n)ψ(t)dt + ω( )|ψ(t)|dt) π n i=1 x

0

i

df

= E1n (ω),

(2.1)

where ψ(t) =

sin t − t cos t , t3

ci and xi , i = 1, ∞, are the zeros of the functions ψ(t), t > 0 and ∞ ψ1 (t) = t

1 sin t cos t ψ(τ )dτ = ( 2 − − sit), t > 0, 2 t t

(2.2)

Section 2

Riesz Sums: A Survey of Available Results

861

respectively, numbered in the ascending order, and ρi (t) are the functions defined on the segments [xi , ci ] by the equalities ψ1 (t) = ψ1 (ρi (x)), xi ≤ t ≤ ci ≤ ρi (t) ≤ xi+1 , ∞ six = −

sin t dt. t

x

If ω(t) is a convex modulus of continuity and the condition 2m

(−1)k ω(

k=0

xk+1 − xk x2m+1 df ) ≤ ω( ), m = 1, 2, . . . , [n/2] + 1, x0 = 0 n n

is satisfied, then 1 E(Hω ; Sn1 ) = E1n (ω) + γn(0) , γn(0) ≤ 0, γn(0) = O( ω(1/n)). n

(2.3)

Moreover, the quantity E1n (ω) admits a representation E1n (ω)

4 = πn

∞

Ψ0 (ψ1 ; t)ω (t/n)dt,

0

where Ψ0 (ψ1 ; t) =

∞

ψ¯1,i (t), and ψ¯1,i (t) is a decreasing permutation of the

i=0

function ψ1,i (t) =

|ψ1 (t)|, t ∈ [xi , xi+1 ], 0,

df

t ∈ [xi , xi+1 ], i = 0, ∞, x0 = 0.

For any modulus of continuity ω(t), the following estimate is true: E1n (ω)

4 ≤ 3π

x1 ω(t/n)dt + 0

4 4 3.34 4 ω(π/n) < ω( )+ ω(π/n). (2.4) πx1 3π n 3.3π

If ω(t) = M tα , 0 < α ≤ 1, then all assumptions of Theorem 2.1 are satisfied and (0) (0) (0) E(M H α , Sn1 ) = E1n (α) + γn,α , γn,α ≤ 0 and γn,α = O(n−(1+α) ),

862

Approximations by Zygmund and de la Vallée Poussin Sums

where E1n (α)

4M α = πnα

Chapter 12

∞ Ψ0 (ψ1 ; t)tα−1 dt. 0

In particular, for α = 1, E1n (1)

4M = nπ

∞

2M |ψ1 (t)|dt = π

0

∞ |

sin t cos t − − sit|dt. t2 t

(2.5)

0

Theorem 2.2. Assume that ω(t) is an arbitrary modulus of continuity. Then E(W

1

Hω ; Sn1 )

2 ≤ πn

t1

∞

2t 4 ω( )ψ1 (t)dt + n πn

xi ω(

i=1 t i

0

qi (t) − t )|ψ1 (t)|dt n

df

= E1n (1; ω), where ψ1 (x) is the function defined by equality (2.2), xi and ti are, respectively, the zeros of the functions ψ1 (x), x > 0, and ∞ ψ2 (t) =

1 sin t Ψ1 (τ )dτ = ( + cos t + tsit), t > 0, 2 t

t

numbered in the ascending order, and qi (t) are the functions defined on the segments [ti , xi ] by the equalities ψ2 (t) = ψ2 (qi (t)), ti ≤ t ≤ xi ≤ qi (t) ≤ ti+1 . If ω(t) is a convex modulus of continuity and, for all i = 1, 2, . . . , [n/4] + 1, 2i

(−1)k ω(

k=1

tk+1 − tk 1 2t2i+1 2t1 ) ≤ (ω( ) − ω( )), n 2 n n

then 1 E(W 1 Hω ; Sn1 ) = E1n (1; ω) + γn(1) , γn(1) ≤ 0, γn(1) = O(n−2 ω( )). n Moreover, the quantity E1n (1; ω) can be represented in the form E1n (1; ω)

2 = πn

t1 0

4 ω(2t/n)ψ1 (t)dt + πn2

∞ 0

Ψ1 (ψ2 ; t)ω (t/n)dt,

Section 3

Zygmund Sums: A Survey of Available Results

where Ψ1 (ψ2 ; t) =

∞

863

¯ 2,i (t), Ψ

i=1

and ψ¯2,i (t) is a decreasing permutation of the function |ψ2 (t)|, t ∈ [ti , ti+1 ], ψ2,i (t) = 0, t∈[ti , ti+1 ], i = 1, ∞. Theorem 2.3. Assume that ω = ω(t) is an arbitrary modulus of continuity. Then, for any r ≥ 2, the following equality is true: E(W r Hω ; Sn1 ) = n−2 Mr−2 (ω) + γn(r) ,

(2.6)

where the quantity Mr−2 (ω) is defined by equality (1.59) and r = 2, O(n−2 ω(1/n)), (r) γn = O(min(n−3 , n−r ln nω(1/n))), r ≥ 3.

3.

Zygmund Sums: A Survey of Available Results

The Zygmund sums were introduced by Zygmund [7] for all s > 0 . In the cited work, Zygmund proved the following assertions establishing the exact orders of deviations of these sums on the classes Wrr and Wrr Hω . Theorem 3.1. Let s be an even nonnegative number. Then, for any f ∈ Wss , f (x) − Zns (f ; x)C ≤ As n−s ;

(3.1)

s−1 for any f ∈ Ws−1 Hω , ω(t) is an arbitrary modulus of continuity and

f (x) − Zns (f ; x)C ≤ As n−(s−1) ω(2π/n).

(3.2)

If s is an odd number, then, for any f ∈ Wss , f (x) − Zns (f ; x)C ≤ As n−s ln(n + 2)

(3.3)

s−1 α H , 0 < α < 1, and, for any f ∈ Ws−1

f (x) − Zns (f ; x)C ≤ As n−(s−1+α) .

(3.4)

864

Approximations by Zygmund and de la Vallée Poussin Sums

Chapter 12

In relations (3.1)–(3.4), As are quantities uniformly bounded in n. ˜ ss , Theorem 3.2. Let s be an odd number. Then, for any f ∈ W f (x) − Zns (f ; x)C ≤ As n−s ; ˜ s−1 Hω , ω(t) is an arbitrary modulus of continuity and for any f ∈ W s−1 f (x) − Zns (f ; x)C ≤ As n−s ω(2π/n).

(3.5)

The investigations carried out by Zygmund were continued by Nagy [7] and then by Telyakovskii [3]. Here, we present only the theorem proved by Telyakovskii because it contains the corresponding result of Nagy. Theorem 3.3. The quantities E(Wβr ; Zns ) = sup f (x) − Zn(s) (f ; x)C , r > 0, s > 0, β ∈ R, f ∈Wβr

have the following asymptotic relations: (i) if r < s, then E(Wβr ; Zns ) = A(μr )n−r + O(n− min(r+1,s) ), where 1 A(μr ) = π

∞ ∞ | μr (v) cos(vt + βπ/2)dv|dt, −∞

(3.6)

(3.7)

0

μr (v) =

v s−r , 0 ≤ u ≤ 1, v −r , v ≥ 1;

(3.8)

(ii) if r = s, then E(Wβr ; Zns ) =

2 βπ ln n + O(n−r ). | sin | π 2 nr

(3.9)

If, in addition, sin(βπ/2) = 0, then E(Wβr ; Zns ) = A(μr )n−r + O(n−(r+1) ), where A(μr ) is given by relation (3.7);

(3.10)

Section 3

Zygmund Sums: A Survey of Available Results

865

(iii) if r > s, then E(Wβr ; Zns ) = sup |f0s (x)|n−s + O(n−r ),

(3.11)

f ∈Wβr

where f0s (x) is a continuous function expandable in the Fourier series ∞

k s Ak (f ; x).

(3.12)

k=1

Relations (3.9) and (3.11) were obtained by Nagy in the cited work. The most general results for the quantities E(Wβr Hω ) =

sup

f ∈Wβr Hω

f (x) − Zns (f ; x)

were obtained by Efimov [10] who proved the following theorem: Theorem 3.4. The following assertions are true: (i) if s < r, then E(Wβr Hω ; Zns ) [n/2] 2| sin(βπ/2)| (k + 1)Δ2 (k s−r )dn,k (ω) + O(n−s ); (3.13) = πns k=0

(ii) if s = r, then E(Wβr Hω ; Zns ) =

2| sin(βπ/2)| dn,0 (ω) + O(n−r ); πnr

(3.14)

(iii) if r < s < r + 1, then E(Wβr Hω ; Zns ) =

[n/2]−1 2| sin(βπ/2)| | (k + 1)Δ2 (k s−r )dn,k (ω)| πns k=0

+ O(n−s

[n/2]−1

(k + 1)−(r+1+s) ω(

k=0

+ O(n

−r

ω(1/n));

1 )) k+1 (3.15)

866

Approximations by Zygmund and de la Vallée Poussin Sums

Chapter 12

(iv) if s = r + 1, then E(Wβr Hω ; Zns )

1

2| cos(βπ/2)| = πnr+1

ω(t) dt+O(n−r ω(1/n)); (3.16) t2

1/(n+1)

(v) if s > r + 1, then E(Wβr Hω ; Zns ) = O(n−r ω(1/n)).

(3.17)

The quantities dn,k (ω) have the same meaning as in Theorem 3.3.

4.

ψ Zygmund Sums on the Classes Cβ,∞

In this section, we present the results dealing with the construction of asymptotic equalities for the quantities ψ ψ E(Cβ,∞ ; Zns ) = sup{|f (x) − Zns (f ; x)| : f ∈ Cβ,∞ }.

These results were mainly obtained by Bushev [1] and Bushev and Stepanets [1]. If τ (v) is a function defined on the semiaxis v ≥ 0 and such that its transformation ∞ 1 τˆ(t) = τˆ(t; β) = τ (v) cos(vt + βπ/2)dv (4.1) π 0

is absolutely integrable on R1 , then, in view of (3.7), we set ∞ |ˆ τ (t)|dt.

A(τ ) =

(4.2)

−∞ ψ The asymptotic equalities for the quantities E(Cβ,∞ ; Zns ) strongly depend on s the behavior of the function g(v) = v ψ(v) at infinity. Here, we assume that this function is convex upward or downward for all v ≥ 1. The condition of convexity of the function g(v) (and the function ψ(v) ) can be imposed not in the entire set v ≥ 1 but in the set v ≥ v0 , where v0 is an arbitrary real number greater than one. In this case, the corresponding asymptotic equalities are also obtained in the cited works but their formulations are fairly complicated and, thus, we do not present them here.

Section 4

ψ Zygmund Sums on the Classes Cβ,∞

867

If the function g(v) is convex for v ≥ 1, then the following cases are possible: (a) g(v) is convex downward and (b) g(v) is convex upward and (c) g(v) is convex downward and (d) g(v) is convex upward and (e) g(v) is convex downward and

lim g(v) = ∞,

v→∞

lim g(v) = ∞,

v→∞

lim g(v) = C > 0,

v→∞

(4.3)

lim g(v) = C > 0,

v→∞

lim g(v) = 0.

v→∞

In formulating the results, these cases should be distinguished, first of all, because the degree of saturation for the method Zns is specified by the quantity n−s (see Chapter 2). In what follows, by O(1) we denote the quantities uniformly bounded in the parameter n, i.e., in the degree of the polynomials Zns (f ; x). The following statements are true: Theorem 4.1. Assume that sin βπ/2 = 0, ψ ∈ M, and g(v) satisfies one of the conditions (a)–(d) in relations (4.3). Then, as n → ∞, ψ E(Cβ,∞ ; Zns ) = ψ(n)A(τn ) + O(1)ψ(n)/n,

(4.4)

where A(τn ) is a quantity defined by equality (4.2) for ⎧ sψ(1) + ψ (1) (1 − s)ψ(1) − ψ (1) ⎪ ⎪ v + , 0 ≤ v ≤ 1/n, ⎪ ⎪ ns−1 ψ(n) ns ψ(n) ⎪ ⎪ ⎨ τn (v) =

v s ψ(nv)/ψ(n), ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ ψ(nv)/ψ(n),

1/n ≤ v ≤ 1,

(4.5)

v ≥ 1.

Moreover, A(τn ) = O(1). If ψ(v) = v −r , 0 < r ≤ s, then equality (4.4) coincides with equality (3.6). In addition, if r < s < r + 1, then the estimate of the remainder in (4.4) is slightly better than in (3.6).

868

Approximations by Zygmund and de la Vallée Poussin Sums

Chapter 12

As examples of the functions ψ(v) for which Theorem 4.1 is true, we can use the functions ψ(v) = v −r lnα (v + c) for r ∈ (0, s), c > 0 and any real α, ψ(v) = v −s arctan u, and ψ(v) = v −s (C − e−v ), c > 0, etc. Theorem 4.1. Assume that sin βπ/2 = 0, ψ ∈ M, the sequence k s ψ(k), k = 1, 2, . . . , is convex downward, and lim k s ψ(k) = 0. Then k→∞

ψ ; Zns ) = n−s sup f s C + O(1)(ψ(n) + nψ (n)) E(Cβ,∞

(4.4 )

ψ f ∈Cβ,∞

as n → ∞, where f s (·) is a continuous function such that S[f s ] =

∞

k s Ak (f ; x).

(4.6)

k=1

It is clear that Theorem 4.1 is an analog of part (iii) of Theorem 3.3. ψ The following statements characterize the quantities E(Cβ,∞ ; Zns ) in the case where sin βπ/2 = 0. Theorem 4.2. Assume that sin βπ/2 = 0, ψ ∈ M , and the function g(v) satisfies condition (a) in (4.3). Then, as n → ∞, ψ E(Cβ,∞ ; Zns ) = ψ(n)A(τn(1) ) + O(1)ψ(n)/n,

(4.7)

(1)

where A(τn ) is a quantity defined by equality (4.2) for ⎧ (s − 1)ψ(1) + ψ (1) 2 ⎪ ⎪ ·v ⎪ ⎪ ns−2 ψ(n) ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ (2 − s)ψ(1) − ψ (1) ⎪ ⎨ + v, 0 ≤ v ≤ 1/n, ns−1 ψ(n) τ (1) (v) = ⎪ ⎪ ⎪ ⎪v s ψ(nv)ψ(n), ⎪ 1/n ≤ v ≤ 1, ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ ψ(nv)/ψ(n), v ≥ 1.

(4.8)

Moreover, βπ 1 2 | sin | π 2 ψ(n)

∞ n

2π ψ(t) dt < A(τn(1) ) < t ψ(n)

∞ n

ψ(t) dt + O(1). t

(4.9)

ψ Zygmund Sums on the Classes Cβ,∞

Section 4

869

In the case where ψ(v) = v −r , r > 0, inequalities (4.9) yield relations (3.6) and (3.9) because ∞ n−r t−(r+1) dt = . r n

In the indicated case, we also have ∞

ψ(t) dt = O(1)ψ(n). t

(4.10)

n

However, equality (4.10) holds not for all ψ ∈ M . Thus, the function ψ(v) = ln−α (v + e) belongs to M for any α > 1 and, at the same time, 1 lim n→∞ ψα (n)

∞

ψα (t) dt = ∞. t

(4.11)

n

Theorem 4.3. Assume that sin βπ/2 = 0, ψ ∈ M , the function g(v) = is convex upward and, in addition,

v s ψ(v)

n ts−1 ψ(t)dt = O(1)ns ψ(n).

(4.12)

1

Then

ψ E(Cβ,∞ ; Zns ) = ψ(n)A(τn(2) ) + O(1)n−s

(4.13)

(2)

as n → ∞, where A(τn ) is a quantity defined by relation (4.2) for ⎧ ψ(1) ψ (1)/ψ(1) s+ψ (1)/ψ(1) ⎪ ⎪ v , 0 ≤ v ≤ 1/n, ⎪ ⎪ ψ(n) n ⎪ ⎪ ⎨ τn(2) (v) = v s ψ(nv)/ψ(n), 1/n ≤ v ≤ 1, ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ ψ(nv)/ψ(n), v ≥ 1.

(4.14)

Moreover, 2 βπ 1 | sin | π 2 π

∞ n

ψ(t) 2π dt < A(τn(2) ) < t ψ(n)

∞ n

ψ(t) dt + O(1), t

(4.15)

870

Approximations by Zygmund and de la Vallée Poussin Sums

Chapter 12

where O(1) are quantities uniformly bounded in n. As examples of the functions ψ(v) satisfying the conditions of Theorem 4.3, we can use the functions ψr (v) = v −r lnα (v + e) for 0 < s − 1 ≤ r < s and α ∈ R and ψα (v) = ln−α (v + e) for α > 1 in the case where s ≤ 1, etc. We now present several more statements related the case where sin βπ/2 = 0. Theorem 4.4. Assume that sin βπ/2 = 0, ψ ∈ M , the function g(v) = v s ψ(v) is convex upward or downward, and lim v s ψ(v) = C > 0.

v→∞

(4.16)

Then ψ E(Cβ,∞ ; Zns )

2 1 = | sin βπ/2| s π n

n ts−1 ψ(t)dt + O(1)ψ(n)

(4.17)

1

as n → ∞. In the case where g(v) is convex upward and lim v s ψ(v) = ∞,

v→∞

(4.16 )

equality (4.17) is true under the same conditions as equality (4.10) and the equality n 1 lim sup s ts−1 ψ(t)dt = ∞. n→∞ n ψ(n) 1

Note that the functions ψ(v) = v −s lnα (v + e), α ≥ 0 and ψ(v) = v −s arctan v are examples of functions satisfying the conditions of Theorem 4.4.

Theorem 4.5. Assume that the sequence k s ψ(k), k = 1, 2, . . . , is convex downward beginning with a certain number k0 , lim k s ψ(k) = 0, and the series k→∞

∞ k=1

k s−1 ψ(k)

Section 5

De la Vallée Poussin Sums on the Classes Wβr and Wβr Hω

871

is divergent. Then 2 | sin βπ/2| k s−1 ψ(k) + O(1)n−s s πn n

ψ ; Zns ) = E(Cβ,∞

(4.18)

k=1

as n → ∞. As an example of functions ψ(v) satisfying the conditions of Theorem 4.5, we can present the function ψ(v) = v −s ln−α (v + e) for α ∈ (0, 1]. Theorem 4.6. Assume that the function v s−1 ψ(v) is integrable on [1, ∞) and the function g(v) = v s ψ(v) is convex downward for all v > a ≥ 1. Then ψ E(Cβ,∞ ; Zns )

1 1 = s sup f s C + O(1)( s n f ∈C ψ n β,∞

∞ ts−1 ψ(t)dt + ψ(n)), (4.19) n

where f s (x) is a continuous function satisfying equality (4.16).

5.

De la Vallée Poussin Sums on the Classes Wβr and Wβr Hω

5.1. The sums Vn,p (f ; x) were first introduced by de la Vallée Poussin [1] in the form 2n−1 1 V2n−1,n (f ; x) = Sk (f ; x), (5.1) n k=n

where Sk (f ; x) are the partial Fourier sums of order k for a function f ∈ L. If tm (·) is a trigonometric polynomial, then (5.1) implies that V2n−1,n (tm ; x) = tm (x) ∀m ≤ n.

(5.2)

Thus, in view of relations (1.3.20), by choosing the polynomial of the best approximation of a given function f ∈ C as tn−1 (·) we get |f (x) − V2n−1,n (f ; x)| ≤ |f (x) − tn−1 (x)| + |V2n−1,n ((f − tn−1 ); x)| ≤ 4En (f )C .

(5.3)

Therefore, the approximation of the function f (·) by the sum V2n−1,n (f ; x) is worse than by the polynomial of its best approximation of order n by a factor of at most four. However, the order of the polynomial V2n−1,n (f ; x) is, generally

872

Approximations by Zygmund and de la Vallée Poussin Sums

Chapter 12

speaking, equal to 2n − 1. Hence, it does not follow from inequality (5.3) that the sums V2n−1,n (f ; x) give a solution of the problem about the construction of a sequence of operators Un∗ (f ; x) for any function f ∈ C that approximates this function f (x) with the best order, i.e., so that f (·) − Un∗ (f ; ·)C = O (En (f )) ,

n → ∞.

(5.4)

In this sense, the de la Vallée Poussin method is not an exception among all other linear processes of approximation. Namely, as follows from the results of Lozinskii [1, 2], none of the linear methods of approximation by polynomials can realize a uniform approximation coinciding in order with the best one for all continuous functions. Nevertheless, as follows from the results presented below, in many cases the sums Vn,p (f ; x) realize an approximation of a function f ∈ L coinciding in order with the best one. The approximation properties of the sums Vn,p (f ; x) are mainly determined by the parameter p, which may depend on the numbers n ∈ N. We set p(n) = θ, θ ∈ [0, 1]. n→∞ n lim

(5.5)

Apparently, the first results for de la Vallée Poussin sums related to the solution of the Kolmogorov–Nikol’skii problem were obtained by A. Timan [1, 5], who proved the following statement: Theorem 5.1. Let θ = 0. If numbers r and α satisfy the inequalities r ≥ 0 and 0 < α ≤ 1, then the following asymptotic equalities hold as n → ∞ : π

E(Wrr H α ; Vn,p ) =

2α+1 π 2 nr+α

ln

n p

2

tα sin t dt + O(

1 ), nr+α

(5.6)

0 π

∼ E(Wrr H α ; Vn,p )

=

2α+1 π 2 nr+α

ln

n p

2

tα sin t dt + O(

1 nr+α

).

(5.7)

0

For α = 1, relation (5.6) yields E(Wrr ; Vn,p ) =

4 1 n ln + g(n, p, r), 2 r π n p

(5.8)

Section 5

De la Vallée Poussin Sums on the Classes Wβr and Wβr Hω

873

where

1 ), n → ∞. nr The following question arises: Does there exist a constant A independent of r and such that the remainder satisfies the estimate g(n, p, r) = O(

g(n, p, r) ≤

A ? nr

The absence of such a constant is a consequence of the following theorem proved by Kulyk [1]: Theorem 5.2. For all integer r, n, and p such that 1 < p < n, the following estimate holds: E (Wrr ; Vn,p ) ≥

1 4 ln n C − 2 r − r, r p(p + 1)(n − p + 1) π n n

where C is an absolute constant independent of n, p, and r, and C < 7. Recall that, for p = 1 in this case, Vn,1 (f ; x) = Sn−1 (f ; x) , r ∈ N, and α = 1, equality (5.6) was established earlier by Kolmogorov [2]; for arbitrary r > 0, it was obtained by Pinkevich [1]; for arbitrary r ≥ 0 and 0 < α ≤ 1, both equalities (5.6) and (5.7) were established by Nikol’skii [3, 6]. A further important step in the investigation of approximations by de la Vallée Poussin sums was made by Efimov [5, 6]. He proved the following theorems: Theorem 5.3. For an arbitrary modulus of continuity ω = ω(t) and all 1 ≤ p ≤ n − 1, the following asymptotic equality is true: 1 E(Hω ; Vn,p ) = An,p (ω) + O(ω( )), n where

⎧ e(n) (ω) n n ⎪ ⎪ ⎪ ln , 0≤p≤ , ⎪ ⎪ π p 2 ⎪ ⎪ ⎨ 1 An,p (ω) = n−p ⎪ n ω(t) 2 ⎪ ⎪ dt, ≤ p ≤ n − 1, ⎪ ⎪ 2 ⎪ π(p + 1) t 2 ⎪ ⎩ 1

(5.9)

(5.10)

n+1

1 e(n) (ω) = sup | f ∈Hω π

π f (x) cos nx dx|. −π

(5.11)

874

Approximations by Zygmund and de la Vallée Poussin Sums

Theorem 5.4. For any 0 ≤ p ≤ equality is true: E(Wβr Hω ; Vn,p ) =

n 2

Chapter 12

and r > 0, the following asymptotic

e(n) (ω) n 1 1 ln + O( r ω( )), n → ∞, r π(n − p + 1) p n n

(5.12)

where e(n) (ω) is defined by (5.11). For β = r and β = r + 1, relation (5.12) yields asymptotic equalities for the ∼

quantities E(Wrr Hω ; Vn,p ) and E(Wrr Hω ; Vn,p ) for any r > 0 and 1 ≤ p ≤ n2 . Note that, according to (5.1.23 ), for any modulus of continuity ω(t) the following estimate is true: 2 π en (ω) < ω( ). (5.13) π n Hence, relations (5.9)—(5.13) imply that Theorem 5.3 gives a solution of the Kolmogorov–Nikol’skii problem on the class Hω for the sums Vn,p (f ; x) whenever lim p(n) n = 0. n→∞

If 0 < θ ≤ 12 , then both terms on the right-hand side of (5.9) have the same order. They also have the same order if 12 ≤ θ < 1 and ω(t) = tα , 0 < α ≤ 1 . In the beginning of the 1960s, Telyakovskii [1–5] obtained a solution of the Kolmogorov–Nikol’skii problem in certain cases where the order of decrease in upper bounds of E(Wβr ; Vn,p ) is equal to n1r , i.e., coincides with the order of decrease in upper bounds of the best approximations of functions from the classes Wβr . He proved the following theorem: Theorem 5.5. For the quantities E(Wβr ; Vn,p ), the following asymptotic equalities hold as n → ∞ : I. If θ = 0, then E(Wβr ; Vn,p ) =

4 1 n 1 ln + O( r ). 2 r π n p n

(5.14)

II. If 0 < θ < 1, then E(Wβr ; Vn,p ) = A(μ1−θ )

1 1 εn + O( r+1 + r ), nr n n

(5.15)

De la Vallée Poussin Sums on the Classes Wβr and Wβr Hω

Section 5

875

where the quantities A(τ ) are defined according to (4.2) and ⎧ 0, 0 ≤ u ≤ 1 − θ, ⎪ ⎪ ⎨ μ1−θ (u) = u−(1−θ) u−r , 1 − θ ≤ u ≤ 1, θ ⎪ ⎪ ⎩ −r u , 1 ≤ u < ∞, p 1 p − θ| ln p , = θ, n | n − θ| n p εn = 0, = θ. n III. If θ = 1 and 0 < r < 1, then εn = |

E(Wβr ; Vn,p ) = A(μ1,r ) where μ1,r (u) =

1 (n − p + 1)1−r + O( ), nr n

(5.16)

1−r u , 0 ≤ u ≤ 1, u−r ,

1 ≤ u < ∞.

IV. If θ = 1 and r = 1, then E(Wβ1 ; Vn,p ) =

2 βπ 1 n 1 | sin | ln + O( ). π 2 n n−p+1 n

(5.17)

If sin βπ 2 = 0, then ∼

E(Wβ1 ; Vn,p ) = A(μ1,1 )

1 1 + O( 2 ) n n

(5.18)

for p = n, ∼

E(Wβ1 ; Vn,p ) = 2A(μ1,1 )

1 1 1 + O( + n n(n − p) n

#

n−p n ln ) n n−p

for n − p → ∞, and ∼

E(Wβ1 ; Vn,p )

∼

1 1 = [A(μ1,1 ) + pE(W ; Vp,p )] + O( n n 1

for n − p = m (m is a fixed number); here, 1, 0 ≤ u ≤ 1, μ1,1 (u) = u−r , 1 ≤ u ≤ ∞.

#

ln n ) n

(5.19)

(5.20)

876

Approximations by Zygmund and de la Vallée Poussin Sums

Chapter 12

V. If θ = 1 and r > 1, then the following relations are true: for n − p = m → ∞, E(Wβr ; Vn,p ) = A(μ1,r )(

1 1 1 1 1 + 2 r−2 + . . . + r−1 ) + O( r + ), (5.21) r−1 nm n m n m n nmr

where μ1,r (u) =

0,

0 ≤ u ≤ 1,

(u − 1)u−r , 1 ≤ u < ∞;

for fixed n − p = m, m ≥ 1, E(Wβr ; Vn,p )

1 = sup | f ∈Wβr π

∞

(r) fβ (t)

−∞

∞

u−m βπ cos(ut + ) du dt| r u 2

0

×(

mr−2 1 1 m + 2 + . . . + r−1 ) + O( r ); (5.22) n n n n

for p = n,

∼

E(Wβr ; Vp,n ) = sup | f (x)| f ∈Wβr

1 1 + O( r ). n n

(5.23)

∼

Relation (5.23) for the classes Wrr , Wrr , and Wβr for integer β was established by Nikol’skii [6, 8] and Nagy [1, 3]. It should be noted that, for the classes ∼

W r with even r and W r with odd r, Nagy [1] proved a formula more exact than (5.23): ∼ 1 1 E(Wβr ; V2n,n ) = sup | f (x)| + O( r+1 ). n n f ∈Wβr Note that equalities (5.14)–(5.23) give a solution of the Kolmogorov– Nikol’skii problem for de la Vallée Poussin sums on the classes Wβr . Furthermore, one can see that, in the cases where 0 < θ < 1 and r > 0 (equality (5.15)), θ = 1 and 0 < r < 1 (equality (5.16)), and θ = 1 and sin βπ ee Poussin sums realize the best 2 = 0 (equalities (5.18)–(5.20)), de la Vall´ order of approximation of functions from Wβr .

Section 6

¯

De la Vallée Poussin Sums on the Classes Cβψ N and C ψ N

877

The solution of the Kolmogorov–Nikol’skii problem on the classes Wβr Hω (r ≥ 0) for the sums Vn,p (f ; x) in the cases where they realize an approximation coinciding in order with the best one became possible only in the end of 1970s due to the method developed by Korneichuk, Dzyadyk, and Stepanets (see, e.g., the book by Stepanets [9, Chapter 6]). However, the application of this method is rather nontrivial and encounters substantial technical difficulties. For this reason, only individual results are available in the case considered. One can judge the character of these results by the following theorem proved by Ostrovetskyi [1]: Theorem 5.6. Let θ = 23 and let ω(t) be a convex modulus of continuity. Then the following asymptotic equality holds as n → ∞ : E(Hω ; Vn,p ) = A(ω, n) + γn,p , where 3 A(ω, n) = πn

π

F (t)ω (

3t ) dt, n

0

F (t) is an extremal function, γn,p < 0, and ⎧ p 2 1 1 p 2 ⎪ ⎪ = , ⎨O(| − | ln p 2 ω( )), n 3 n n 3 |n − 3| γn,p = ⎪ p 2 ⎪ ⎩O( 1 ω( 1 )), = , n n n 3 π π 0.56ω( ) < A(ω; n) < 1.42ω( ). n n Note that the function F (t) is determined with the use of the lemma proved by A. Timan in [3]. One can get an idea of this function by noting the following properties of it: F (t) is continuous and monotonically decreasing on [0, π], F (0) = 2 + π2 , and F (π) = 0.

6.

De la Vallée Poussin Sums on the Classes ¯ Cβψ N and C ψ N

First, note that the approximation properties of the sums Vn,p (f ; ·) have already been considered in Sections 5.22 and 5.23. In this section, we present results

878

Approximations by Zygmund and de la Vallée Poussin Sums

Chapter 12

related to obtaining asymptotic equalities for the quantities E(Cβψ N; Vn,p ) = sup f (·) − Vn,p (f ; ·)C f ∈Cβψ N

and E(C ψ N; Vn,p ) = sup f (·) − Vn,p (f ; ·)C . f ∈C ψ N

The first results in this direction were obtained by Rukasov in [1, 2]. He proved the following statements: Theorem 6.1. Suppose that ψ ∈ MC and θ = 0. Then the following asymptotic equality holds as n → ∞ : ψ ; Vn,p ) = E(Cβ,∞

4 n ψ(n) ln + O(ψ(n)). 2 π p

(6.1)

Theorem 6.2. Suppose that ψ ∈ MC and 0 < θ < 1. Then the following asymptotic equality holds as n → ∞ : (n)

ψ ; Vn,p ) = A(τ1−θ )ψ(n) + O(αn ψ(n) + E(Cβ,∞

where

1 ψ(n)), n

(6.2)

⎧ ⎪ 0, 0 ≤ v ≤ 1 − θ, ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ v − (1 − θ) ψ(nv) , 1 − θ ≤ v ≤ 1, (n) τ1−θ (v) = θ ψ(n) ⎪ ⎪ ⎪ ⎪ ψ(nv) ⎪ ⎪ , 1 ≤ v < ∞, ⎩ ψ(n) ⎧ p p 1 ⎪ ⎨| n − θ| ln p , n = θ, | −θ| αn = n p ⎪ ⎩ 0, = θ, n

and (n)

A(τ1−θ ) = O(1),

n → ∞.

Equalities (6.1) and (6.2) always give a solution of the Kolmogorov–Nikol’skii ψ problem for the sums Vn,p (f ; x), 0 ≤ θ < 1, on the classes Cβ,∞ , ψ ∈ MC . In particular, it follows from equality (6.2) that, in the case where 0 < θ < 1, the ψ sums Vn,p (f ; x) guarantee the best order of approximation of the classes Cβ,∞ , ψ ∈ MC .

¯

De la Vallée Poussin Sums on the Classes Cβψ N and C ψ N

Section 6

879

ψ The behavior of the quantities E(Cβ,∞ ; Vn,p ) in the case θ = 1 was considered by Novikov and Rukasov in [1]. One of the results of a more general theorem (Theorem 1) from the work indicated can be formulated as follows:

Theorem 6.3. Suppose that θ = 1, ψ ∈ M , and the functions gn (v) = are monotonically increasing and convex upwards for v ≥ n − p. Then the following asymptotic equality holds as n → ∞ : (v−n+1)ψ(v) n

ψ E(Cβ,∞ ; Vn,p )

n

2 βπ 1 = | sin | π 2 p

(v − n + p)ψ(v) dv v

n−p+1

βπ | + O(ψ(n) + | sin 2

∞

ψ(v) dv). (6.3) v

n

For ψ(v) = v1 , equality (6.3) coincides with equality (5.17). Examples of α functions for which Theorem 6.3 is true are the functions ψ(v) = ln (v+c) , 0< vr r ≤ 1, α > 0, c > 1. Note one more result of Rukasov and Novikov [1] presented below. Theorem 6.4. Suppose that ψ(k) = exp(−αk), k = 1, 2, . . . , α > 0, and β ∈ R. Then the following asymptotic equality holds as n → ∞ : ψ E(Cβ,∞ ; Vn,p ) =

4ψ(n − p) ψ(n) ψ(n − p) + O( + ). πp(1 − ψ(2)) p p(n − p)

(6.4)

ψ and The rate of convergence of de la Vallée Poussin sums on the classes C∞ C ψ Hω was investigated by Rukasov, Novikov, and Chaichenko in [1]. Corollaries of the general theorems of the works indicated can be formulated as follows:

Theorem 6.5. Suppose that ψ1 ∈ M0 , ψ2 ∈ MC , and 0 ≤ θ < 1. Then the following asymptotic equalities hold as n → ∞ : ψ ; Vn,p ) = E(C∞

4 n ψ(n) ln + O(1)ψ(n) 2 π p

(6.5)

and π

E(C ψ Hω ; Vn,p ) =

2θω n ψ(n) ln 2 π p

2

ω( 0

2t 1 ) sin t dt + O(1)ψ(n)ω( ), (6.6) n n

880

Approximations by Zygmund and de la Vallée Poussin Sums

Chapter 12

where ψ(n) = ψ12 (n) + ψ22 (n), θω ∈ [ 23 ; 1], θω = 1 if ω(t) is a convex modulus of continuity, and O(1) are quantities uniformly bounded in n. Theorem 6.6. Suppose that ψi ∈ F, numbers p = p(n) are chosen so that n − p ∈ [η −1 (ψi ; n); n], i = 1, 2, and there exist constants K1 and K2 such that η(ψ1 ; n) − n 0 < K1 ≤ (6.7) ≤ K2 < ∞, n = 1, 2, . . . . η(ψ2 ; n) − n Then the following asymptotic equalities hold as n → ∞ : ψ ; Vn,p ) = E(C∞

η(n) − n 4 ψ(n) ln+ + O(1)ψ(n) 2 π p

(6.8)

and E(C ψ Hω ; Vn,p ) π

=

η(n) − n 2θω ψ(n) ln+ 2 π p

2

ω(

2t 1 ) sin t dt + O(1)ψ(n)ω( ), (6.9) n n

0

where ψ(n) = ψ12 (n) + ψ22 (n), η(n) is either η(ψ1 ; n) or η(ψ2 ; n), θω ∈ [ 23 ; 1], θω = 1 if ω(t) is a convex modulus of continuity, and O(1) are quantities uniformly bounded in n. It is easy to see that, in certain important cases, equalities (6.5), (6.6), and (6.9) give a solution of the corresponding Kolmogorov–Nikol’skii problem; it is also clear that many interesting problems remain unsolved in this direction.

BIBLIOGRAPHICAL NOTES (PART II)

Chapter 6 The results of Section 2 in the case of the classes Lψ β L2 and the statements of Section 3 were presented in the work by Stepanets [22]; the results of Section 4, except Theorem 4.2, were given in the work by Chernykh [1]. Inequality (4.1) was proved by Jackson [1]. Theorem 5.1 was established by Marcinkiewicz [1]. Its simplified version was presented by Zygmund [6]. Imbedding problems for subsets of Lψ β were also considered by Stepanets and Zhukina [1]. Corollary 7.1 is well known. It can be found, e.g., in the works of Korneichuk [5] and Tikhomirov [1]. The results of Corollary 8.3 were presented by Stepanets [21, 29]. Chapter 7 In Section 1, we used the book of Dzyadyk [5]. In Sections 2 and 3, we used the monograph by Korneichuk [5]. These works contain detailed bibliographical notes. We only note here that almost all results of Section 3 related to the space L were proved by Nikol’skii [8]. The method related to representation (3.8) is taken from the work of Stechkin [10]. The results of Section 4 in the case where ψ ∈ Mc ∪ M+ ∞ were established by Stepanets and Kushpel [1, 2]. These results −r for ψ(k) = k , r > 0, and β = r, were known earlier (see, e.g., Tikhomirov [1]). Theorems 5.1 and 5.2 belong to Nagy [1]. For integer values of β, Theorem 7.3 follows from the works by Krein [1] and Nikol’skii [8] (in the spaces C and L, respectively). For fractional β, this theorem was proved by Bushanskii [1] (see also Shevaldin [1]). Proposition 8.1 can be found in the work of Stepanets and Kushpel [2]. Proposition 8.2 and all statements of Section 9 were proved by Stepanets [44]. 881

882

Bibliographical Notes (Part II) Chapter 8

For more detailed information on integration of functions by polynomials, see Zygmund [5], Goncharov [1], etc. Formula (1.13) dates back to Euler and Gauss. Theorems 2.1 and 2.2 were proved by Nikol’skii [2, 4, 6]. The results of Sections 3 and 4 were established by Stepanets and Serdyuk [3, 4]. The results of Section 5 were obtained by Stepanets [46]. Chapter 9 All information about entire functions of exponential type used in this chapter is taken from the book by Akhiezer [1], where one can find an extensive bibliography in this direction. The main material of Sections 1–11 is based on the works of Stepanets [25, 26, 28]. Assertions (4.21) and (4.22) of Theorem 4.3 were presented in the book by Akhiezer [1]. The results of Sections 12 and 13 were obtained by Stepanets in [23] and [45], respectively. The results of Section 14 were established in the work by Stepanets and Pachulia [1]. The work of Stepanets, Wang Kunyang, and Zlang Xirong [1] is devoted to the extension of the results of Sections 1–11 to analogs of ¯ ψ-integrals. Basic information on approximation by entire functions can be found in the books of Nikol’skii [10] and A. Timan [5]. Chapter 10 The main material of this chapter is the combination of the results obtained by Stepanets [27] and Savchuk [1, 2]. The auxiliary material used in this chapter is taken from the books by Dzyadyk [5], Smirnov and Lebedev [1], Suetin [1], and Golusin [1]. Theorem 4.1 was proved by Landau [1]. In the case where ψ(t) = Γ(t + 1)/Γ(t + r + 1), Theorem 5.1 was proved by Stechkin [5]. For information about duality relations, see, e.g., the book by Duren [1]. Chapter 11 The main material of Sections 1–5 was published in the works of Stepanets [41, 42]. Sections 6–10 are based on the work by Stepanets [43]. The results of Section 11 are taken from the paper by Stepanets and Serdyuk [6]. The proof of Theorem 3.2, in fact, coincides with the arguments of Tikhomirov [1, Section 4.4], who found the widths of ellipsoids in a Hilbert space,

Bibliographical Notes (Part II)

883

i.e., the widths of sets that coincide in the notation of the present monograph with the closure of the sets ψUϕ2 . In the case of the Hilbert space of functions defined on a segment, statements analogous to Theorems 4.1 and 4.2 were proved earlier by Stepanets [22]. The quantity en (f )p defined by equality (2.9) for p = 2 was introduced by Stechkin [6] in connection with the problem of finding a criterion for the absolute convergence of orthogonal series. The facts concerning convex programming used in Subsection 7.7 are taken from the book by Tikhomirov [1]. In the one-dimensional case, the Hausdorff–Young theorem mentioned in Subsection 10.2 was presented with complete proof in several monographs (see, e.g., Bari [1] and Zygmund [5, 6]). For the multidimensional case, see, e.g., the books by Stein and Weiss [1], Edwards [1, 2], etc. Inequality (11.30) was established by Chernykh [2]. He also proved the unimprovability of this inequality. For p = 2 and k = 1, Theorem 11.1 was proved by Babenko [1]. The proof of Theorem 11.1, for the most part, repeats the arguments of the work indicated and of the works by Chernykh [1, 2]. The proof of the duality relation formulated as Proposition 11.1 can be found, e.g., in the book of Korneichuk [6]. For the spaces Lp , inequalities of the form (11.71 ) were obtained by M. Timan [1] (see also A. Timan [5, p. 351]). The condition (Br ) for natural r was introduced by Bari [2] and Bari and Stechkin [1]. Chapter 12 Bibliographical notes are mainly presented in the text of the chapter. Additional information in this direction can be found in the works by Bari [1], Baskakov [1, 2], Efimov [1–13], Nagy [1–7], Nikol’skii [1–10], Sorich and Stepanets [1], Stepanets [9, 21, 29, 40], Stechkin [4–8], Telyakovskii [1–14], A. Timan [1–5], Cesari [1], etc.

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Xiung, Yungshen [1] On the best approximation of periodic differentiable functions by trigonometric polynomials, Izv. Akad. Nauk SSSR, Ser. Mat., 23, No. 1, 67–92 (1959). [2] On the best approximation of periodic differentiable functions by trigonometric polynomials (second report), Izv. Akad. Nauk SSSR, Ser. Mat., 25, 143–152 (1961). Zaderei, N. N. [1] Approximation by Fourier sums on classes of periodic functions of many variables determined by differential operators of even order, Ukr. Mat. Zh., 32, No. 2, 212–219 (1980). Zaderei, N. N., and Stepanets, A. I. [1] Approximation by Fourier sums on classes of periodic functions determined by polyharmonic operators, Mat. Zametki, 27, No. 4, 49–59 (1980). Zaderei, P. V. [1] On approximation of periodic functions of many variables by positive polynomial operators, in: Problems in the Theory of Approximation of Functions and Its Applications [in Russian], Institute of Mathematics, Ukrainian Academy of Sciences, Kiev (1978), pp. 86–90. [2] On multidimensional analog of one result by Boas, Ukr. Mat. Zh., 39, No. 3, 380–383 (1987). [3] On the convergence rate of Fourier series of (ψ, β)-differentiable functions of two variables, in: Investigations in the Approximation Theory of Functions [in Russian], Institute of Mathematics, Ukrainian Academy of Sciences, Kiev (1987), pp. 24–34. [4] On asymptotic equalities for norms in the metric of L of functions determined by multiple trigonometric series, Dokl. Akad. Nauk SSSR, Ser. Mat., 302, No. 6, 1305–1308 (1988).

916

References (Part II)

[5] On the convergence of Fourier series in the mean, Ukr. Mat. Zh., 41, No. 4, 562–568 (1989). [6] Asymptotic equalities for integrals of modulus of the sum of multiple series of sines, Analysis Mathematica, 15, No. 3, 245–262 (1989). [7] On the conditions of integrability of multiple trigonometric series, Ukr. Mat. Zh., 44, No. 3, 340–365 (1992). [8] Approximation of (ψ, β)-differentiable periodic functions of many variables, Ukr. Mat. Zh., 45, No. 3, 367–377 (1993). Zamanski, M. [1] Classes de saturation de certains procédés d’approximation des séries de Fourier des fonctions continues et application a` quelques problémes d’approximation, Ann. Sci. Ecole Norm. Sup., 66, No. 1, 19–93 (1949). [2] Classes de saturation des procédés de sommation des séries de Fourier et applications des séries trigonométriques, Ann. Sci. Ecole Norm. Sup., 67, 161–198 (1960). Zygmund, A. [1] On the convergence and summability of power series on the circle on convergence, Proc. London Math. Soc., 47, 326–350 (1941). [2] The approximation of functions by typical means of their Fourier series, Duxe Math. J., 12, 695–704 (1945). [3] On the degree of approximation of functions by Fejér means, Bull. Amer. Math. Soc., 51, No. 4, 274–278 (1945). [4] Trigonometric Series [Russian translation], Vol. 1, Mir, Moscow (1965). [5] Trigonometric Series [Russian translation], Vol. 2, Mir, Moscow (1965). [6] Trigonometric Series [Russian translation], GONTI, Moscow, Leningrad (1939). [7] Smooth functions, Duke Math. J., 12, 47–76 (1945).

Index Curve Jordan, 680 rectifiable, 681

Abel transformation, 12, 565 Approximation best, 430 n-term, 755

Derivative (ψ, β)-derivative, 120 ¯ (ψ, β)-derivative, 132 ¯ ψ-derivative, 138 Weyl–Nagy, 119 Duality relation, 735

Best linear method, 520 Classes of functions Ar , Ar Hω , 112 C ψ (T)+ , 702 Cβψ N, 126

Fourier–Lagrange coefficients, 555 Function Bernoulli, 518 conjugate, 117 entire, of exponential type, 609 η(t) = η(ψ; t), 147 η¯(t) = η¯(ψ; t), 151 ηa (t) = ηa (ψ; t), 160 Lagrange, 789 μ(t) = μ(ψ; t), 147 Zhukovskii, 683

Cβψ¯ N, 133 Hω [a, b], Hω , 108 L0 , 113 ˆ ψ N, 598 L β N, 130 Lψ β¯ Lψ p (T)+ , 696 ¯ ψ L N, 432 Lp , 101 Lip α, H α , 109 M , 101 W r , W r Hω , 112 r H , 118 ˜ r, W W ω r Wβ , Wβr N, 120 Weyl–Nagy, 119 Condition A∗n , 497 Constant Favard, 519 Landau, 713 Convolution of two functions, 122

Hyperbolic cross, 822 Inequality Bernstein, 539 Jackson, 444, 481, 486, 583 Lebesgue, 188 Lebesgue–Landau, 724 Minkowski, 334, 483 Young, for convolutions, 285 917

918

Index

Integral Cauchy-type, 679 cosine, 217 elliptic, 295 Poisson, 295 sine, 213 Integral representation first, 165 Interpolation polynomial, 553

Operator Fσ,c (f ; x), 608 Sσ,c , 610

Kernel Dirichlet, 2 of operator (or method), 2 Poisson, 295, 536 Kolmogorov–Nikol’skii problem, 188

Quasiconvex sequence, 28

Lebesgue constants of de la Vallée Poussin method, 18 of Fejér method, 17 of Fourier method, 15 of Rogosinski method, 19 of Zygmund method, 20 Lemma Boas, 31 Fejér–Stechkin, 297 Korneichuk–Stechkin, 197 on a convex majorant, 106 Methods(processes), of summation, linear, 2 Modulus of continuity, 102 in the spaces Lp , 110 Multiplicator, 449 Operator Faber, 738 Fσ,c , 613

Parseval equality, 742 Polynomial Chebyshev, 684 Faber, 682 of the best approximation, 430

Saturation class, 81, 98–100 order, 81, 98–100 sufficient conditions, 82 Saturation of method, 79, 98 Set of functions A, A0 , AC , A∞ , 625 Eσ , 609 F , 153 F0 , 625 L, 140 M , 174 M0 , M∞ , MC , 148 + M+ 0 , M∞ , 148 M∞ , 468 M∞ , 466 M+ ∞ , 470 W, W 2 , Wσ2 , 609 Space Hardy, Hp , 685 ˆ p , 598 L Sϕp , 741 Sum Abel–Poisson, 690 de la Vallée Poussin, 3, 847 Fejér, 2, 848 Fourier, 1

Index Sum Riesz, 3, 860 Rogosinski, 3 Zygmund, 4, 847 Theorem Banach, 5 Banach–Steinhauz, 88 Bernstein, 822 Chebyshev, 490 de la Vallée Poussin, 491 direct, 437 Dzyadyk–Stechkin–Xiung Yungshen, 522 Fejér, 2 Hardy–Littlewood, 451 Hausdorff–Young, 818 imbedding, 452 inverse, 437, 545 Jackson, 822 Karamata–Tomic, 27 Korneichuk, 524 Landau, 713

919 Theorem Marcinkiewicz, 97, 449 Nagy, 8 Nikol’skii, 6, 557 on convolution, 123 Privalov, 118 Riemann, 681 Riesz, F., 685 Riesz, F.–Riesz, M., 694 Salem, 143 Serdyuk, 525 Smirnov, 694 Telyakovskii, 45 Weierstrass, 5 Wiener–Paley, 611 Transfinite diameter, 682 Width, 747 p-ellipsoid, 747 ψ-derivative, 440, 697, 745 ψ-integral, 697, 745 ¯ ψ-integral, 137

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