FOURIER ANALYSIS AND APPROXIMATION Volume I
This is Volume 40 in PURE AND APPLIED MATHEMATICS A Series of Monographs and Textbooks Editors: PAULA. SMITHAND SAMUEL EILENBERG
A complete list of titles in this series appears at the end of this volume
FOURIER ANAL YSIS AND APPROXIMATION Volume 7
OneDimensional Theory
Paul L. Butzer
Rolf J. Nessel
Technological University of Aachen
Academic Press New York and London 1971
0 1971, BY BIRKH~USER VERLAG BASEL. (LEHRB~CHER UND MONOQRAPHJEN AUS DEM GEBIETE DER EXAKTEN WISSENSCHAFTEN, MATHEMATJSCHE REJHE, BAND40) ISBN 3 7643 0520 7 (BirkhiIuser Verlag)
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to
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At the international conference on ‘Harmonic Analysis and Integral Transforms’, conducted by one of the authors at the Mathematical Research Institute in Oberwolfach (Black Forest) in August 1965, it was felt that there was a real need for a book on Fourier analysis stressing (i) parallel treatment of Fourier series and Fourier transforms from a transform point of view, (ii) treatment of Fourier transforms in LP(Rn)space not only for p = 1 and p = 2, (iii) classical solution of partial differential equations with completely rigorous proofs, (iv) theory of singular integrals of convolution type, (v) applications to approximation theory including saturation theory, (vi) multiplier theory, (vii) Hilbert transforms, Riesz fractional integrals, Bessel potentials, (viii) Fourier transform methods on locally compact groups. This study aims to consider these aspects, presenting a systematic treatment of Fourier analysis on the circle as well as on the infinite line, and of those areas of approximation theory which are in some way or other related thereto. A second volume is in preparation which goes beyond the onedimensional theory presented here to cover the subject for functions of several variables. Approximately a half of this first volume deals with the theories of Fourier series and of Fourier integrals from a transform point of view. This parallel treatment easily lends itself to an understanding of abstract harmonic analysis; the underlying classical theory is therefore presented in a form that is directed towards the case of arbitrary locally compact abelian groups, which are to be discussed in the second volume. The second half is concerned with the concepts making up the fundamental operation of Fourier analysis, namely convolution. Thus the leitmotiv of the approximation theoretic part is the theory of convolution integrals, the ‘smoothing’ of functions by such, and the study of the corresponding degree of approximation. Special as this approach may seem, it not only embraces many of the topics of the classical theory but also leads to significant new results, e.g., on summation processes of Fourier series, conjugate functions, fractional integration and differentiation, limiting behaviour of solutions of partial differential equations, and saturation theory. On the other hand, no attempt is made to present an account of the theory of Fourier series or integrals per se, nor to prepare a book on classical approximation theory as such. Indeed, the theory of Fourier series is the central theme of the monumental treatises by A. ZYGMUND(1935, 1959) and N. K. BARI(1961). With respect to the theory of Fourier integrals we have aimed to bring portions of the fine treatises by E. C. TITCHMARSH (1937) and S. BOCHNJBK.CHANDRASEKHARAN (1949) up to date, yet complementingthem with Fourier transforms on the circle. Furthermore, a number of excellent books giving a broad coverage of approximation theory has appeared in
viii
PREFACE
the past years; for the classical ones we can single out N. I. ACHIESER (1947) and I. P. NATANSON (1949). In contrast, the present volume is meant to serve as a detailed introduction to each of these three major fields, emphasizing the underlying, unifying principles and culminating in saturation theory for convolution integrals. Whereas many texts on approximation treat the matter only for continuous functions (and in LPspace, if at all, separately), the present text handles it in the spaces C,, and LS,, 1 I p < m, simultaneously. The parallel treatment of periodic questions and those on the line, already mentioned in connection with Fourier transforms, is a characteristic feature of the entire material as presented in this volume. This exhibits the structures common to both theories (compare the treatment of Chapters 6 and 11, for example), usually discussed separately and independently. Whenever the material would be too analogous in statement or proof, emphasis is laid upon different methods of proof. However, the reader mainly interested in the periodic theory can proceed directly from Chapters 1 and 2 to Chapter 4 and from there to the relevant parts indicated in Chapter 6 and Sec. 7.1. He may then turn to Chapter 9, Sec. 10.110.4, Sec. 11.41 1.5, Sec. 12.112.2. On the other hand, Chapters 4 and 5, together with the basic material in Chapters 1 and 3 (Sec. 1.11.2, 1.4,3.13.2) may serve as a short course on (classical) Fourier analysis; for selected applications one may then consult Chapters 6 and 7. As a matter of fact, Chapter 7 gives the first and bestknown application of Fourier transform methods, namely to the solution of partial differential equations. In Chapters 1013 these integral transform methods are developed and refined so as to handle profounder and more theoretical problems in approximation theory. A brief introductory course on classical approximation theory for periodic functions may be based on Chapters 1 and 2. The present treatment is essentially selfcontained; starting at an elementary level, the book progresses gradually but thoroughly to the advanced topics and to the frontiers of research in the field. Many of the results, especially those of Chapters 1013, are presented here for the first time, at least in bookform. Although the presentation is completely rigorous from the mathematical point of view, the lowest possible level of abstraction has been selected without compromising accuracy. In many of the proofs intricate analysis is required. This we have carried out in detail not only since we believe it is more important to save the reader’s time than to save paper, but because we believe firmly that the student reader should be able to follow each step of a proof. Despite the virtues of elegant brevity in the presentation of proofs, many recent texts have gone to the extreme of sacrificing understanding to the cost of all but the more expert in their fields. Although we have attempted to range both in depth and breadth, it remains inevitable that several themes have been slighted in a subject of rapidly increasing diversity and development. Presumably no apology is necessary for the fact that we have been guided in our selection by pursuing those topics which have caught our imagination; however, in the process we have attempted to illustrate a variety of analytical techniques. With this stepbystep development the volume can be readily utilized by senior undergraduate students in mathematics, applied mathematics, and related fields such as mathematical physics. It is also hoped that the book will be useful as a reference for workers in the physical sciences. Indeed, the central theme is Fourier analysis and
PREFACE
ix
approximation, subjects of wide importance in many of the sciences. The principal prerequisites would be a solid course on advanced calculus as well as some working knowledge of elementary Lebesgue integration and functional analysis. To make the presentation selfcontainedthese foundations are collected in a preliminary Chapter 0. Following each section there is a total of approximately 550 problems, many consisting of several parts, ranging from fairly routine applications of the text material to others that extend the coverage of the book. Many of the more difficult ones are supplied with hints or with references to the pertinent literature. The results of problems are often used in subsequent sections. Each chapter ends with a section on ‘Notes and Remarks’. These contain historical references and credits as well as detailed references to some 650 papers or books treating or supplementing specific results of a chapter. Important topics related to those treated but not included in the text are outlined here. Although we have tried our best to give everyone his full measure of credit, we apologize in advance for any oversights or inaccuracies in this regard. Here, as well as in the subject matter, we will appreciate any corrections suggested by the reader. The second volume will deal with more abstract aspects of the material. Special emphasis is placed upon the theory in Euclidean nspace. Fourier transforms will be discussed in the setting of distribution theory, and a systematic account given of those parts of approximation theory concerned with functions of several variables. Included will be characterizations of saturation classes of singular integrals with radial or product kernels by Lipschitz conditions, Riesz transforms and fractional integrals, Bessel potentials, etc. by means of embedding theorems. The material presented here first took form during several onesemester courses on Fourier series, on Fourier transforms, and on approximation theory given at the Technological University of Aachen during the past decade by one of the authors and assisted by the other. The third and final typewritten version was begun in the summer of 1966, as a joint effort of both authors. We have been especially fortunate with the assistance of several members of our team of collaborators. Dr. EBERHARD L. STARK read and checked the whole manuscript, gave helpful suggestions, edited every chapter, assisted in reading the proofs, and prepared the index. It is certain that without his patient and unstinting work nothing on the scale of the volume could have been comand WALTER TREBELS gave valuable advice and criticism, pleted. Drs. ERNSTGORLICH read parts of the manuscript and set the authors straight on many a vital point; several portions of the manuscript were written in collaboration with Dr. TREBELS, including Chapter 11. Mr. FRIEDRICH ESSERassisted in reading the proofs. We are particularly indebted to our secretary Miss URSULACOMBACH who typed the final version cheerfully and with painstaking care; the earlier version had been capably typed by Mrs. KARINKOCHand Mrs. DORISEWERS. We also wish to thank Mr. C. EINSELE of Birkhauser Verlug for his patience, and the staff of William Clowes and Sons Ltd. for their skill and meticulous care in the production of this book. Aachen, February, 1970
P. L. BUTZERand R. J. NESSEL
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0 Preliminaries 0.1 Fundamentals on Lebesgue Integration . . . . . . 0.2 Convolutions on the Line Group . . . . . . . . . 0.3 Further Sets of Functions and Sequences . . . . . 0.4 Periodic Functions and Their Convolution . . . . . 0.5 Functions of Bounded Variation on the Line Group . . 0.6 The Class BVan . . . . . . . . . . . . 0.7 Normed Linear Spaces, Bounded Linear Operators , . . 0.8 Bounded Linear Functionals, Riesz Representation Theorems . 0.9 References . . . , . . . . . . . . . . .
. . .
. . . . . . . . .
24
. . . . . . .
25
, . . . . . . . . . . . . . . . . . .
29 30
. . . . . . . . . . . .
39
.
.
. .
.
Part I Approximation by Singular Integrals
1 Singular Integrals of Periodic Functions 1.0 Introduction . . . . . . . 1.1 NormConvergence and Derivatives 1.1.1 NormConvergence 30 1.1.2 Derivatives 33
1.2 Summation of Fourier Series 1.2.1 1.2.2 1.2.3 1.2.4 1.2.5 1.2.6 1.2.7
. . .
Definitions 39 Dirichlet and Fejer Kernel 42 Weierstrass Approximation Theorem 44 Summability of Fourier Series 44 RowFinite &Factors 47 Summability of Conjugate Series 47 FourierStieltjes Series 49
1.3 Test Sets for NormConvergence
. . . . .
,
1.3.1 Norms of Some Convolution Operators 54 1.3.2 Some Applications of the Theorem of BanachSteinhaus 1.3.3 Positive Kernels 58
. . . .
.
Modulus of Continuity and Lipschitz Classes 67 Direct Approximation Theorems 68 Method of Test Functions 70 Asymptotic Properties 72
. . . . .
20
54
55
. . . . . . . . . .
1.4 Pointwise Convergence . . . 1.5 Order of Approximation for Positive Singular Integrals 1.5.1 1.5.2 1.5.3 1.5.4
1 4 6 8 10 14 15
. . . . . . . .
61 67
xii
CONTENTS
. .
79
. . . . . . . . . . . . . . . .
86 89
1.6 Further Direct Approximation Theorems, Nikolski? Constants 1.6.1 Singular Integral of FejCrKorovkin 79 1.6.2 Further Direct Approximation Theorems 80 1.6.3 Nikolskil Constants 82
1.7 Simple Inverse Approximation Theorems 1.8 Notes and Remarks . . . . . . .
2 Theorems of Jackson and Bernsteln for Polynomials of Best Approximation and for Singular Integrals 2.0 Introduction . . . . . . . . . . . . . . . . . 94 2.1 Polynomials of Best Approximation . . . . . . . . . . 95 2.2 Theorems of Jackson . . . . . . . . . . . . . . 97 2.3 Theorems of Bernstein . . . . . . . . . . . . . . 99 2.4 Various Applications . . . . . . . . . . . . . . 104 2.5 Approximation Theorem for Singular Integrals . . . . . . 109 2.5.1 Singular Integral of AMPoisson 109 2.5.2 Singular Integral of de La V a l k Poussin
112
. . . . . . . . . . . . .
2.6 Notes and Remarks
. . 116
3 Singular Integrals on the Line Group 3.0 Introduction . . . . . .
. . . . . . . . . . . . . . . . . . . . . . . . . .
3.1 NormConvergence 3.1.1 3.1.2 3.1.3 3.1.4
3.2 3.3 3.4 3.5 3.6
119 120
Definitions and Fundamental Properties 120 Singular Integral of FejCr 122 Singular Integral of GaussWeierstrass 125 Singular Integral of CauchyPoisson 126
Pointwise Convergence . . . . . . Order of Approximation . . . . . Further Direct Approximation Theorems Inverse Approximation Theorems . . . Shape Preserving Properties . . . . 3.6.1 Singular Integral of GaussWeierstrass 3.6.2 Variation Diminishing Kernels 154
. . . . .
Fourier Transforms
.
.
.
.
.
.
a
150
. . . . . . . . . . . . . . .
163
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.1.1 Fundamental Properties 167 4.1.2 Inversion Theory 171 4.1.3 Fourier Transforms of Derivatives
4.2 Lh,Theory, p > 1
.
132 136 142 146 150 158
4 Finite Fourier Transforms 4.0 Introduction . 4.1 Li,Theory . .
.
. . . . . . . . . * . . . . . . . . . . . . . .
. . . . . . . . . .
3.7 Notes and Remarks
Part II
. . . . . . . .
172
. . . . . . . . .
4.2.1 The Casep = 2 174 4.2.2 The Casep # 2 177
.
. . . . .
167 167
174
CONTENTS
4.3
Finite FourierStieltjes Transforms
. . . . . . . . . .
4.3.1 Fundamental Properties 179 4.3.2 Inversion Theory 182 4.3.3 FourierStieltjes Transforms of Derivatives
4.4 Notes and Remarks
xiii 179
183
. . . . . . . . . . . . . . .
5 Fourier Transforms Associated with the Line Group 5.0 Introduction . . . . . . . . . 5.1 L1Theory. . . . . . .
. . . . .
. . . . . . . . . . . . . .
185
188 188
5.2
5.1.1 Fundamental Properties 188 5.1.2 Inversion Theory 190 5.1.3 Fourier Transforms of Derivatives 194 5.1.4 Derivatives of Fourier Transforms, Moments of Positive Functions, Peano and Riemann Derivatives 196 5.1.5 Poisson Summation Formula 201 LPTheory, 1 < p I 2 , 5.2.1 The Casep = 2 208 5.2.2 The Case 1 < p < 2 209 5.2.3 Fundamental Properties 212 5.2.4 Summation of the Fourier Inversion Integral 213 5.2.5 Fourier Transforms of Derivatives 214 5.2.6 Theorem of Plancherel 216
. . . . . . . . . .
208
5.3
FourierStieltjes Transforms
. . . . . . . . . . . .
2 19
. . .
5.3.1 Fundamental Properties 219 5.3.2 Inversion Theory 222 5.3.3 FourierStieltjes Transforms of Derivatives
5.4 Notes and Remarks
224
. . . . . . . . . . . . . . .
6 Representation Theorems 6.0 Introduction . . . . . . . 6.1 Necessary and Sufficient Conditions
. . . . . . . . . . . . . . . . . . . .
227
231 232
Representation of Sequences as Finite Fourier or FourierStieltjes Transforms 232 6.1.2 Representation of FunctionsasFourieror FourierStieltjesTransforms 235
6.1.1
6.2 Theorems of Bochner 6.3 Sufficient Conditions 6.3.1 6.3.2 6.3.3 6.3.4
. . . . . . . . . . . . . . . . . . . . . . . . . . . .
QuasiConvexity 246 Representation as LizTransform 249 Representation as L1Transform 250 A Reduction Theorem 252
6.4 Applications to Singular Integrals
. . . . . . . . . . .
6.4.1 General Singular Integral of Weierstrass 257 6.4.2 Typical Means 261
6.5 Multipliers 6.5.1 6.5.2
241 246
. . . . . . . . . . .
256
. . . . .
266
. . . . . . . . . . . . . . .
273
,
Multipliers of Classes of Periodic Functions 266 Multipliers on LP 268
6.6 Notes and Remarks
7 Fourier Transform Methods and SecondOrder Partial Differential Equations 7.0 Introduction . , . . . . . . . . . . . . . . . 278
CONTENTS
xiv
. . . . . . . . . . .
7.1 Finite Fourier Transform Method
281
7.1.1 Solution of Heat Conduction Problems 281 7.1.2 Dirichlet's and Neumann's Problem for the Unit Disc 284 7.1.3 Vibrating String Roblems 287
.
7.2 Fourier Transform Method in L1
. . . . . . . . . .
7.2.1 Diffusion on an Infinite Rod 294 7.2.2 Dirichlet's Problem for the HalfPlane 7.2.3 Motion of an Infinite String 298
Part 111 Hilbert Transforms
297
. . . . . . . . . . . . . . .
7.3 Notes and Remarks
. . . . . . . . . . .
8 Hilbert Transforms on the Real Line 8.0 Introduction . . . . . . 8.1 Existence of the Transform . .
294
. . .
300
.
303
. . . . . . . . . . . . . . . . . . . . , .
305 307
8.1.1 Existence Almost Everywhere 307 8.1.2 Existence in L1Norm 310 8.1.3 Existence in LpNorm, 1 < p < to 312
8.2 Hilbert Formulae, Conjugates of Singular Integrals, Iterated Hilbert Transforms . . . . . . . . . . . . . . . . 316
.
8.2.1 8.2.2 8.2.3 8.2.4
Hilbert Formulae 316 Conjugates of Singular Integrals: 1 c p < to 318 Conjugates of Singular Integrals: p = 1 320 Iterated Hilbert Transforms 323
8.3 Fourier Transforms of Hilbert Transforms
. . .
. . . . .
324
8.3.1 Signum Rule 324 8.3.2 Summation of Allied Integrals 325 8.3.3 Fourier Transforms of Derivatives of Hilbert Transforms, the Classes (W)~P,(V")~P 327 8.3.4 NormConvergence of the Fourier Inversion Integral 329
. . . . . . . . . . . . . . .
8.4 Notes and Remarks
9 Hilbert Transforms of Periodic Functions 9.0 Introduction 9.1 Existence and Basic Properties . .
. . . . . . . .
9.1.1 Existence 335 9.1.2 Hilbert Formulae
338
9.2 Conjugates of Singular Integrals 9.2.1 The Case 1 < p c to 341 9.2.2 Convergence in Can and L,:
. . . . . . . .
33 1
.
334 335
. . . . . . . . . . .
34 1
,
. . . . . . . . .
341
9.3 Fourier Transforms of Hilbert Transforms
. . . . .
. . .
347
9.3.1 Conjugate Fourier Series 347 9.3.2 Fourier Transforms of Derivatives of Conjugate Functions, the Classes (w&2s,(v&2n 349 9.3.3 NormConvergence of Fourier Series 350
9.4 Notes and Remarks
. . . . . . . . . . . . . . .
353
xv
CONTENTS
Part IV Characterization of Certain Function Classes
. . . . . .
10 Characterization in the Integral Case 10.0 Introduction . . . . . . . . . . . . . . 10.1 Generalized Derivatives, Characterization of the Classes W:,n.
. . . . .
355 357 358
10.1.1 Riemann Derivatives in Xz,Norm 358 10.1.2 Strong Peano Derivatives 361 10.1.3 Strong and Weak Derivatives, Weak Generalized Derivatives 363
Characterization of the Classes VC2, . . . . . . . . . . Characterization of the Classes (V);zn . . . , . . . . . Relative Completion . . . . . . . . . . . . . . Generalized Derivatives in LPNorm and Characterizations for 1 5 p I 2 . . . . . . . . . . . . . . . . . 10.6 Generalized Derivatives in X(R)Norm and Characterizations of the Classes WitR)and V;(R) . . . . . . . . . . . . . . 10.7 Notes and Remarks . . . . . . . . . . . . . . .
10.2 10.3 10.4 10.5
11 characterizationin the Fractional Case 11.0 Introduction . . . . . . . 1 1 . I Integrals of Fractional Order . . 11.1.1 Integral of RiemannLiouville
366 37 1 373 376 382 389
. . , . . . . . . . . . . . . . . . . .
39 1 393
IU~~],
.
400
. . . .
.
409
393
11.1.2 Integral of M. Riesz 396
11.2 Characterizations of the Classes W[Lp; Iula], V[Lp;
1 I p I2 11.2.1 Derivatives of Fractional Order 400 11.2.2 Strong Riesz Derivatives of Higher Order, the Classes V[Lp; Iul"] 405 11.3 The Operators Rp)on Lp, 1 Ip I 2 . . . . . 11.3.1 Characterizations 409 11.3.2 Theorems of BernsteinTitchmarsh and H. Weyl 414
11.4 The Operators Rp)on X,, . . . . . . . . . . . . . 416 1 1.5 Integral Representations, Fractional Derivatives of Periodic Functions 419 11.6 Notes and Remarks . . . . . . . . . . . . . . . 428 Part V Saturation Theory
. . . . . . . . . . . . . . . .
12 Saturation for Singular Integrals on X,, and Lp, 1 I p I 2 12.0 Introduction . . . . . . . . . . . . . . 12.1 Saturation for Periodic Singular Integrals, Inverse Theorems 12.2 Favard Classes . . . . . . . . . . . . .
43 1
. . . . . . . . .
433 435 440
. . . . . . . . . . . .
452
12.2.1 Positive Kernels 440 12.2.2 Uniformly Bounded Multipliers 441 12.2.3 Functional Equations 446
12.3 Saturation in Lp, 1 5 p 5 2
12.3.1 Saturation Property 452 12.3.2 Characterizations of Favard Classes: p = 1 455 12.3.3 Characterizations of Favard Classes: 1 < p 5 2 459
xvi
CONTENTS
. . . . . . .
463
. . . . . . . . . . . .
471
12.4 Applications to Various Singular Integrals
,
12.4.1 Singular Integral of FejCr 463 12.4.2 Generalized Singular Integral of Picard 464 12.4.3 General Singular Integral of Weierstrass 465 12.4.4 Singular Integral of BochnerRiesz 467 12.4.5 Riesz Means 469
12.5 Saturation of Higher Order
12.5.1 Singular Integrals on the Real Line 471 12.5.2 Periodic Singular Integrals 474
12.6 Notes and Remarks
13 Saturation on X(R)
. . . . . . . . . . . . . . .
. . .
. . . . . . . . . . . . . . . . . . .
13.0 Introduction . 13.1 Saturation of D,(A x ; 7 ) in X(R), Dual Methods 13.2 Applications to Approximation in Lp, 2 c p c co
478
.
483 485 488
. . . . . . . . . . . . . .
493
. . .
,
.
13.2.1 Differences 488 13.2.2 Singular Integrals Satisfying (12.3.5) 489 13.2.3 Strong Riesz Derivatives 490 13.2.4 The Operators RP) 491 13.2.5 Riesz and Fej& Means 492
13.3 Comparison Theorems
13.3.1 Global Divisibility 493 13.3.2 Local Divisibility 495 13.3.3 Special Comparison Theorems with no Divisibility Hypothesis 498 13.3.4 Applications to Periodic Continuous Functions 500
13.4 Saturation on Banach Spaces
. . . . . . . . . . . .
13.4.1 Strong Approximation Processes 502 13.4.2 SemiGroups of Operators 504
. . . . . . . List of Symbols . . . . , . . . . Tables of Fourier and Hilbert Transforms . . Bibliography . . . . . . . . . . Index . . . . . . . . . . . . 13.5 Notes and Remarks
502
. . . . . . . .
507
. . . . . . . . . . . . . , . . . . . . . . . . . . . .
51 1
. . . .
5 15 521 547
0 Preliminaries
In this chapter a number of fundamental concepts and results on real variable theory, Lebesgue integration, convolutions, functions of bounded variation, normed linear spaces, bounded linear operators, and functionals are listed, together with some of the conventions and terminology that will be adhered to throughout this book. For the convenience of the reader the results are set down in a form actually used in the text, and to that extent only. In order to refer to the concepts and propositions by number later on, some are presented in various different forms although unified formulations would be possible. Several of the more important facts are given with proof or also with explicit references; this will permit the reader to review the material. The reader is also referred to the list of symbols.
0.1 Fundamentals on Lebesgue Integration
Let R be the set of all real numbers, also called the real line or line group, and C the set of all complex numbers, the complex plane. Unless specified otherwise, all functions f,g, . . . of this section are defined on R and assumed to be (complexvalued?) Lebesgue measurable (on 08). C denotes the set of all functions which are uniformly continuous and bounded on R,endowed with the norm (0.1.1)
l l f l l c = Xsug E
If(4l.
is the set of functions which are Lebesgue integrable to the pth power over R if 1 5 p c co, and essentially bounded (bounded almost everywhere) on R if p = 00. ForfE Lp LP
t It should be emphasized that, for the purposes of this text, the extension from real to complexvalued functions is clear : A certain property holds for a complexvalued function if and only if it holds for the real part Re (f)as well as for the imaginary part Im (f). ~F.A.
2
PRELIMINARIES
(0.1.2)
i f p is such that I 5 p < co, and in casep = co
Ilf
(0.1.3)
Ilm
= ess SUP If(x)l. xeR
Thus LP consists precisely of those functions f for which the norm 11f IIp is afinite number. The constant factor (I/.t/%) in (0.1.2) shall turn out to be convenient in Fourier analysis. In this connection, NL1 is the set of those f E L1 which are normalized by J"f(u) du = 4%.If any confusion may occur, the more systematic notations C(R), LP(W), I l f I I L ~ , etc. are employed. Moreover, instead of 1 f 11 the symbol IIf(o)II is also used, thus indicating the variable relative to which the norm is taken (e.g h)ll). Iffhas anelementary representation, we write Ilf(x)li (e.g. "(1 x2)l/1).
.(fI[
+
+
Since R is noncompact, there are counterexamples (see Problem 5.2.1) proving that f E Lp does not necessarily imply f~ Lq for any other value q, 1 I q Ico. However, if f~ Lp n L", an elementary estimate yieldsf€ Ls for every q with p Iq I03. More generally, i f f € Lp n Lq for some 1 Ip < q Ico,thenfE Lafor everyp Is Iq. Indeed, setting El
= {x E
I If(x)l
one has El v Ez = R, and for q
11,
00
In this text, X(W) always denotes one of the spaces C or Lp, 1 Ip < 03. For = C, and almost everywhere in case X(R) = Lp, 1 I p < co,i.e., if 11f  gllxtw,= 0. In this event we also write f = g in X(W). A realvalued function f is said to be even, odd, positive (nonnegative), and strictly positive iff (x) = f ( x), f ( x ) = f ( x), f ( x ) 2 0, and f (x) > 0, respectively. It is called monotonely increasing (on W) if f ( x l ) If ( x a ) for x1 Ix2, strictly monotonely increasing if f ( x l ) < f ( x z ) for x1 < xz, monotonely decreasing if f ( x l ) 2 f ( x J for x1 I xz. A sequence of functions {fn(x)};cl is said to be monotonely increasing if fn(x) I fn+ l(x) for every x and n = 1,2, . . .. Proposition 0.1.1. (Fatou's lemma). Let {fn};= be asequence of positive functions on W. Iflim inf,,, fn(x) = f ( x ) a.e., then
f, g E X(R) we write f ( x ) = g(x) (a.e.) if equality holds for all x E R in case X(R)
Idmm
f (u) du I lim inf n+ m
j fn(u)du. m
Proposition 0.1.2. (Lebesgue's monotone convergence theorem). Let {f,,}:= be a monotonely increasing sequence of positive functions. Zflimn+mfn(x) = f ( x ) a.e., then
(0.1.4)
lim
n m
/Im
fn(u) du =
/am
f ( u ) du.
This result is also known as Beppo Levi's theorem, see also Prop. 0.3.3.
3
PRELIMINARIES
Proposition 0.1.3. (Lebesgue's dominatedconuergence theorem). Let {fn};= c L1, and suppose that limn+mf n ( x ) = f ( x ) a.e. Zfthere exists g E L' such that 1fn(x)I I g(x) a.e. for all n, then f belongs to L', and (0.1.4) holds. Proposition 0.1.4. (Fubini's theorem). Let x, y E R, and f ( x , y ) be a (complexvalued) function of two (real) variables defined and measurable on the twodimensional Euclidean space R2. (i) Suppose that f E L'(R2), i.e., the double integral $?a J Z m f (x, y ) d x dy is absolutely convergent. Then, for almost all x,f (x, y ) is absolutely integrable over R with respect to the variable y , i.e. f (x, E L1(R) a.e. Moreover, j? f y ) dy E L1(R) and 0)
(ii) Suppose that j:m and Jm m
{jTm I f ( x , y)l
f(x9 Y ) d x dr
=
(0,
dy) d x exists as afinite number. Then f
1"
 m {J""f(XY
}
Y ) dY dx
=
E
L1(Ra)
lmm {lmm Y ) dx} dY.
The second part of Prop. 0.1.4 is also associated with the name TonelliHobson. Proposition 0.1.5. (Minkowski's inequality). L e t f , g E X(R). Then cf
I l f + gllxtw, I I l f
+ g ) E X(R) and
+
Ilgllxta). If p is such that 1 I p I coy the conjugate number p' is defined through (I/p) + (I/$) = 1 in case 1 < p < co,p' = co i f p = 1, andp' = 1 i f p = co. IIX(IR)
Proposition 0.1.6. (Hofder's inequality). Let f E Lpy 1 < p I03, and g E Lp'. Then .fg E L' and llfglll II l f llpllgllp~. Let us emphasize that the assumption f E Lp, 1 Ip I coy means that f belongs to L P for some fixed p with 1 Ip I 00 (and not for all p of this interval). This will be the notation throughout this text. Forp = 2 the assertion of Prop. 0.1.6. is also known as the inequality of CauchySchwarzBunjakowski. Proposition 0.1.7. (HofderMinkowski inequality). Let f ( x , y ) be defined and measurable on R2. If I/f (0, y ) I x ( w ) E L', then
This is also known as the generalized Minkowski inequality (see HARDYLITTLE[ l , p. 1481). WOODP~LYA Proposition 0.1.8. (Completeness of X(R)). Let {fn}:=l c X(R) be a Cauchy sequence in X(R), i.e., limn,,,,+" 11 f n  fmIIxtw,= 0. Then there exists f E X(R) such that limn+" 1 f n  f = 0. The same assertion holds in L". Let S, f l yf 2 , . . . E Lp, 1 Ip 2 00. If limn+m/If n  f IIp = 0, then f is called the strong limit or Lplimit or limit in the mean o f order p of the sequence {fn}: = ;one says that the sequence {f n } converges in Lpnorm or in the strong topology towards f, and
/Ix(w)
4
PRELIMINARIES
writes f = slimn,,fn vergence.
(P)
= I.i.m.f,.
In Cspace, strong convergence is uniform con
Proposition 0.1.9. (Continuity in the mean). Iff belongs to X(R), it follows that Ilf(0 + h) f(o)Ilxta, = 0. The latter assertion does by no means hold if C is replaced by L“. Thus, considering strong convergence, we usually deal in this text with the spaces X(R).
h z + o
be Lebesgue measurable functions such that Proposition 0.1.10. ( i ) Let f, {.f,},”= f n + f i n measure, i.e., for each S > 0 lim [meas {XI If(x)  fn(x)I > S}]
n m
=
0.
Then there exists a subsequence {fnk} such that lirn,+ fn,(x) = f ( x ) a.e. (ii) Let f, {fn};= belong to Lp, 1 I p < co. If limn.+ I l f  fnllp = 0, then there exists a subsequence {fn,} such that limk. fnk(x)= f ( x ) a.e.
Proposition 0.1.11. Let f, {fn}:= belong to Lp, 1 < p < co. I f there exists a constant M such that llfnllp I M for all n = 1, 2,. . ., and iflimn,,fn(x) = f ( x ) a.e., tlzenjor every g E LP’
0.2 Convolutions on the Line Group
Let f, g be two (complexvalued) functions defined and measurable on R. The expression (0.2. I )
is called the convolution off and g. Proposition 0.2.1. Let f belongs to C, and
(0.2.2)
E
Lp, 1 5 p
I
co,andg E Lp’. Then ( f * g)(x)exists everywhere,
Ilf *gllc 5 IlfIIPllSllP~.
Moreoiler, if 1 < p < 00, then f* g E Coyi.e. f * g The same is true for p = 1 i f , in addition, g E C,.
E
C and limlxl+m (f*g)(x) = 0.
Proof. Let 1 5 p < 00. Since I(f*g)(x)l 5 IlfllPllgllP.by Holder’s inequality, the convolution (f*g)(x)exists for every x E R. Furthermore, I(f*g)(x + 11)  (f*g)(x)l 5
[email protected] + h)  f(0)llPllSllP~.
and thereforef* g E C by the continuity offin the mean. Clearly, (0.2.2) holds. If p the rciles offand g may be interchanged.
=
m,
5
PRELIMINARIES
Let 1 < p < w . Given
E
> 0, there exists a finite interval  a 5 u 5 a, such that
If x E R is such that 1x1 > 2a, then the interval x  a Iu 5 x ( u ( > a, and hence
+ a is contained in the set
Thus f * g vanishes at infinity, givingf* g E Co.The same method of proof applies in case p = 1, g E
co.
Proposition 0.2.2. LetfE X(R) andg E L’. Therz (f*g)(x) exists (a.e.) ar an absolutely conaergent integral, f* g E X(R), and
Ilf* gllxm, 5
(0.2.3) Proof. Let X(R)
=
Lp,
IlfIlxtR~llgll1~
1 5 p < co. Since for almost all N
whicht belongs to L’, it follows that
exists as a finite number. Therefore by Fubini’s theorem (0.2.5) exists as well and is equal to Ilfilgllglll. This implies that (0.2.6) exists for almost all x E R and belongs to L’. This proves the assertion for p = I. For 1 < p < 03 Holder’s inequality delivers
t The fact that I/(x  u)lplg(u)l is a measurable function on Ra is a rather delicate result of measure theory: see e.g. HEWITTSTROMBERG [ l , p. 3961, WILLIAMSON [ l , p. 651.
6
PRELIMINARIES
This shows by (0.2.6) that ( f * g)(x) exists almost everywhere as an absolutely convergent integral. Moreover, by (0.2.4) and (0.2.5)
=
l l ~ l l ~ ~ p ' l l f l l Pgll:'p ll
= IlfllPllglll.
Finally, if X(R) = C, it follows as in the proof of Prop. 0.2.1 that (f*g ) ( x ) exists for all x , belongs to C, and satisfies Ilf*gllc I Ilfllcllglll. If one of the above hypotheses is satisfied, it follows by an elementary substitution that convolving two functionsf, g is a commutative operation, i.e. (f* g)(x) = ( g * f ) ( x ) (a.e.). Obviously, convolution is distributive, i.e.f* ( g + h) = f * g + f* h. Suppose now that f, g , h belong to L'. Prop. 0.2.2 then implies f * g E L1. Thus (f* g ) * h is welldefined as an element in L1, and it follows by Fubini's theorem that convolution is also associative, i.e. (f* g ) * h = f * ( g * h). The rtimes convolution (product) f * . . *f of f E L1 with itself is denoted by [f*Ir. It exists as a function in L'. Convolution is an operation which leaves many of the structural properties of each of its members invariant (compare Sec. 1.1.2, Problems 5.1,1(iv), 6.3.5(vi)). Fundamental is the translationinvariance of convolutions. If T, denotes the operation of translation by a E R, i.e. (T,f)(x) = f ( x a), this means that Taw*g ] = T ,f * g = f * Tag. As an important consequence, convolution is a smoothness increasing operation (compare Problems 3.1.43.1.6).
+
0.3 Further Sets of Functions and Sequences
Let Z be the set of all (positive and negative) integers, P the set of all positive integers including zero, and N the set of all naturals 1, 2, . , .. Let r E P. C' denotes the class of all functionsf E C which are rtimes differentiable on R and for which f'" E C for each 1 I j I r. Obviously, one sets Co = C and C" = Cr. AC (= ACD)is the set of all absolutely continuous functions f, i.e. f ( x ) admits for every x E Iw the representation f ( x ) = J'T g(u) du for some g E L1. Correspondingly, AC'' denotes the set of all functions which are (r  1)times absolutely continuous, i.e. f E AC'' means that there exists g E L' such that for every x E R
nrm=o
(0.3.1)
f( x ) =
m
:/
dul/"  m du2 . . .
g(u,) du,,
each of the iterated integrals, possibly apart from the first, defining a function in L1. The subscript '0'stands for 'zero at infinity', whereas the subscript '00'means compact = 0 for support. Thus, CL denotes the set of those f E C' for which limlxl+mf(~)(x) each integer j with 0 s j I r, Cl,, the set of those f E Cl, which have compact support. The subscript 'loc' stands for 'locally'. Thus Clocis the set of all locally continuous functions, i.e.fis continuous on every finite interval. The class AC,,, is defined as the set of locally absolutely continuous functions, thus f E AC1,, means that there exists
7
PRELIMINARIES
g E Lfoc such that f ( x)  f (0) = [t g(u) du for every x E R. More generally, f E AC;,,' means that there exists g E Lfocand constants a,, a,, . . ., such that for every x E R (0.3.2) f ( x ) = a,
+
IOx + 1' + dul[al
[a2
+
* * *
du,, [a,'
+
/our1
g(ur>du,].
. .]It.
I f [a,b],(a, b), [a, b), (a, b] denote the sets of those x for which a _< x Ib, a < x < by I x < b, a < x 5 b, respectively (a, b may be infinite), then X[a, b] is the set C[a, b] of functions continuous on [a,b] or Lp[a,b], 1 I p < co,the set of functions Lebesgue integrable to the pth power over [a,b]. A point x E R is called a Dpoint of the functionfif
a
(h + 0).
(0.3.3)
D stands for differentiability; for, the Dpoints of an integrable f are precisely those points at which the indefinite integral off is differentiable to the valuef(x). If rh
then x is called a Lebesgue point or Lpoint o f f . Evidently, every Lpoint off is a Dpoint o f f b u t not conversely, and every point of continuity of an integrable f is an Lpoint. Moreover, Proposition 0.3.1. Iff thus Dpoints, o f f .
E
L'(a, b), then almost allpoints of the interval (a, b) are Lpoints,
Proposition 0.3.2. (Mean value theorem). Let A; f (x) be the rth (onesided) difference o f f at x (see Problem 1.5.2). I f f ~ C c : , , , then lim,,.+, h  ' d ; f ( x ) = f ( * ) ( x )at every x E R, and in fact uniformly on every compact interval. I f f E C', then lim Ilh'A;f(o)
f(r)(o)l[~
=
h+O
0.
Proposition 0.3.3. (Theorem of& Leui). Let{fn},"= c L'(a, b). Ifz:n"= Ji 1 fn(x)I dx 0 such that llfllx IM for allfg B. A sequence {f,}of a normed linear space X is called a (strong) Cauchy sequence in X if 1imm,,+ l fm  frill = 0. The space X is said to be complete if every Cauchy sequence is convergent to some element in X, in other words, if limm,n,ml fm  frill = 0 implies  frill = 0. A complete normed linear the existence of somefE X such that limn,m space is called a Banuch space. Several examples of Banach spaces have already been introduced. Thus the spaces C and C,, of continuous functions are Banach spaces under the norms (0.1.1)and (0.4.l), respectively, strong convergence being equivalent to uniform convergence. Likewise, the Lebesgue spaces LP or Lg, are Banach spaces under the norm (0.1.2),(0.1.3)or (0.4.2), (0.4.3) with the usual convention that the elements of Lebesgue spaces are considered as equivalence classes consisting of those functions which are equal almost everywhere; the zero element is then the set of all functionsffor whichf(x) = 0 a.e. Further examples of Banach spaces are the classes Ip with norms given by (0.3.6), (0.3.7)and BV with norm (0.5.1). On the other hand, BV,, is not a normed linear space since IIplIsv2, = 0 only implies p(x) = const. However, if one considers equivalence classes consisting of all functions in BV,, which differ only by a constant, then the set of these equivalence classes defines a Banach space BV;,; in particular, each equivalence class contains a representative which is normalized at  n to zero. In BVg, the convolution (0.6.4) is commutative (recall (0.6.7)). A Banach space X is called a Hilbert space if the norm is induced by an inner product, i.e., to each pair f,g E X there is associated a (unique) complex number (f,g), the innerproduct, subject to the following conditions: (i) (alfi a2f2,g) = alcfi, g) a2(f2,g) for any scalars ul, a,, (ii) (f,g) = (g,f)t, (iii) (f,f) = llfIl2. Examples of Hilbert spaces are given by La, Li, and l a with inner products
+
+
[If
[If
+
(0.7.2) (0.7.3)
k=00
t If c is a complex number, then E denotes the complex conjugate of c.
+
17
PRELIMINARIES
Let X and Y be two norined linear spaces which may be identical or distinct, and D c X a subset. One says that T is an operator of D into Y if for eachfE D there is determined a unique g E Y, denoted by g = Tf [or g = T(f)], called the value of T at$ The terms: operator, mapping, transformation are used synonymously. The set D = D(T) is called the domain of T. If D1G D, the set { T O D1} is the image of D, under the mapping Tand denoted by T(Dl); in particular, T(D) is called the range of T. If one takes D = X, then Tis said to be a mapping of X into Y if T(X) c Y, and onto Y if T(X) = Y. T is said to be onetoone if Tfl = Tf2 impliesf, = f a . If T is the mapping of X into itself defined by Tf = ffor allfE X, T is called the identity operator (of X) and denoted by 1. The mapping T of X into Y is said to be continuous at the point fo E X if to each E > 0 there is a 6 > 0 such that llTf  Tfol1y < E for all f E X with I l f  follx < 6. T is a continuous transformation of X into Y if Tis continuous at every point of X. An operator T on some function space X is positive iff(x) 2 0 for all x E R implies (Tf)(x) 2 0 for all x E R. An operator T of X into Y is called linear if T(otlfl a2f2)= alTVl) aZT(f2) for a11fi,f2 E X and complex numbers a,, a2.T is called bounded if there exists a constant M 2 0 such that IITflly 5 Mllfl(x for a l l y € X. The smallest possible value of M satisfying this inequality is said to be the norm or bound of T and denoted by llTll. From this definition it follows that
+
(0.7.4) for every f E X, and
+
IlTfllY 2 IlTll llfllx
A linear operator T of X into Y is continuous if and only if T is continuous at a single point fo E X, or, if and only if T is bounded. The linear system of all bounded linear transformations of X into Y, endowed withnorm (0.7.5), is again a normed linear space, denoted by [X, Y]. Consequently, the more precise notation ~ ~ T ~ is~ ~ sometimes x,y, employed for the norm (0.7.5). If, in addition, Y is complete, thus a Banach space, so is [X, Y]. A sequence of operators {T,}:= c [X, Y] is said to converge strongly to the operator T E [X, Y] if limn+mIIT,,f  Tfll~= 0 for eachfE X; {T,} is strongly Cauchy conoergent (on X) if Iim,,,,+ 11 T,f  T,,flly = 0 for eachfe X. If {T,} converges in the norm of [X, Y] towards T, i.e. limn+a llT,  Tllrx,y,= 0, the sequence {T,} is said to converge uniformly, thus in the uniform operator topology, towards T; equivalently, limn.m IIT,,f Tflly= 0 uniformly for allfE X with llfllx 2 1. Evidently, uniform convergence of bounded linear operators implies strong convergence but not conversely. The transformation T E [X, Y] is said to be isometric if it preserves norms, i.e. llTflly = llfllx for every f~ X; T defines a contraction if llTflly I llfllx for every SEX. A subset A of a normed linear space X is said to be dense in X if to eachfe X and E > 0 there exists g E A such that  gllx c E ; A is said to be fundamental in X if the set of all finite linear combinations of elements of A is dense in X. A normed linear space X is called separable if it contains a denumerable dense subset. The spaces X, are separable (see Theorem 1.2.5) while L,”, is not.
\If
2F.A.
18
PRELIMINARIES
Proposition 0.7.1. Let A be a dense linear subset of a Banach space X, and suppose that To is a bounded linear transformation of A into the Banach space Y with bound llToI([A,YI. Then To can be uniquely extended to a bounded linear transformation T of X into Y having the same bound, i.e. Tf = Tof for all f E A and ~ ~ T ~ =~ IJToIJIA,YI. ~x,y, Proposition 0.7.2. (Uniform boundedness principle). Let {Tn};= be a sequence of bounded linear operators of the Banach space X into the normed linear space Y. If { llTnf IIy} is bounded for each f E X separately, i.e., iffor each f E X there exists a constant M , such that for all n E N (0.7.6)
IlTnf IIY
Mf,
then the sequence { IITnIIrx,yl} is bounded, i.e., there exists a constant M such that IITnfIly I M l l f l l x f o r a l l n ~ Na n d f E X . Proof. Suppose that the sequence { llTnll} is not bounded. Replacing, if necessary, { 1 Trill} by a subsequence (which is also denoted by { 11 Tn 1 }), one may then assume that (0.7.7) According to (0.7.5), there exists a sequence
(0.7.8) and therefore by (0.7.7) (0.7.9)
{fn},"p1
IITnfnII > +llTnll,
lim
nr m
c X such that for all
llfnll
n
= 1,
IITnfnII = m.
Now, let {ak}T= be a sequence of positive numbers such that
(0.7.10) (for instance, ak = 6  k ) .By (0.7.7)one may find a positive integer nl such that a111 Tnl11 > 5. This determinesf,, and therefore Mnl of (0.7.6). By (0.7.7)again, one may choose n2 with na > nl such that .211TnaII > 5(c(lMn1 2). Proceeding in this way, one arrives at a subsequence {Tnk}= {Tn}such that n k + l > n k and
+
For the corresponding subsequence llfnll = 1, for i > j
{ f n k } C {fn}
set up
sj
=
2Ll akfnk.Then,
since
which converges to zero for i , j P 00 by (0.7.10). Thus the elements sj E X form a Cauchy sequence, and since X is a Banach space, they converge to an element g E X. Therefore a k h k is welldefined as an element in X, and g =
19
PRELIMINARIES
Since by (0.7.6) and (0.7.11)
and by (0.7.8) and (0.7.10)
it follows by (0.7.8) that for all positive integers m
This implies limm+ 11 T,,g 11 = 00, a contradiction to the assumption (0.7.6). This completes the proof (see also LORENTZ[3, p. 95 f]).
As an immediate application one has a theorem on the convergence of sequences of bounded linear operators; this theorem states in particular that the strong limit of a sequence of continuous linear transformations is continuous.
Proposition 0.7.3. (Theorem of BanachSteinhaus). (a) A sequence {T,}:= I of bounded linear operators of a Banach space X into a Banach space Y converges strongly to a bounded linear operator T of X into Y if and only if there exists (i) a constant A4 > 0 such that IIT,II[x,YI5 M for all n E N , and (ii) a dense subset A of X such that the sequence {T,} is strongly Cauchy convergent on A. (b) Let X be a Banach space and {T,}:= a sequence of bounded linear operators of X into itself. Thenfor each f E X lim llT,,f f l l x = 0
(0.7.12)
nt m
if and only i f there exists (i) a constant M > 0 such that 1 T,,1 [x,x, 5 M for all n E N , and (ii) a dense subset A of X such that (0.7.12) holds for each g E A.
Proof. To prove (a), let {T,,}c [X, Y] converge strongly to T E [X, Y], i.e., for each f E X, 1 T,, f  Tflly= 0. Then the sequence { IITnfIly}is bounded for each f E X, and thus IITnllrx,yl I M by Prop. 0.7.2. Moreover, limm,n+mIITmf T,,fllr = 0 for each f E X, and { T,}is strongly Cauchy convergent on X, particularly on every subset of X. Conversely, let conditions (i) and (ii) be satisfied. Since A is dense in X, givenfE X and e > 0, there exists g E A such that [ I f  gllx c e. By the triangular inequality, the linearity of T,,, and by (i) IITmf
Tnflly
I IITm(f g ) l l Y + I l T m g  TngllY 5 2ME + I J T m g  TngIIv.
+
1ITA.f g)lly
Thus, since g E A, it follows by (ii) that limm.nrm IITmf  Tnflly = 0 for each f E X, and { T,,} is strongly Cauchy convergent on the whole space X. Since Y is complete, this implies
the existence of an element TfE Y such that IIT,,f  Tflly= 0 for each f E X. Evidently, the operation T is linear. T is also bounded; for, since ~ ~ T , ,I f ~Mllfllx ~ , . uniformly for all n E N, it follows that llTflly 5 M l l f l l x for all f E X. This establishes (a). The proof of the second version of the theorem of BanachSteinhaus, namely (b), follows along the same lines. Let X, Y be two normed linear spaces. The (Cartesian) product X x Y is the set of
20
PRELIMINARIES
all ordered pairs ( f , g ) with f E X, g E Y; thus X x Y = { ( f , g ) I f E X, g E Y}. A linear T f ) If E D} is a transformation T of D 5 X into Y is said to be closed if its graph closed subspace of the product space X x Y as endowed with the norm Il(f, g)llxxy = llfllx Ilglly, i.e., whenever for a sequence {fn}c D there existfE X and g E Y such that limn+" fnllx = 0 and limn+" llg  Tfn1ly = 0, t h e n f e D and T f = g. A bounded linear transformation of a closed domain D G X into Y is closed.
{v,
+
11s
Proposition 0.7.4. (Closed graph theorem). A closed linear transformation of a Banach space X into a Banach space Y is bounded, thus continuous. The domain D(T) of a closed linear operator T of a Banach space X into itself be= (( f IIx + llTf l l ~ . comes a Banach space under the norm Ilf/lDcn A (complex) Banach space X is said to be a (complex) commutative Banach algebra if to each pair f, g of elements in X there exists an element fg, called the product, such that (i) 1 fgll 5 llfll llgll for everyf, g E X, and (ii) X becomes a commutative (complex) algebra under this multiplication on X, i.e., the associative law f ( g h ) = (fg)h, the distributive law f ( g + h) = f g + fh, the commutative lawfg = gf, and the relation (0rf)g = cr(fg) hold for anyf, g, h E X, a E C. If the algebra contains an element e such that ef = ffor everyfE X, then e is called the unit element of X. Examples of Banach algebras are given by the Banach spaces L1, BV, Lin, BVg, with convolution as multiplication (compare (0.2.3), (0.5.5), Prop. 0.4.1 (ii), (0.6.5), respectively). L1, L:, are commutative Banach algebras without unit element (see Prop. 5.1.12, 4.1.4), BV, BVg, are those with unit element (see (5.3.3), (4.3.2); but note that BVzn is not a commutative Banach algebra, see (0.6.7)). The Banach spaces C, Co, CZn,L", L&, I", :I are commutative Banach algebras if multiplication is defined through a pointwise product (e.g., if f, g E C , then ( f g ) ( x ) = f ( x ) g ( x ) ; if {h}z m r {gk}g= E I", then f g = { j i g k } h m). The algebras C, Can,L", Lam, I" do possess a unit element (namely e(x) = 1 and {ek}P= with ek = 1, respectively), whereas Co,:I do not. A further example of a not necessarily commutative Banach algebra is given by the space [X, X I of all bounded linear operators of a Banach space X into itself, multiplication being defined through the composition ( T S ) f = T ( S f ) for T, S E [X, XI, f~ X. This algebra has a unit element as given by the identity operator. Speaking informally, two algebraic systems of the same nature are said to be isomorphic if there is a onetoone mapping, called isomorphism, of one onto the other which preserves all relevant properties. For instance, two linear systems are isomorphic if there is a onetoone linear mapping of one onto the other. An isomorphism T of a Hilbert space H1 onto a Hilbert space Hais a onetoone linear mapping of H1 onto HZwhich also preserves inner products, i.e. (Tf,Tg) = (f,g ) for allf, g E HI; thus Hilbert space isomorphisms are always isometric mappings. Likewise, an (algebra) isomorphism T of a Banach algebra Al into a Banach algebra Aa is a onetoone linear mapping of Al onto the range of Tin Az which preserves products, i.e. T ( f g ) = (Tf)(Tg) for all f, g E Al.
0.8 Bounded Linear Functionals, Riesz Representation Theorems
Let X be a normed linear space. A bounded linear operator of X into (the Banach space) R or C of real or complex numbers is called a bounded linear functional. Functionals are denoted by f*,g*, . . , the value off * at f~ X by f *(f)(which is a real or complex number), and the totality of all bounded linear functionals on X by
.
PRELIMINARIES
21
X*, the adjoint or dual or conjugate of X. X* becomes a Banach space under the norm (compare (0.7.4), (0.7.5))
(0.8.1)
Ilf*II
= S U P If*(f>l Ilf II = 1
cf* E X*).
Proposition 0.8.1. Let X be a normed linear space. I f f E X is such that f *(f ) = 0for all f * E X*, then f = 8. Since the dual X* of a normed linear space X is again a normed linear space (and even a Banach space), one may consider the dual of X*, called the second dual of X and denoted by X**. X** is a Banach space. To eachfb E X one may associate an elementf,** E X** via f ,**(f *) = f *(So),f * E X*, which establishes a natural mapping f o +f :* of X onto a linear manifold Xz* of X**. This mapping is onetoone and normpreserving, and thus an (isometric) isomorphism of X onto X;*. In this sense, X c X**. If X = X** under the natural mapping, then X is called reflexive. Examples of reflexive Banach spaces are given by Lg, and L P for 1 < p < a.However, Can, Li,, L,",, C, L1, La are not reflexive. A sequence {fn}:=l of elements o f a normed linear space X is said to converge weakly, thus in the weak topology, to f E X, in notation: wlimn+mf n = f , if limn+a f * ( f n ) = f * ( f ) for every f * E X * . Proposition 0.8.2. Let X be a normed linear space and{fn}:= c X, f E X. (i) Ifthe weak limit exists, it is unique. (ii) I f {f n }converges strongly t o f , it also converges weakly to$ (iii) I f { f n }converges weakly t o f , then 11 f jlx Ilim inf,,, 1 fnIIx* Proposition 0.8.3. A sequence {fn}:= of elements of a normed linear space X converges weakly to f E X fi and only if there exists (i) a constant M > 0 such that I(fnllx 5 M for all n, and (ii) a dense subset A of X* such that limn+ f *(f n ) = f *cf> for every f * E A. This is an immediate consequence of the theorem of BanachSteinhaus. Proposition 0.8.4. (Weak compactness theorem). If X is a reflexive normed linear space, then each bounded sequence in X contains a weakly convergent subsequence. Analogously one may consider the weak topology of the dual space X* of X. However, this topology turns out to be less useful than that obtained when restricting the functionals to X = X**; the latter defines the weak* topology of X*. Thus a sequence {f=:}: of bounded linear functionals on X converges weakly* to f * E X*, in notation : w*lim,+, f : = f * , if limn+mf : ( f ) = f * ( f ) for every f E X; f * is then called the weak* limit of this sequence. If X is reflexive, then the weak and weak* topology of X* coincide. Applying the uniform boundedness principle, one immediately has Proposition 0.8.5. Let X be a Banach space and{f n*},"=l c X*,f* E X*. (i) Ifthe weak* limit exists, it is unique. (ii) I f {f :} converges weakly* to f *, then { 11f ,* IIx*} is bounded. (iii) I f { f X} converges weakly* to f *, then 11f *IIx* I lim infntm1 f n*IIx*. Proposition 0.8.6. A sequence {f=:}: of bounded linear functionals on a Banach space X converges weakly* to f * E X* if and only if there exists (i) a constant M > 0 such that IIf:Ilx* 5 M for all n, and (ii) a dense subset A of X such that limn+mf : C f ) = f *(f)for every f E A.
22
PRELIMINARIES
For the following result (as well as for Prop. 0.8.4) one may consult TAYLOR [I, p. 2091, ZAANEN [l, p. 155 ff].
Proposition 0.8.7. (Weak* compactness theorem). If the normed linear space X is separable, then every sequence {f:} c X* having unifbrmly bounded norms (i.e. 1 x* < M,all n) contains a weakly* convergent subsequence.
:fI[
The foregoing implications of the BanachSteinhaus theorem together with the above selection principles (Prop. 0.8.4,0.8.7) are particularly useful in connection with analytic formulae giving a representation of all bounded linear functionals on a given space X. A list of four such results, unspecifically referred to as (F.) Riesz representation theorem, is added. It will be clear from the context, i.e., from the underlying Banach space, which result applies. Obviously, the integrals in the propositions to follow represent bounded linear functionals on the spaces in question. Proposition 0.8.8. I f f * is a bounded linear functional on Can,then there exists p E BV,, such
that (0.8.2)
for every f more,
E C2=.Thefunction p E
BVznis uniquely determined by f * up to a constant. Further
(0.8.3)
Thus (C2,)* is isometrically isomorphic to BVg, ( see also Prop. 0.6.1). Proposition 0.8.9. To each bounded linear functional f unique g E Lgb such that
* on LZ,,
1I p
0 such that Ipn(a)l IM and [Var pn]f:5 M for all n E N. Then there exists a subsequence {p,,,} and p E BV[a, b] such that lim,, pnk(x)= p(x) for all x E [a, bl. (ii) (Thereom of HellyBray for BV,). Let {p,,} C BV, and suppose that Ilpn\le.va,, IM for all n E N. Then there exists a subsequence {pnk}and p E BVzn such that for every s E Czn
Moreover,
lIpIIBVan
Ilim
IIhIIsVz..
Proposition 0.8.14. (Weak* compactness theoremfor Lp, 1 < p 5 m). Let { fn}:=l C Lp for some 1 < p I03, and let there exist a constant M > 0 such that 1 fnllp 5 M f o r all n E N. Then there exists a subsequence {fnk} and g E Lp such that for every s E Lp r m
r m
lim
km
Moreover, llgllp I lim infk,,
J m ~
fnk(u)s(u) du =
J m
g(u)s(u) du.
~ ~ f n k ~ ~ p .
Proposition 0.8.15. (Theorem of HellyBray for BV). Let {pn}c B v and suppose that
24
PRELIMINARIES
IIpnllBv 5 M for all n E N, Then there exists a subsequence {pnk} and p E BV such that for every s E Co
Moreooer, IIpllBv5 lim infk., IIpflkIlBv. The latter result is also referred to as the weak+ compactness theorem for BV.
Finally a result concerning the boundedness of sequences of functions in Lpspaces is given which indeed is an immediate consequence of the uniform boundedness principle (compare Prop. 0.8.5(ii), and the Riesz representation theorems (see also ZYGMUND [71, p. 1661). Proposition 0.8.16. Let {f,,};=l c Lg, for some 1 5 p i a, and suppose that s(u)f,(u) dul is bounded in n E N for each s E Lgk separately. Then there exists a constant M > 0 such that llfnllp I M f o r all n E N. A related result holds in Lpspace.
0.9
References
General references to the material of the whole chapter are HEWITTSTROMBERG 111, ROYDON[l], RUDIN[4]. Specifically,for the results on real variable theory and Lebesgue integration (Sec. 0.10.6) see ASPLUNWBUNGART [l], HALMOS[l], HILDEBRANDT [l], MCSHANEBOTTS [I], MUNROE [l], TAYLOR [2], WILLIAMSON [l], ZAANEN [2]. For the basic facts on functional analysis (Sec. 0.7, 0.8) see BANACH[l], DUNFORDSCHWARTZ [I], GARNIRDE WILDESCHMETS [I], HALMOS [2], LJUSTERNIKSOBOLEV [ 11, RIESZSZ.NAGY [l], TAYLOR111, YOSIDA[l], ZAANEN[l].
Part I Approximation by Singular Integrals
One of the fundamental problems of analysis is to approximate a given function f in some sense or other by functions having certain properties, and generally, by functions which have 'better' properties thanf. It is to be expected that the betterbehaved functions are to be constructed from the givenfby some smoothing operation on f itself. The approximation off by singular convolution integrals is of special interest. Given f~ Can,a convolution integral of the type
(f*xn)(x) = (1/24
j' f(x  u)xn(u) du R
will be called a singular integral provided that the sequence {x,(x)}n"= is a (periodic) kernel, specifically,Y,, E L:, with ,x,,(u) du = 2~ for each n E N. Such a sequence is said to be an approximate identity if, in addition, llx,,lll I M for all n and J d s , u , s n Ix,,(u)I du = 0 for each 0 < 6 c n. If the functions x,, happen to be positive, they usually have the familiar bellshaped graph: the area under the curve y = X,,(U) is equal to 2 ~whereby, , for increasing n, the peak at u = 0 becomes higher and narrower in such a way that the area under the curve near u = 0 comes out equal to 2 ~ . The name approximate identity is justified by the fact that forfe C,, the sequence {f*x,,} tends uniformly toffor n + m. Indeed, in view of the properties of {x,,(x)}, the magnitude of the convolution integral f * xn for large n essentially depends upon the value of its integrand near u = 0 (fbeing bounded). Since f(x  u ) is then near to f ( x ) (fbeing continuous), (f*x.)(x) is roughly equal tof(x)(l/27r) x,,(u) du, which tends tof(x) for large n. One of the important features of the convolution integral f* x,, is that the 'best' properties of each of its factors are inherited by the product itself. This is due to the translationinvarianceand commutativity of convolutions.Thus, iff€ C,, is convolved with xn E C,, for which xz)E C,,, then the convolution product is rtimes continuously differentiable. Special as these convolution integrals may seem, they nevertheless subsume a large
yd
26
APPROXIMATION BY SINGULAR INTEGRALS
number of integrals of analysis; these occur in the theory of Fourier series, as solutions of partial differential equations, and in approximation theory. Each of these associations may be outlined briefly. The theory of singular integrals is intimately connected with the theory of Fourier series. Thus the nth partial sum of the Fourier series of a functionfmay be written as a singular integral with the Dirichlet kernel, while the sequence of arithmetic means of these partial sums form a convolution integral with the FejCr kernel. It turns out that the FejCr kernel is an approximate identity, whereas the Dirichlet is not. The procedure of taking the first arithmetic means is known as CesAro summation of the Fourier series off. There are many other methods of summation, the kernels of which form approximate identities and which therefore sum Fourier series effectively. Some specific examples are provided by the Abel, Weierstrass, Riemann, and Riesz methods. Thus convolution is a vital concept in summability problems of the theory of divergent series. A second connection is concerned with initial and boundary value problems in the theory of partial differential equations. Thus the solution of Dirichlet’s boundary value problem for the unit disc is a singular integral having as approximate identity the AbelPoisson kernel {p,(x)}. It was FATOUwho showed that iff’(xo) exists, then the derivative (f*pr)’(xo)of the solution converges tof’(xo) when r + 1  . One of the vital problems of approximation theory is to estimate the error, or discrepancy, (f*xn)(x)  f ( x ) under varying hypothesis upon f and xn. As a rule, the smoother the function, the faster the error tends to zero for n 4 co. These are direct theorems of approximation theory. Conversely, inverse theorems infer smoothness properties off from the smallness of the error (f*xn)(x)  f ( x ) . The smoothness properties uponfare usually given in terms of differentiability or Lipschitz properties. One speaks of an equivalence theorem in case the direct and inverse theorems are exact converses of each other. These problems lead to the study of the classical theory of approximation of periodic functions by trigonometric polynomials, as associated with the names of D. JACKSON and S. N. BERNSTEIN. At this stage, the fundamental concept of best approximation of a periodic function by polynomials comes into play. The Jackson result is a direct theorem; the kernels that are used in its proofs are positive approximate identities. Not only do singular convolution integrals deserve study on the circle group, but also on the line group. GivenfE Lp, 1 I p c 00, such an integral has the form
with kernel { ~ ( xn)} ; satisfying x(“; n) E L’, J?m x(u; n) du = 6 for each n E N. If, in addition, Ilx(0; n)lll IMfor alln and JdSlu, Ix(u; n)l du = 0 for each S > 0, then { ~ ( xn)} ; is called an approximate identity. However, it is important that detailed study can be restricted to the case x(x; n) = nx(nx) with x E L’, J?m x(u) du = d z ; in other words, x ( x ; n ) is generated by a function x(x) of one variable through a
APPROXIMATION BY SINGULAR INTEGRALS
27
simple scale change. These kernels are said to be of Fejtr's type; correspondingly, the convolution integral
J ( f ; x ; n) =
Sm n d2a
f(x  u)x(nu) du.
m
It is basic that every kernel of FejCr's type is an approximate identity, and our remarks here are confined to them. As for the circle group, the graph y = n,y(nu) for n +co reveals that {nx(nu)} is a peaking kernel approaching the (Dirac) delta measure. Thus J ( f ; x; n) tends in Lpnorm tofas n f co,and one is interested in examining direct, inverse, and equivalence theorems for the approximation off by J ( f ; x ; n). These problems again include a number of exciting questions. As an example we may mention the limiting behaviour of solutions of the heat equation for an infinite rod. Indeed, its elementary solution is Green's function G(x, t) = (2t)l/, exp { xa/4t}, and, according to the generalized superposition principle, the general solution is then given by the convolution of G(o,t) with f,f being the initial temperature distribution. As we are dealing with integrals of convolution type, it is to be expected that Fourier transforms enter into the discussion. Indeed, the powerful techniques of Fourier analysis will be fully exploited. The Fourier transform of the (convolution) product J ( f ; x ; n) is the (pointwise) product of the transforms offand nx(n o), i.e.f^(u)x"(u/n), thus separating the functionfand the kernel. For large n the transform x^(u/n) is close to f(0) which takes on the value 1 in view of the normalization of x; on the other hand, the Fourier transform of the delta measure is equal to 1. Consequently, on the basis of the transform methods to be applied in our later approximation theoretical investigations, the behaviour of the transform ~"(u)near u = 0 will be the decisive one. Furthermore, iff has period 2a and xA(u) = 0 for IuI 2 I (which implies that the transform of nx(n vanishes for IuI 2 n), then J ( f ; x ; n) has Fourier coefficients which vanish for Ikl 2 n, and is therefore a trigonometric polynomial of degree less than n. This gives the connection with the approximation of periodic functions by trigonometric polynomials, It is supplemented by the fact that every kernel x of Fejtr's type generates a periodic approximate identity via 0)
ForfE C,, it then follows that
1 1'f(x 2a x
 u)xt(u) du = J m
4%
mf(
x  u)x(nu) du.
Since many important examples of periodic kernels are generated by some x E L', we can, for the corresponding singular integral of periodic functions, take our choice of either expression. The righthand one has the advantage that the kernel is determined by scale change from one single generating function x, whereas the periodic kernel {x:(x)} may have a more complicated functional dependence on the parameter n (though the Fourier coefficients of x: are given by ~"(kln)).
28
APPROXlMATlON BY SINGULAR INTEGRALS
Having outlined some of the problems that are to be treated in Part I, we may conclude the introduction with a brief but more systematic outline of the individual chapters. Chapter 1 is exclusively concerned with singular integrals on the circle group, their fundamental properties, norm and pointwise convergence, direct approximation theorems, and asymptotic expansions. Included is a section on the classical theory of Fourier series. Chapter 2 is devoted to the theorems of Jackson and Bernstein for polynomials of best approximation. The basic properties of such polynomials, in particular their existence, are considered briefly. The chapter concludes with inverse theorems for convolution integrals which need not necessarily be polynomial summation processes nor have orders of approximation as good as polynomials of best approximation. Chapter 3 is devoted to a detailed study of singular integrals on the line group. In many respects, the treatment is parallel to that for the circle. On the other hand, it complements the periodic results by those of Sec. 3.1.2. The theory of Fourier transforms is only introduced in Part I1 and subsequently applied.
1 Singular Integrals of Periodic Functions
1.O Introduction
Following our discussion of the purpose of the study of singular integrals in the introduction to Part I, the scope of this chapter may now be outlined briefly. Sec. 1.1 deals with basic properties of singular integrals such as their convergence in the norms of the spaces Can, LE,, 1 I p < 00. Convolution integrals of type x,,* dp, p E BV,,, are examined and a short discussion on strong and weak derivatives is included. In Sec. 1.2 the emphasis is upon the fundamental facts of trigonometric and Fourier series, summability of Fourier series, the FejCr and AbelPoisson means. Summability of conjugate series is considered and FourierStieltjes series are introduced. Sec. 1.3 is concerned with necessary and sufficient conditions assuring normconvergence of the singular integral &(Ax ) towardsffor allfE Xz,. The fundamental BanachSteinhaus theorem delivers such conditions in case of C,, or L:,space. For this purpose, the norms of convolution operators are determined in Sec. 1.3.1. In case the associated kernels are positive, an interesting theorem of BOHMANKOROVKIN (1952/53) states that normconvergence takes place for all f E X,, if and only if it does so for the two particular test functions cos x and sin x. In Sec. 1.4 pointwise convergence of singular integrals is studied, various hypotheses upon the kernel being taken into account; also the convergence almost everywhere of x,, * dp is examined. An application to the AbelPoisson integral leads to the theorem of Fatou. Sec. 1.5 is concerned with questions on the order of approximation of periodic functions by positive singular integrals. Prop. 1.5.10 is a recast of the BohmanKorovkin theorem in a quantitative form: the rapidity of convergence of I0(Ax ) to f ( x ) is estimated in terms of the rapidities of convergence of I,,(cos u ; x ) to cos x, Z,,(sin u ; x ) to sin x. Subsection 1.5.4 is devoted to asymptotic expansions of positive singular integrals. Sec. 1.6 treats direct approximation theorems with applications to the integrals of FejkrKorovkin and FejCr. There is also a short discussion on the best asymptotic or Nikolskii constants for the measure of approximation. Sec. 1.7 is
30
APPROXIMATION BY SINGULAR INTEGRALS
devoted to simple inverse smallo approximation theorems for singular integrals generated by rowfinite &factors. Finally, there is a result by KOROVKIN on the critical order of approximation by positive integrals of this type.
1.1 NormConvergence and Derivatives
1.1.1 NormConvergence Let us begin with the definition of a kernel which is to generate our singular integrals. Definition 1.1.1. Let p be aparameter ranging over some set A which is either an interval (a, b) with 0 5 a < b 5 +a or the set N, and let po be one of the points a, b or +a.A set of functions {xP(x)} will be called a (periodic) kernel i f x DE L:, for each p E A and
[" x,(u)du
(1.1.1)
J
=
2~
A
(P E A).
We call the kernel {xD(x)} real ifxp(x)is a realfunction, bounded ifx,, E L,",, continuous i f x,,(x) E C,,, and absolutely continuous if x&) is absolutely continuous for each p E A. A real kernel {xp(x)} is said to be even ifxp(x)= xD( x) a.e., positive i f x D ( x )2 0 a.e. for each p E A. Instead of condition (1.1.1) one often assumes that (1.1.2)
But there is no loss of generality to suppose that the functions xP(x)are normalized by (1.1.1) from the beginning. Definition 1.1.2. Let f form
E
X,, and {xD(x)}be a kernel. Then we call an expression of the
a (periodic) singular integral (or convolution integral). W e say that the singular integral is positive, continuous if the corresponding kernel is positive, continuous. Thus to eachfE X,, we associate a set of functions {ZD(fi x)} for which we have in view of Prop. 0.4.1 Proposition 1.1.3. I$f and (1.1.4)
E
X,, and {x,,(x)) is a kernel, then ZD(fi x) E Xa, for each p E A
I I U f i o)llxp,
Moreover, if the kernel is bounded, then belongs to Can.
I l X P I I l I l f IIX&
ZD(Ax) is a continuous function of
x, thus
SINGULAR INTEGRALS OF PERIODIC FUNCTIONS
31
Setting I,,(j; x) = [I,,f](x) for p E A, the integrals (1.1.3) define bounded linear transformations I,, of X,, into X,,, determined by the kernel {x,(x)}. Of fundamental importance are theorems guaranteeing the convergence of Io(J x) towards a given f as p 4 po. In this respect we are mainly interested in theorems on normconvergence, in other words, in theorems giving the strong convergence of the operators I,, towards the identity operator in X,,. Assumptions assuring this fact lead one to the following definition of an approximate identity.
Definition 1.1.4. A kernel {xp(x)}is called a beriodic) approximate identity if, with some constant M > 0, (1.1.5)
lim
(1.1.6)
P+Do
1
(P
llxDIl1
E
A),
(0 < 6 < ?I).
Ix,,(u)l du = 0
dslulsn
We call an approximate identity even, positive, bounded or continuous fi the kernel is even, positive, bounded or continuous. Before proceeding to the main convergence theorem let us mention that, instead of (1.1.6), we shall often assume the property
(0 < 6
0 there is a 6 > 0 such that  u)  f(o)IIXzn IE for all IuI I 6. This implies I eM. Now take 6 fixed. Then
.(fI[
12
1 2 llfllxan5
1
dslulsn
IXP(4l
4
32
APPROXIMATION BY SINGULAR INTEGRALS
which tends to zero as p > po according to (1.1.6). Thus (1.1.8) holds in case X,, = C2n, If X,, = Lg,, 1 Ip < 00, we proceed by the HolderMinkowski inequality to deduce (1.1.10). Sincef is continuous in the mean (compare Sec. 0.4), the proof follows as before. If we replace X,, by Lb, (1.1.8) need no longer hold since f is not necessarily continuous in this norm. Then we have Proposition 1.1.6. Let f
E
L .,;
If the kernel of the integral (1.1.3) is an approximate identity,
rhen for every s E ,L: (1 .l. 11)
:%!
I:,
[IpCr; X)
 ~ ( x ) ] s ( xdx) = 0.
Proof. Again ( I . 1.9) holds, and hence by Fubini's theorem for every s E Lin and 0 < 6 < T
say. But since s E L:, is continuous in the mean, for each IIs(0 + u)  s(o)llr Ie for all IuI 5 6. Thus
Furthermore, la 5 2
1
IIflImIbll~
e 7
0 there is a 8 such that
Ixo(u)l du,
d5lulsn
which proves (1.1.11) in view of (1.1.6). Analogously to (1.1.3) one may also assign to each function integral
pE
BV,, a singular
(1.1.12) for which the following proposition holds. Proposition 1.1.7. Let P E A \ and
pE
(1.1.13)
BV,, and k p ( x ) }be a kernel. Then ID(dp;x ) E Lk, for each IlWG
0)111
IIXPII1IIPIIBV2,.
If the kernel {xD(x)) is moreouer an approximate identity, then for every s E C,, (1.1.14)
lim
PPO
j:zs(x)Zo(dp;x) dx
=
L
s(x) dp(x).
Proof. (1.1.13) follows by Prop. 0.6.3. To establish (1.1.14) we have by Fubini's theorem
SINGULAR INTEGRALS OF PERIODIC FUNCTIONS
33
Since the expression in curly brackets tends unifornily to s(u) by Theorem 1.1.5, (1.1.14) follows. Let us conclude with an example. The (first) integral means (also called moving average or Steklov function, or last not least, singular integral of RiemannLebesgue) are defined by
1 x
(1.1.15)
Ah(f;
=

h
+ (hl2)
x(h/2)
/
f ( u ) du = 1
h/2
f(x
 u ) du.
h/2
Here h is a positive parameter ranging over A = (0,27r) and tending to O + . The integral (1.1.15) is of type (1.1.3). Indeed, setting p = h, the kernel {xp(x)}is given as the set of those 2rperiodic functions which, for each fixed h, coincide on [T, T) with (2~/h)q  , b , 2 , h / 2 1 ( ~ (recall ) definition (0.3.5) of characteristic functions). Obviously, this kernel is an even, positive, and bounded approximate identity. Therefore by Prop. 1.1.3, Theorem 1.1.5
Corollary 1.1.8. L e t f E X,,. Then &(f; x ) belongs to Xznn C2,such that I IlfIlx,,for all h E (0,2r) and
I I A h e o)llx2,
Thus every integrable function f can be approximated in the mean arbitrary closely by continuous functions, namely by the &(f; x). These are furthermore differentiable almost everywhere as stated by Prop. 0.3.1. Further examples of singular integrals will be submitted during the course of the subsequent sections, see in particular Sec. 1.2. 1.1.2 Derivatives While Theorem I . 1.5 was concerned with normconvergence of singular integrals towardsf, we shall now discuss approximation of derivatives. By the way, we shall illustrate our earlier remarks regarding convolutions as smoothness increasing operations. Roughly speaking, we shall show that if either factor is differentiable, so is the convolution product. To this end, let us first introduce some classes of functions. In what follows, r always denotes a natural number. We set
Although the derivatives in this definition are to be taken in the pointwise sense, the fact that all derivatives occurring belong to the underlying space X,, actually implies that the classesW;C, may be precisely described through strong derivatives.To develop this point, we commence with
Definition 1.1.9. I f f o r f ~X,, there exists g E X2, such that lim Ilh"f(. ha0
3P.A.
+ h) f(.)l
 g(o)llx2,
=
0,
34
APPROXIMATION BY SINGULAR INTEGRALS
then g is called the uniform deriilative o f f if X,, = C,,, and the derivative o f f in the mean of order p if X,, = Lin,1 Ip c 00. In short, we shall speak of these derivatives as the (first ordinary) derivative in X,,norm or strong derivative, and denote g by Di’lf. For any r E N, the rth strong? derivative o f f E X,, ir then defined successively by Dr’f = Di1)(Dr 
As an immediate consequence we state Proposition 1.1.10. f E Wk,, implies the existence of the rth strong derivative Dp’foff. I f X,, = Can, then (DP’f)(x)= f(”(x)for all x, and if X,, = Lg,, 1 s p c do, then (DP’f)(x)= +(I)(x)a.e., where E AC‘,; with +(I) E Lg, is such that f ( x ) = +(x)a.e.
+
Proof. Let X,, = Lg,, 1 Ip c (1.1.17)
f(x
h, h
03.
The assumption assures that for each h E R
 f ( x )  +’(x)=
iloh[+‘(x+
u)
 #(x)]du a.e.
Therefore by the HolderMinkowski inequality and the continuity of 4’ in the mean
Thus the first strong derivative off exists, in fact DilY = 4’. The proof is now completed by an obvious induction. Indeed, let k c r, and suppose that the kth strong derivative o f f exists with D 3 ’ f = +( k) . Since + ( k ) is absolutely continuous, for each hER
This implies by the continuity of +(lC
l)
in L$,norm that
establishing the existence of the (k + 1)th strong derivative off and Dkk l’f = l). In Canspacethe proof is similar, but more elementary. In Sec. 10.1.3 we shall show that also the converse of Prop. 1.1.10 is valid. Surprisingly, this is even true for the (seemingly) more general concept of a weak derivative. +
Definition 1.1.11. If for f
E
X,, there exists g
E
X,, such that
t Strong derivatives are denoted by DE’/ to distinguish them from usual pointwise derivatives f“). The subscript s stands for strong.
SINGULAR INTEGRALS OF PERIODIC FUNCTIONS
for every 7 E BV,,
if X,,
=
C,,, and
35
I:,
h,  f ( x ) s(x) dx = g(x)s(x) dx h for every s E LEk if X,, = Lg,, 1 Ip < co, then g is called the (jirst ordinary) weak derivative ofS, and denoted by D$Y For any r E N, the rth weak derivative o f f E X,, is defined successively by DYY = D:"( 0sly). As is to be expected, one has immediately f(x
+
Proposition 1.1.12. If the rth strong derivative o f f E X,, exists, so does the rth weak derivative. In fact, we then have D$'f = Dry As a consequence of the preceding propositions we have that, iff E W;C,,, then the rth weak derivative o f f exists and (DP'f)(x)= f'"(x) if X,, = C,,, for example. For the converse see Sec. 10.1.3. For purposes of illustration let us here state the following result which may be regarded as a fundamental theorem of the calculus involving strong and weak derivatives. Proposition 1.1.13. Let f
E Xz,.
If thefirst weak derivative o f f exists and is zero, thenf ( x ) =
const (a.e.). For a proof as well as for further results, including strong and weak generalized derivatives, see Sec. 10.1. We now come to the actual problem of this subsection. However, instead of treating derivatives of the convolution of two arbitrary functions (to be found in Problem 1.1.8), we shall immediately turn to singular integrals (1.1.3), i.e., to convolutions for which one factor is a kernel.
Proposition 1.1.14. Let f E X,,. Ifthe kernel {xP(x)} of the integral ZP(Ax ) is continuous such that x,, E Wk,.l for each p E A,then Zp(fi x) is an rtimes continuously diferentiable function of x. In particular, for every x and p E A (1.1.18)
[I,(fi
o)](')(x)
=
1'
L f ( x  u)x;)(u) du. 27r ,
Proof. Let r = 1. Since x,, E Can,it follows that x,, E LQanfor every 1 Iq I 00. Therefore, for each p E A, ZO(Ax) exists everywhere as a function in C,, by Prop. 0.4.1. also ( f * x J ( x ) exists everywhere, and hence for arbitrary
Since f
E
X,, implies f E Liz, we have
36
APPROXIMATION BY SINGULAR INTEGRALS
for every x. As xp E Wian,the first uniform derivative of xp exists by Prop. 1.1 .lo. Thus it follows that [Ip(f; o)]’(x)exists at every x and that (1.1.18) is valid for r = 1. If r E N is arbitrary, the proof is completed by a straightforward induction.
Proposition 1.1.15. Let {xP(x)}be the kernel of the integral Io(f; x). Iff E W!&,, then
IpV, x ) E W;, for each p E A and (1.1.19)
[ZP(f;
O ) l ( W
= Ip(.P; x)
for every x. Moreover, if{xP(x)}is an approximate identity, then (1.1.20)
Proof. Again we restrict ourselves to the case r = 1. We may proceed as in the proof of the preceding proposition to deduce
This proves (1.1.19), the r6les off and xp being interchanged in comparison with (1.1.18). If {xp(x)}is an approximate identity, (1.1.20) now follows by Theorem 1.1.5 as applied to f r ) .
Proposition 1.1.16. Let the kernel (xp(x)}of the integral I p ( f ;x ) be bounded. I f f E WLgn, 1 I p < 00, then Zpcf; x ) is an rtimes continuously differentiable function of x, and for every x and p E A (1.1.21)
[Ip(f; o ) ] ( ~ ) ( x= ) Ip(DrX X ) = Ip(#r); x),
where 4 E AC;; with #r) E Lbn is such that $(x) = f ( x ) a.e. Moreover, if{xp(x)}i~an approximate identity, then (1.1.22)
lim I [Zp(f;
PPO
0)IYo)

(~P’f)(o>llp
= 0.
Proof. Let r = 1. Since xp E Lamn implies xP E LI, for every 1 I q I 00, the convolution Ipcf; x ) exists everywhere as a function in Canby Prop. 0.4.1. Furthermore, Dil’fexists by Prop. 1.1.10, and hence by Holder’s inequality
for every x and p E A. Since the righthand side tends to zero as h 0 by the definition of a strong derivative, (1.1.21) follows for r = 1. An obvious induction delivers the general case. Finally, Theorem 1.1.5 as applied to Dry gives (1.1.22). Certainly, further results on derivatives of convolutions may be obtained by varying and even weakening the hypotheses (see also the Problems of this section). However, the above three propositions suffice in our later investigations on approximation by convolution integrals I,( f ; x). They nevertheless illustrate the fact that a convolution is differentiableif either of the factors is differentiable. Here differentiability is actually taken in the strong sense, This enables one to use Holder’s inequality, thus avoiding f
37
SINGULAR INTEGRALS OF PERIODIC FUNCTIONS
the (sometimes more delicate) criteria for the interchange of integration and differentiation (with respect to a parameter) known from real variable theory. Prop. 1.1.14 is applicable to many of the examples of singular integrals that are to follow. Indeed, one reason for the interest in the approximation o f f by specific integrals Z,(A x ) is that the approximating functions I,(A x) are smoother than f. This is achieved by the fact that many kernels belong to C&. Therefore we assumed x,, to be continuous in Prop. 1.1.14. But the kernel of the integral means (1.1.15) is not continuous; here we have Problem 1.1.7(ii) which depends upon the particular structure of the integral Ah(S; x). On the other hand, we may apply Prop. 1.1.15, 1.1.16, provided f E W;C,. Since the kernel of the integral means AI1(S; x ) is a bounded approximate identity, we have
Corollary 1.1.17. Let f E W;C,. Then A,,(A x ) is an rtimes continuously diyerentiable function of x and [Ah(f;o ) ] ( ‘ ) ( x ) = Ah(Dr’f; x) for every x and h E ( 0 , 2 ~ )Moreover, . lim
h+O+
[ I [ A h d f ; o)I’(o)
 ( ~ f Y ) ( o ) I l ~ ~ , ,= 0.
Problems 1. (i) Let {x,(x)} be a kernel. Show that J’!,I,(f; x ) dx = J’!,f(x) dx for every f E Xzn and p E A. (ii) Show that if {x,(x)} is a kernel [approximate identity], so is {x,(  x)}. 2. Prove Theorem 1.1.5 for approximate identities satisfying (1.1.2) instead of (1.1.1) (compare also EDWARDS[lI, p. 601). 3. Let {fn};I be a sequence of functions in X,, which are bounded and continuous in X,,norm, uniformly with respect to n E N. If (x,,(x)} is an approximate identity, show that lirn,,,, IlI,,(f,; 0)  fn(o)llxan = 0 uniformly for n E N. 4. (i) Let {x,(x)} be a kernel. Show that {k,* x,)(x)} is again a kernel, known as the iterated kernel. (ii) Let (x,(x)} be an approximate identity. Show that the iterated kernel again has the
property (1.1.5). What can one say concerning (1.1.6)? (iii) Let the iterated singular integral be defined by ( 1.1.23)
If {x,(x)} is an approximate identity, show that I,”(f; x ) = Z,,(I,V, x ) (a.e.) and llIz(f; 0)  f(o)IIXln = 0 for every f E Xzn. Extend to iterates of higher order. (Hint: Use the estimate 0);
lim,,,,
llI3A
 f(0)IlXan
5
I l I A I D e 0)

+
IlZJf; 0)  f ( ~ ) I l x l n ) 5 . Let the rfhintegral means A;(A x ) be defined successivelyby A;(f; x ) = kfh(Ai ‘cf; 0 ) ; x). Show that A L e x ) is of type (1.1.3) with kernel given by the rtimes product 0)
f(0);
0)llx2n
Furthermore, [ ( 2 ~ / h )~hla.h/a]*]‘. [
Show that A K f ; x ) E Can for every h E (0, 2 4 and limh,o IIAicf; every f E Xzn (and r E N).
0)
 f(o)llxan
=0
for
38
APPROXIMATION BY SINGULAR INTEGRALS
6. Let f be defined in a neighbourhood of a point x E R. For (sufficiently small)! E R the first central diference o f f a t x with respect to the increment h is defined by Ag(x) = f(x (h/2))  f(x  (h/2)),and the higher differences by E u ( x ) = &$!&'f(x). Show (by induction) that
+
7.
(i) Let f~ Xan. Show that limh,o Aicf; x) = f ( x ) almost everywhere, in particular at all points of continuity off(for r = 1 compare with Prop. 0.3.1). (ii) Let f E Xzn. Show that AaU, x) has absolutely continuous derivatives of order (r  l),and [ A b ( f ; o ) ] ( ~ ) ( x )= h'E;f(x) (a.e.). (iii) Let f~ W',,,, and let 4 E AC',;' with +(I) E Xzn be such that 4(x) = f(x) (a.e.1.
Show that A L ( # ~ ) x) ; = AL((D:"~; x ) = [ A L ( ~o;) ] ( W = krqf(x) (a.e.1. In particular, show that A f V , x ) E Wxi: and [AEV,O ) ] ( ~ + ~ ) ( X )= ha[#r)(x+ h) +(r)(x h)  2+(r)(x)] (a.e.). [l, p. 2541, TIMAN [2, p. 163 ff]) (Hint: GRAVES 8. (i) Let f~ Wtln and g E Lin. Show that If*g](l)(x) = (f"' * g ) ( x ) for every x. (ii) Letfs W:pn, 1 Ip < Q), and g E LE;. Show that ( f * g)(x) is rtimes continuously differentiable and that for every x
+
If* gl(r)(x)= ( X ' f* g )(x) = (+(r) * g)(x), where E AC',;' with +(I) E LIn is such that +(x) = f ( x ) a.e. (iii) Let f~ W:!,, 1 5 p < 03, and g E W:;, k E N. Show that (f*g)(x) is (r + k)times continuously differentiable and that for every x [f*g]"+"'(x) = (Dry*g)'"(x) = (Dry* Dy'g)(x). (iv) LetfE W',,, andg E Lln. Show that the rth strong derivative (in Xzn)off* g exists and that Dr)Lf* g ] = (Dry*g). (Theorem 10.1.12 indeed gives f * g E Wkln, in other words, if f e WL1, and g E Lin, then also f * g E WL,,,.) (v) Let f E Xan and g E W;;,. Show that the rth strong derivative (in X2,) o f f * g exists and that D:)If* g ] = f * Dr)g, Dr)g being the rth L?j,derivative (in fact f * g E WLln by Theorem 10.1.12). 9. LetfE W',,,, and let {xp(x)} be a continuous kernel such that xp E Wt,, for each p E A. Show that I,V, x) is an (r + k)times continuously differentiable function of x. Moreover, for every x and p E A
+
[ I D V , 0)Y' + k ) ( x ) = [ZO( DP'f; o)]'k'(x) = (Dry* xLk')(x). 10. Apply Prop. 1.1.6, 1.1.7 to the integral means. Show in particular that if p E BV,,, for every s E Can
11.
then
x e Can be such that x(0) = 1 and 0 I x(x) < 1 for all x E [ T,T I , x # 0. Show that (2n IIx"II1)' j?,x"(u)du = 1 for each (fixed) 0 < 8 IT. Verify that {(~~xn~~dlxn(x)l is a positive bounded approximate identity. (Hint: See also KOROVKIN [5, p. 19 fl) (ii) Let the singular integral x ) be defined by (n E N) (i) Let
where x is as in (i). Show that
llI,,(fi
0)
 f(0)11~,,
=
0 for every f~ X2,.
39
SINGULAR INTEGRALS OF PERIODIC FUNCTIONS
+
(iii) As an example consider x(x) = (1 cos x)/2. (Hint: This leads to the singular integral of de La VallQ Poussin as treated in Sec. 2.5.2; see also KOROVKIN [S, p. 541, RUDIN14, p. 89 f])
1.2 Summation of Fourier Series 1.2.1 Definitions
A trigonometricpolynomial (or in short a polynomial) t,(x) of degree n, n E P, is an expression of the form n
(1.2.1) where the coeficients ak, bk are arbitrary (complex) numbers independent of x. It will soon be clear why it is convenient to provide the constant term of (1.2.1) with the factor +.The set of all trigonometric polynomials of degree not exceeding n will be denoted by T,. If t , E T, and (la,/ lbnl) # 0, we say that t , is strictly of degree n. Trigonometric series are series of the form
+
(1.2.2) with arbitrarily given coefficients ak,bk.The trigonometric series(l.2.2) may or may not converge. It converges at x if the sequence of partial sums (1.2.3)
+
s,(x) = +ao
{akcos kx
+ bk sin kx}
k= 1
converges at x . In any case, if the coefficients {ak,b,} are given, we may assign to them the trigonometric series (1.2.2), at least formally. Applying Euler’s formulae ,lkX
cos kx =
(1.2.4)
+ ,tkx
2
efkx
sinkx =

,fkX Y
2i
we may write the nth partial sum (1.2.3) of (1.2.2) in the form S,(X) = &7o k
3
n
2
{(ak  ibk) d k x k (ak
k=l
+ ibk)
C?fkx}.
This suggests that one defines a,, bk for k E b by the convention (1.2.5) Then, if we set (1.2.6)
ak
= Uk,
bk = bk,
bo = 0
(k E N).
40
APPROXIMATION BY SINGULAR INTEGRALS
we obtain for s,,(x) (1.2.7)
k = n
and thus the nth partial sum of the trigonometric series (1.2.2) corresponds to the nth symmetric partial sum of the series (1.2.8) Conversely, any series (1.2.8) with (complex) coefficients ck can be written in the form . call (1.2.2) the real form (1.2.2) by setting ak = CI, ck and b, = i(ck  c  ~ ) We and (1.2.8) the complexform of a trigonometric series. The adjectives refer to the real trigonometric system {cos kx, sin kx}, Ep and the complex trigonometric system {etkx}k z. In addition, we may usually suppose that the coefficients in (1.2. I), (1 2.2) are real; if they are complex, the real and imaginary parts of (1.2. I), (1.2.2) can be taken separately. A complex trigonometric series (1.2.8) may be written in the form (1.2.2) with real coefficients ak, bk if and only if C  k = C, for all k E Z. Note that whenever we speak of convergence or summability of series (1.2.8) we shall always mean the limit, ordinary or generalized, of the symmetric partial sums (1.2.7). Suppose now that the series (1.2.2) converges uniformly in x to a function f which is necessarily continuous. Then (see Problem 1.2.2) the coefficients ak, b, of (1.2.2) are uniquely determined by the sum f and given by
+
(1.2.9)
A
f ( u ) cos ku du,
bk =
11' f ( u ) sin ku du. A
On the other hand, the last formulae are meaningful for every f E Lin. Thus we may evaluate coefficients a,, b, by (1.2.9) and form the corresponding trigonometric series (1.2.2) for any f E Lin. Trigonometric series which are generated in this way are called Fourier series. The formal definition is given by
Definition 1.2.1. For any f E Liz, the numbers
f"(k)
=
1j' 7r
(1.2.10)
h"(k)= I
7r
f ( u ) cos ku du
A
s'
f ( u ) sin ku du
A
are called the real Fourier coeficients o f f , and the numbers (1.2.1 1)
f"(k) =
1
j' f ( u ) etkudu
(k E a
A
the complex Fourier coeficients o f f . The corresponding trigonometric series (1.2.12) (1.2.13)
f
N
+f:(O)
+
2 {f"(k) cos k x + &''(k)
k=
1
sin kx),
SINGULAR INTEGRALS OF PERIODIC FUNCTIONS
41
are called the real and complen Fourier series off, and denoted by Syl.Finally we write the nth partial sum of the Fourier series o f f as (1.2.14) Sn(f; x ) = +fC(O)
n
n
k=1
k= n
+ 2 {fC(k) cos kx + L''(k) sin kx} = 2
f"(k) eik".
Note that the adjectives 'real' and 'complex' again refer to the trigonometric and exponential functions occurring. Thus real Fourier coefficients are complex numbers if f i s complexvalued. The Fourier series of a function f E ,:L is a trigonometric series but not conversely (see Problem 4.1.3). Therefore we changed the notation of the coefficients and wrote f , ( k ) ,AA(k),f"(k) instead of ak, bk, ck to distinguish between arbitrary coefficients and Fourier coefficients of a function f. Since the Fourier coefficients (1.2.10) and (1.2.1 1) in particular satisfy (1.2.5) and (1.2.6), the same notation Slfl is justified for the real and complex Fourier series of f. A similar remark applies to (1.2.14). Finally, the sign in (1.2.12) or (1.2.13) denotes an equivalence relation and simply means that to each f E ,:L we associate its Fourier series S M . This does not say anything about its convergence or representation for$

Let us now indicate the connections between Fourier series and harmonic functions in order to develop the notion of a conjugate Fourier series. Let z = 6 + iv = re'* be a complex variable, and let f E L;, be a realvalued function (so that the real Fourier coefficients are indeed real numbers andfn(  k ) = f"(k)). Since the Fourier coefficients (real or complex) are bounded by Ilflll, the power series (1.2.15)
F(z) = fA(0)
defines a function F(z)
+2
2 f^(k)rk
k 1
= u(r, x ) + iu(r, x), holomorphic in Iz]
< 1. Thus the real part
and the imaginary part u(r, x) =
(1.2.17)
2 rk{fr(k)sin kx  f,^(k)cos k x )
k=l
of F(z) are conjugate harmonic functions. Formally, u(r, x) tends to the (real) Fourier series off as r 1 . Concerning the conjugate harmonic function u(r, x), its (formal) limit for r +1  yields a series which we may call the conjugate Fourier series off. This series may again be associated with f in the same way as its ordinary Fourier series. f
More specifically we have
Definition 1.2.2. Let (1.2.12) be the real Fourier series of a function f m
(1.2.1 8)
2 {f:(k) k1
E
L;,. Then we call
sin kx  fF(k) cos kx}
the conjugate (or allied) Fourier series off. If we consider (1.2.12) in its complex form (1.2.13), the conjugate Fourier series off takes on the form m
42
APPROXlMATlON BY SINGULAR INTEGRALS
With S yl denoting the Fourier series and Sncf;x ) its nlh partial sum, the corresponding notation for the conjugate Fourier series will be S[nand S;(f; x). Analogously, one may define conjugate trigonometric series (to general trigonometric series (1.2.2) and (1.2.8)). Note that S  " [ f l = S[n + f*(O), the equality only indicating that the (at least formal) series on both sides have the same coefficients.We shall return to this question in Chapter 9.
Dirichlet and FejC Kernel In order to study the convergence of the Fourier series associated with a function f E Liz, we consider its partial sum. According to (1.2.10) we have 1.2.2
Sn(f;
X)
= =
+ " f(u)[l + 2 *
j_"..u)[+
1 2Tr
{COS ku cos kx
k1
k1
cos k(x
I
+ sin ku sin kx}
du
 u ) ] du.
As (see Problem 1.2.5) (1.2.20) D,(x)
= 1+2
n
2 cos kx = k= 1
sin (42)
x = 2j7r
jE&
we obtain (1.2.21)
1 " Sn(A x ) = 5 f,f(x
 u)Dn(u)du
In view of (1.2.20) and Def. 1.1.1, the functions D,(x) define a kernel with parameter p = n E P and po = 00. {D,(x)} is called the Dirichlet kernel, and the corresponding singular integral (1.2.21), i.e., the nth partial sum of the Fourier series ofJ the singular integral of Dirichlet. Thus Prop. 1.1.3 may be applied to the integral of Dirichlet. But it is crucial here that the Dirichlet kernel fails to be an approximate identity (and so Theorem 1.1.5 cannot be applied). This is shown by
Proposition 1.2.3.
If we define L, = 11 D,ll
as the Lebesgue constants (of the Fourier
series), then (1.2.22) Proof. Obviously,
4 L, =  log n na
+ O(1)
(n + a).
SINGULAR INTEGRALS OF PERIODIC FUNCTIONS
Since the function (l/u) 
cot I(
1 ~ I1
is bounded for
du
=
+
742, we have
:lo 7 + 'In
43
sin nu
du
O(1)
But in view of the inequality
valid for 0 5 u
I n/n, it
follows that
c (kr/n) + u = n { l o g n + 0(1)}.
n1
k=l
The desired estimate (1.2.22) may now be obtained immediately. The situation is very different if we replace the partial sums Sn(Jx ) of the Fourier series off by the (first) arithmetic (or Fejkr) means
Making use of the representation (see Problem 1.2.6)
n + 1,
x = 2jn
we have
(1.2.25)
f(x
 u)F,(u) du.
This is Fejkr's singular integral belonging to$ {Fn(x)}is called the FejPr kernel; it is an even, positive, and continuous kernel with parameter p = n and po = 00. Moreover, {Fn(x)}is an approximate identity satisfying (1.1.7) because for any fixed 0 < S < 7r d sSUP l x l r n IFn(x)l
(n
+
1 1 ) sinz (812)'
Thus Theorem 1.1.5 may be applied to give
Corollary 1.2.4. I f f (1.2.26)
E
X2n,then
lim
n m
IImn(f;
o)IIxan
II.n(f;
0)
5 Ilf(Ixan and
 f(o)IIx,,
=
0.
44
APPROXIMATION BY SINGULAR INTEGRALS
1.2.3
Weierstrass Approximation Theorem At this stage we shall give some applications of the results obtained so far. According to (1.2.23) the integral of Fejkr may be rewritten in the form (1.2.27)
.,,(A
X)
=
+f:(O) +
= k= n
(1
(1
k  ){f;(k)
cos kx
k 1
+ far\(k)sin k x )
 m) lkl f"(k)eikx
which shows that it is a polynomial of degree ti. Thus Cor. 1.2.4 contains the celebrated Weierstrass approximation theorem for periodic functions, namely Theorem 1.2.5. The set of all trigonometricpolynomialsforms a dense subset Of Xzn. In particular, i f f E C2n,then, given any e > 0, there exists a natural number n = n(e) and a trigonometricpolynomial tn(x)of degree n such that If ( x )  fn(x)I c e uniformly for all x. Weierstrass' theorem is actually only an existence theorem as it states that there exist polynomials having the desired property. But in the proof of Theorem 1.2.5 via Cor. 1.2.4 we have at the same time constructed a welldefined sequence of polynomials, namely the FejCr means uncf; x ) of the Fourier series off. Proposition 1.2.6. The real trigonometric system is closed in XZn,i.e., i f f E XZn,then
[" f ( u ) cos ku du = 0,
J n
J
f ( u ) sin ku du = 0
(k E P)
R
impliesf ( x ) = 0 (a.e.). The same is truefor the complex trigonometric system. Proof. The hypothesis means that all real Fourier coefficientsf$(k), h"(k) off vanish. Thus, by (1.2.27), .,(A x) = 0, from which ~ ~ f ~ ~=x0, by , n (1.2.26).
E Xzn
As an important consequence we state Corollary 1.2.7. A function f E X,, is uniquely determined by its real (complex) Fourier coeflcients. In other words, i f the real (complex) Fourier coeflcients of two functions of class X2, are equal for all k E P (k E Z), then thesefunctions are equal in Xzn. Thus a trigonometric series cannot be the Fourier series of more than one function in Xan. 1.2.4
Summability of Fourier Series
We already saw that Theorem 1.1.5 does not assure normconvergence of the partial sums of Fourier series off towardsf since the Dirichlet kernel does not constitute an approximateidentity. But this theorem can be applied to the first arithmetic means of a Fourier series since they furnish a singular integral the kernel of which is an approximate identity. This suggests that we place more emphasis upon summability of Fourier series in order to produce normconvergence in some generalized sense. We do not
SINGULAR INTEGRALS OF PERIODIC FUNCTIONS
45
intend to treat summation of general sequences and series here but only consider the problem with respect to Fourier series in the following sense: Let A be a parameter set as in Sec. 1.1 and p E A. {8,(k)} is called a 8factor if, for each p E A, O,(k) is a real function on h satisfying (1.2.28)
e,(o)
e,(k) E 11,
=
1,
e,(k)
=
e,(k).
Note that if A = P, p = n, po = 00, we may arrange the numbers 8,(k) as an infinite matrix {O,(k)} with rowparameter n. For f E 1:" we may form the &means m
(1.2.29)
V,(f;x )
=
+f:(O)
+ 2 B,(k){f,"(k)cos kx + fA(k) sin kx} k=l
of the real Fourier series off. Since 8, E 'I and the Fourier coefficients are all bounded by Ilfl l, the series on the righthand side of (1.2.29) converges absolutely and uniformly in x , and thus defines a function V,(f; x ) E C,, for each p E A. If we begin with the complex form (1.2.13) of the Fourier series, we again arrive at the same V,(S;4,thus m
since 8, is an even function on Z. If we substitute (1.2.10) into (1.2.29), then (1.2.31)
V,(f;
=
j"f(x  u ){1 + 2 2 8,(k)cos 21r 1 "  j f ( x  u)C,(u) du, 21r " OD

k=l
+
=
where we have set m
(1.2.32)
CD(X)= 1
+ 2 k2  1 0,(k) cos kx,
the interchange of integration and summation being justified by the uniform convergence of the series. According to (1.2.28), the functions C,(x) define an even and continuous kernel. The assumption 8,(0) = 1 corresponds to (1.1.1); the weaker condition (1.1.2) would read lim p,oo 0,(0) = 1. The hypothesis that a 0factor and thus the corresponding kernel {C,(x)} is even, reflects the fact that the convergence of a Fourier series in its, e.g., complex form is defined by the convergence of the symmetric partial sums (1.2.14), and it implies that (1.2.29) and (1.2.30) represent the same function for each p E A. If the &means U,(f;x ) of the Fourier series o f f € Liz converge in some sense (pointwise, in norm, etc.) to a limit as p + po, and if this limit coincides with the usual sum of the series in case the Fourier series converges in the ordinary sense, we call {0,(k)} a convergencefactor with respect to the limit notion under consideration. The &factor then defines a summation process; we say the Fourier series is 0summable and call the limit its &sum. As a first result we obtain by Theorem 1.1.5
46
APPROXIMATION BY SINGULAR INTEGRALS
Proposition 1.2.8. Let {BP(k)}satisfy (1.2.28) such that the corresponding kernel {Cp(x)}of (1.2.32)forms an approximate identity. Then for each f E X,, lim II UAf;
(1.2.33)
PtPO
0)
 f(f(0)IIXan
=
0,
i.e., the Fourier series o f f is 0summable to f in X,,norm. As an example we take ( 1.2.34)
Br(k) = rlkl, p = r, A
=
[0, l),
po = 1.
Obviously, {rlkl} satisfies (1.2.28). It is called the AbelPoisson factor. If we form the corresponding means (1.2.29), then we again arrive at (1.2.16). It will be convenient to redefine the function represented by the series (1.2.16) as Pr(f;x), thus m
(1.2.35)
Pr(f; x ) = f.fc^(O)
+ 2 rk{f,^(k)cos k x + f,^(k) sin k x } k= 1
m
=
2
k=  m
rlkyA(k)elkx.
Pr(f;x) are called the AbelPoisson means of the Fourier series off. On account of (1.2.31) we also have
where in view of (1.2.32) and Problem 1.2.18(i) the kernel is given by m
(1.2.37)
p,(x) = 1
 r2 + 2 21 rkcos k x = 1  2r1 cos x + r2' k=
{pr(x)}is called the AbelPoisson kernel, and Pr(f;x) the singular integral of AbelPoisson. In Problem 1.2.18 we have collected some elementary facts concerning the AbelPoisson kernel. {p,(x)}turns out to be even, positive, and continuous. Moreover, it is an approximate identity satisfying (1.1.7). Thus Prop. 1.2.8 yields
Corollary 1.2.9. The Fourier series o f f E X,, is AbelPoisson summable in X,,norm to  f(0)IIXan= 0.
f, i.e2 1imrI IIPv(f; 0)
Let us return for a moment to (1.2.16). We already know that for realvaluedfE Xznthe integral P,(f; x ) is the real part of the function F(z) of (1.2.19, holomorphic in IzI c 1. Therefore P,(f; x ) is a harmonic function in the interior of the unit disc, that is, it satisfies Laplace's equation (1.2.38)
if we take its polarcoordinate form, or (1.2.39)
if we prefer Cartesian coordinates and define f = r cos x , 7 = r sin x. If we start off with an arbitrary function f ( x ) E X2, and assume it to be defined on the unit circle {(r, x ) I r = 1,  n I x I n},then we may interpret Cor. 1.2.9 in the following sense:
SINGULAR INTEGRALS OF PERIODIC FUNCTIONS
47
Proposition 1.2.10. The integral Pr(f ; x) o f f E X,, is a solution of Diriclrlet's problem for the unit disc, the boundary value f being attained in X,,norm. In other words, Pr(f; x ) satisfies Laplace's equation in the interior of the unit disc and takes on the prescribed boundary values in the sense of Cor. 1.2.9. At this stage, the definition of the Dirichlet problem or first boundary value problem of potential theory for the unit disc has, in a sense, only informal character; it will be made more precise later on. In particular, it will be shown that the singular integral of AbelPoisson gives the unique solution of the problem (Sec. 7.1.2).
1.2.5 RowFinite &Factors Let us now discuss 0factors for which the parameter is discrete, i.e., A = P, p = n, po = 00, and for which the functions 0,(k) have compact support on H for each n E P. This means that the corresponding matrix { 0,(k)} is rowfinite. In this case (1.2.29) reduces to the finite sum
where m(n) increases with n on P and is such that 0,(k) = 0 for lkl > m(n). We shall call such 6factors rowJinite. Of course, the first condition of (1.2.28) is now trivially satisfied, and the kernel (1.2.32) is the even trigonometric polynomial m(n)
(1.2.41)
C,(X) = 1
+ 2 2 8,(k) cos kx. k=l
The associated singular integral Un(f; x ) of (1.2.40) then defines an operator which transforms the space Xzninto the set Tmc,)of polynomials of degree m(n) at most. We call such operators (trigonometric) polynomial operators of degree m(n). We have already considered two examples of rowfinite &factors. Thus (1.2.42) (1.2.43)
'
Ikl n give the partial sums of the Fourier series and the FejCr means, respectively. Whereas the Dirichlet factor (1.2.42) corresponds to a kernel which fails to be an approximate identity, the FejPr factor (1.2.43) produces an approximate identity. For further examples of 0factors we refer to the Problems. 1.2.6 Summability of Conjugate Series If we apply a &factor to the conjugate Fourier series (1.2.18) off E XSn,we obtain co
(1.2.44)
V ; ( f i x) =
2 0,(k){f;(k)
k=l
sin kx
 f"(k)
cos kx}.
48
APPROXIMATION BY SINGULAR INTEGRALS
According to (1.2.28), the series on the righthand side converges absolutely and uniformly in x for each p E A and thus defines a function of class C2, which we shall, corresponding to (1.2.29) and Def. 1.2.2, denote by U ; g x). If we prefer the complex form (1.2.19) of the conjugate Fourier series off, we obtain the same functions
U; (f;x )
(1.2.45)
m
=
2
e,(k){  i sgn k}f"(k) elk*.
k m
Proceeding as in the case of ordinary Fourier series it follows that (1.2.46)
where we have set m
(1.2.47)
C;(x)
=
2
2 O,(k) sin kx k 1
(P E A).
Wecall C,"(x)the conjugatefunction of C,(x) and V;(f; x ) the conjugate of the singular integral U,(f; x). Note that the functions C;(x) do not define a kernel since condition (1.1.1) is not satisfied, Indeed, s"" C;(u) du = 0 for every p E A. Thus the summation of conjugate Fourier series possesses special features the discussion of which shall be left to Sec. 9.3.1. In Chapter 9 we shall also clear up the notion "conjugate" and "conjugate of" by giving a precise interpretation to this formal definition. At this stage, let us substitute the factors of Dirichlet, FejCr, and AbelPoisson into (1.2.47), obtaining cos((2n + 1)x/2) 2 sin kx = cot (x/2) sin (42) k=l sin(n + 1)x 2(n + 1) sin2 (x/2)' 2r sin x = 2 2 rk sin kx = 1  2rcosx + r2' n
(1.2.48)
D;(x) = 2
)2 (1.2.49) F ~ ( x=
k= 1
W
(1.2.50)
p;(x)
k=l
Note that the conjugate of the singular integral of Dirichlet is nothing but the nth partial sum of the conjugate Fourier series off, namely (1.2.51) S;Cf;x) = 
" f ( x  u)D;(u) du =
2
k =n
{  i sgn k}f"(k) elkx.
In the AbelPoisson case the integral P;cf; x ) is often denoted by QrVyx), i.e., (1.2.52) which turns out for realvaluedfto be the imaginary part of F(z) of (1.2.15). Since F(z) is holomorphic in IzI < 1, Qr(f; x ) defines a harmonic function in the open unit disc. Moreover, Qr(Ax ) is a conjugate harmonic to the integral P,(f; x ) of AbelPoisson, in the usual sense of the theory of functions of a complex variable.
49
SINGULAR INTEGRALS OF PERIODIC FUNCTIONS
1.2.7
FourierStieltjes Series
We briefly introduce FourierStieltjes series. Definition 1.2.11. For any p E BV, the numbers (1.2.53)
/
1 "
cos ku dp(u), = ,
pF(k) = 
py(k)
/'
=
sin ku dp(u)
A
(k E P)
are called the real FourierStieltjes coefiients of p, and p"(k) = 27r
(1.2.54)
j'
(k E Z)
elkudp(u)
the complex FourierStieltjes coeflcients of p. The corresponding trigonometric series (1.2.55)
dp

+
+p;(O)
5 {py(k)cos k x + py(k) sin kx}, dp  2 p"(k) elkx k
1
m
(1.2.56)
k=m
are called the real and complex FourierStieltjes series of p and denoted by S [dp].The nth partial sum of the FourierStieltjes series of p is indicated by S,,(dp; x). Conjugate FourierStieltjes series are defined in an obvious way as for Fourier series. Concerning the meaning of the symbol N ,the definition of convergence of FourierStieltjes series, etc., the same remarks hold as for Fourier series. Note that the integrals in (1.2.53) and (1.2.54) are RiemannStieltjes integrals. Proceeding just as for Fourier series, we obtain for the nth partial sum of the FourierStieltjes series of p E BV, n
(1.2.57)
2
Sn(dp;x ) =
p"(k) elkx = 
k=n
27r
1'
n
D,(x
 u)dp(u),
{Dn(x)}being Dirichlet's kernel. Thus Sn(dp;x) is a singular integral of type (1.1.12). Moreover, if we sum up FourierStieltjes series by a &factor, we have (in correspondence with the notation in (1.1.12), (1.2.31)) m
(1.2.58)
Uo(dp;X ) =
2
O,(k)p"(k) elkX = 27r
k=  m
1",
C o b  U)+(U),
{C,(X)}being the kernel defined by (1.2.32). Thus Prop. 1.1.7 is applicable to the summation of FourierStieltjes series and gives Proposition 1.2.12. Let {B,(k)} satisfy (1.2.28) such that the corresponding kernel {Co(x)}of (1.2.32) is an approximate identity. I f p E BV, then for every s E C,, (1.2.59)
lim 000
11,
s(x)U,(dp; x ) dx =
For a more detailed discussion of FourierStieltjes series we refer to Sec. 4.3. ~F.A.
50
APPROXIMATION BY SlNGULAR INTEGRALS
Problems 1. (i) (1.2.1) is called the real form of a trigonometric polynomial of degree n. Show that every t, E T n may be written as rn(x) = I&=, Ck elkx, the complex form of a trigonometric polynomial. (ii) Show that if 1, E Tny r, E T,, then tntm E T n t m . (iii) Show that if t, E T , is even, thus a cosinepolynomial, then tn can be represented in the form pn(cos x), where pn(x) is an algebraic polynomial of degree n, and conversely. (iv) If tn E T, is even, show that t,(arc cos x) is an algebraic polynomial of degree n. 2. (i) Show that cos mu sin nu du = 0, n, m = n # O n, m = n # O sin mu sin nu du = cos mu cos nu du = 277, m = n = 0 0, m f n 0, m # n
c,,
{
for m,n E P. Also verify that for k, 1 E Z elku ellu
du =
k = 1 0, k # 1.
(ii) Suppose that the trigonometric series(l.2.2)converges uniformly in x to a functionf. Show that the coefficients are determined by f through (1.2.9). (Hint: Insert the series (1.2.2) into f ( u ) cos ku du, interchange the order of summation and integration, and use (i)) (iii) Suppose that the trigonometric series (1.2.8) converges uniformly in x to a function f. Show that the coefficients ck are determined by f through 2 n q = f ( u ) elk" du. (iv) Suppose that the trigonometric series (1.2.2) converges to a function f ( x ) a.e., and the partial sums (1.2.3) remain bounded. Show that (1.2.9) is valid. (Hint: Lebesgue's dominated convergence theorem ; see also ASPLUNDBUNOART 11, P. 4241) 3. Let f E L:, and c E @. Show that
r,
S,(L x )  c =
f'~ ( +xu ) + j ( x   2c]~,(u)du. 2n id)
0
Thus the Fourier series off converges at point xo to the value c if and only if one has limn+a,J:g(xo, u)Dn(u)du = 0, whereg(x0, u) = f(xo u) f ( x o  u)  2c. 4. Establish the following inequalities: 1  cosx I x2/2 (i) (sinxl I1x1. (x E R), (ii) sin x 2 2x/7rY 1  cos x 2 2xa/na (0 Ix I4 2 ) , (iii) if m, n E M with m I n, then IC&=,,,eikxl I [sin(x/2)1l for all x with x # 2nj, j E Z. (Hint: Use the formula for the nth partial sum of a geometric series: see also Problem 1.2S(i)) (iv) Show that ( r x )  l sinvx 2 (1 + x2)l(1  xa) for all X E R. (Hint: See REDHEFFER [I]) 5. (i) Prove (1.2.20) and (1.2.48). (Hint: Either proceed by mathematical induction or use
+ +
(ii) Show that (Dn(x)l I l/sin(S/2) uniformly for all x of 0 c S I 1x1 I n E
n
and
N.
(iii) Show that the Dirichlet kernel {D,(x)} cannot satisfy (1.1.7). (Hint: Consider the value of D,(x) at x = nk/n for k = 1, 2, , , ,n )
.
51
SINGULAR INTEGRALS OF PERIODIC FUNCTIONS
6. (i) Prove (1.2.24) and (1.2.49). (Hint: See the preceding Problem) (ii) Show that Fn(x) I l/sin (6/2) uniformly for all x of 0 < 6 I1x1 IT and nEN.
+
(iii) Show that (n I)Fn(X) = I ~ ; = o eikxI2. (iv) Show that Fn(x) In + 1 for all x. (Hint: Use that ID,(x)( I2n + 1 for all x ) (v) Show that F,(x) n2n for 1x1 Il/n, n 2 1, and F,(x) I #/(n 1)x2 for I/n < 1x1 I'IT (see Problem 1.2.4(ii)). Thus, setting F,*(x) = 27r2n/(l + nax2),then Fn(x)IF:(x) for 1x1 I'IT and n 2 3. Show that f , F,*(u) du I 2n3 for all n. ! 7. (i) Let ao, al, a2,. . . and bo, b,, b2,.. . be any complex numbers, and define A  l = 0 and Ak = zF=oa, for k 2 0. Show that for 0 5 m In
+
This formula is known as Abel's transformation or formula for partial summation. (ii) Show that Iz;=, Al,etkrl 5 h,lsin (x/2)1' for all x # 2 7 r j , j ~N,provided is real and Am 2 A,,, 2 A, 2 0. (Hint: Use (i) and Problem 1,2,4(iii)) (iii) Let {hk};=O be a sequence of positive real numbers which decrease monotonely to sin kx converge zero as k co. Show that the series zF= I\k cos k x and Z b except possibly at x = 2nj, j E Z. The convergence is indeed uniform in every interval [S, 27r  61, 0 < S < T . (iv) Let {hk} be as in (iii) and set f
m
m
f(X)
=
+ k2= l
(A&)
COS
kx,
&Sin kx.
g(X) = k= 1
As tof, show that i f f € Lin, then the series is the Fourier series off; in other words, for k E P
f ( u ) c o s kudu
=
Ak,
8.
j' f(u) sin ku du
1 'IT
=
0.
n
State and prove similar assertions for g. (Hint: HARDYROGOSINSKI [l, p. 321, HEWITT [l, p. 591; compare also with Sec. 6.3.2, particularly Problems 6.3.2, 6.3.3) (i) Let f be an integrable function over an interval (a, b). Show that ) pu du = 0 = lim lim I Q b f ( u cos P+
o+ m
m
IQb
f(u) sin pu du.
(Hint : Prove the above relations for characteristic functions of bounded intervals, proceed to step functions, and then use the fact that fmay be approximated in the mean arbitrarily closely by step functions, see also ASPLUNDBUNGART [l, p. 4251) (ii) Let f E Lin. Show that limk+mfF(k) = limk+,far(k) = limk+mfn(k)= 0. (Hint: Use (i); see also Prop. 4.1.2) 9. (i) Show that the Fourier series of f~ L:, converges at a given fixed point xo to the value c if and only if for some 0 < 8 < 'IT
This is known as Riemann's principle of localization. (Hint: Setting g(x0, u) as in Problem 1.2.3, then sin ((2n l)u/2) du S,(f; xo)  c = 
+
U
+ However, lim,,,,
1
jbnsin ((2n + 1142) du = Il + I2 + I,.
(Iz + I,)
= 0
by Problem 1.2.W))
APPROXIMATION BY SINGULAR INTEGRALS
52
(ii) Let f e Lin. If there is a point xo and 0 < 8 < x such that f ( x o + u, + f ( x o  u,  2f(xo) du < Q),
so" I
U
I
show that limn4 Saw, xo) = f(xo).This is known as Dini's condition for the convergence of Fourier series. (Hint: Use (i) and Problem 1.2.8(i)) (iii) I f f e L&, is differentiable at xo, show that the Fourier series offat xo converges to f(xo). (Hint: ASPLUNDBUNOART [I, pp. 43M311) 10. (i) Show that there is a constant M > 0 such that I J: (sin u/u) dul I M for all a, b E R. (Hint: Use the second mean value theorem: Iff is integrable and g positive and monotonely increasing on [a, b], then there is a 6 e [a, b] such that g(lrlf0) du = g(b  1J:fWdu) (ii) Letfe Lin and supposethatfis of bounded variation in an interval [xo  6,xo S] around a point xo. Show that the Fourier series off converges at this point xo to the sum V(x0+) f ( x o )]/2. This is Jordan's theorem. (Hint: By Problems 1.2.3, 1.2.9(i) and the Jordan decomposition it suffices to show that for some 0 < 6 < T
s:
+
+
where g(x) is positive and monotonely increasing on [0,6] with g(O+) = 0. To this end, by the second mean value theorem sin ((2n l)u/2) du
+ sin ((2n + 1142) du.
U
U
Now apply (i) and Problem 1.2.8(i)) (iii) Letfe BV1,, be 2xperiodic. Show that the Fourier series offconverges everywhere to the value f ( x ) . This is referred to as Jordan's criterion. (Hint: ASPLUNDBUNGART [l, pp. 4324361) 11. Let f E BVl,, be 2xperiodic and denote the total variation off over [  x , 74 by V. (i) Show that If^(k)l 5 V/(24kI) for every k # 0. (ii) Show that IS,,(f; x)l Il l f l l m Vfor every x and n E N. (Hint: Show that
+
SnW x )
= an(f; x )
+ n+l I c5= ,l / c l f ~ ( ketkx. )
Now use (i) and Cor. 1.2.4, see also ACHIESER[2, p. 991) (iii) Show that the partial sums of the Fourier series off converge boundedly to f, i.e., Sn(f;x ) = f(x) a.e. and there is a constant M such that IS,,(f; x)l I M uniformly for all n E N. 12. Letfbe a continuous function on the finite interval [a, b] and E > 0. Show that there exists a natural number n = n(e) and an algebraic polynomial p,,(x) of degree n such that If(x)  p,(x)I < E, uniformly for all x E [a, bl. (Hint: Reduce by a linear substitution to the case [ .rr/2,~/2],definef outside this interval so as to become a function in Can, apply Theorem 1.2.5, and expand the trigonometric polynomial by Taylor's theorem into a power series; see also HARDYROGOSINSKI [l, p. 221) 13. (i) Let f e Can. Iff*(k) = 0 for Ikl > n, show that f E T,. (ii) Show that if the Fourier series offe Can converges uniformly in x, then it representsf(x) at each x. (Hint: Use Cor. 1.2.7) 14. (i) If {xp(x)}is a kernel, show that Z,,(e'"";x ) = x;(n) etnx. (ii) Let {xp(x)}be a kernel and In a complex trigonometric polynomial with coefficients Ck. Show that zp(tn;X) = 22, , $(k)Ck etkx.
53
SINGULAR INTEGRALS OF PERIODIC FUNCTIONS
(iii) If t, is given as in (ii), show that S,,,(t,; x ) = tn(x)for m 2 n. (iv) If {xp(x)}is an approximate identity, show that lim,,,, xph(k) = 1 for each k E Z. (Hint : Use Theorem 1.1.5) 15. (i) Let f, g E Liz. Show that if one of the factors f or g is a trigonometric polynomial of degree n, so is the convolution product f * g. (ii) Let {x,(x)} be a continuous kernel with parameter n E N. The singular integral In(f;x ) is called polynomial if it maps Xan into T,, thus if In(f;x ) E T, for every f E Xzn. Show that a singular integral is polynomial if and only if x,,E T,. (Hint: If I,,(f; x ) is polynomial, show that x;(k) = 0 for lkl > n ; use Problems 1.2.13, 1.2.14, see also BUTZERNESSELSCHERER [1I) 16. (i) The (first) Cesciro means of a sequence {a,}:= of complex numbers are defined as a, = a, then the sequence {A,,}, A, = (l/(n + 1)) z E = o ak. Show that if also limn,.,, A, = a. (ii) Show that, at a point xo, the Fourier series of a function f E Czneither converges to f(xo) or diverges. (Hint: Use (i) and Cor. 1.2.4) 17. The modified partial sums S:(f; x) of the Fourier series o f f E Ll,, are defined (see ZYGMUND [71, p. SO]) by
+ n%l {LA@) cos kx + K ( k ) sin kx) k=l + +{f$(n) cos nx + LA@) sin nx} = + fA(n) elnx} 2 f A ( k ) + f { f A ( n)
S:(f; x ) = f f A ( 0 ) n1
,Inx
elkx
k s 0Il)
= +tSn(f;X ) + snl(f; x)I. (i) Show that S:(f; x ) = (1/2n) f(x  u)Dx(u) du, where the modlfied Dirichlet kernel {D;(x)} is given through D,*(x) = D,(x)  cos nx. (ii) Show that D,*(x) = cot ( 4 2 ) sin nx for all x # 2nj, j E H. (iii) Show that limn+ IIS,(f; 0 )  S,*cf, o ) I I ~=~ 0~ for every YE Xan. (Hint: Use Problem 1.2.8(ii)) 18. (i) Prove (1.2.37) and (1.2.50). (Hint: Use the fact that 1 2 2;=1 zk = (1 + z)/(l  z ) for IzI < 1 , set z = retx, and consider the real and imaginary part of both sides) (ii) Show that 1  ra 1  ra = ( 1  r)a + 2r(1  cos x )  ( 1  r)a 4r sins (x/2)* Hence, for each 0 I r < 1 , p,(x) is an even, positive, and continuous function of x . In fact, p,(x),p;(x) E C;,, for each 0 I r < 1 . Show that pi(x) S 0 for x E [ O , n ] , thus pr(x)is a monotonely decreasing function of x on [0, n] for each 0 < r < 1. (iii) Show that ( 1  r)/(l + r ) Ip,(x) I (1 + r)/(l  r), thus p,(x) I2/(1  r ) for all x and 0 Ir < 1. Furthermore, p,(x) Ina(l  r)/2rxa for 0 Ix In and 0 < r < 1. Show that the kernel {p,(x)} satisfies (1.1.7). (iv) Show that for all x and 0 5 r < 1 " 1 P cos kx = +log ( 1  2r cos x P).
+
+
2
k1
+
19. Let f E Can be continuously differentiable. Show that
(ii) limr+l P;(f; x ) = f ' ( x ) (i) limn+" uL(f;x) = f'(x), uniformly for all x. (Hint: Use Prop. 1.1.15) 20. Let {O,(k)} be a rowfinite &factor with m(n) = n. Setting B,(n + 1 ) = 0, suppose that, for each n E N, AaOn(k) = O,(k + 2 )  28,(k + 1) + O,(k) 1 0 for all k = 0, 1,. . ., n  1. Show that IIC,lll If + M (n + 1)18,,(n)\, the kernel {C,,(x)] being given by
54
APPROXIMATION BY SINGULAR INTEGRALS
(1.2.41). If &(n) 2 0, one may take the constant M to be zero. (Hint: Apply Abel's [2] and Sec. 6.3.2). transformation twice to C,,(x); see also GERONIMUS 21. For each n E N let t, E T, be such that t,, is positive and does not vanish identically. Suppose that to each E > 0,O c 8 < T there corresponds an no such that
r
*
t,(u) du
0, and that {O,(x, t ) } constitutes a kernel with parameter t > 0, t 0 +. (ii) Show that {e,(x, t ) } is an even, positive, continuous approximate identity. (Hint: As to the positivity of &(x, 1) for each t > 0, either use the representation
n (1  eWzkt)(I + 2 e  ( a k  l ) cos r x + ea(akl)c m
&(x, t ) =
)
k1
(see ERDBLYI[IIII, p. 1771) or use (3.1.36), (5.1.61). As to condition (1.1.6), use Problem 1.3.8, Theorem 1.3.7) (iii) Show that limr,o+ 1 W,V,  f(o)IIxln = 0 for every f E Xan. 0)
1.4 Pointwise Convergence
Up to the present we only considered normconvergence of the singular integral
Z,(A x ) towardsfe X,, as p +po. In trivial cases only does this give information about pointwise convergence. For instance, if X,, = C,, and the kernel {x,(x)} is an approximate identity, then Theorem 1.1.5 implies lirn,,,, Z,(A x) = f ( x ) uniformly and hence pointwise. A slight generalization is Proposition 1.4.1. Let f E X,, and the kernel {x,(x)} of the integral Z,cf; x) be an approximate identity satisfying (1.1.7). (a) At every point xo of continuity off, lirn,,,, Z,(A xo) = f(xo). (b) Iff is continuous on (a  q, b q) for some 7 > 0, a < b, a, b E R, then lirn,,,, Z,(A x ) = f ( x ) uniformly on [a,b ] . (c) If the kernel {x,(x)} is, in addition, even, and xo is such that lirn,,,, [f ( x o + h) + f ( x o  h)] = 2c exists, then lirn,,,, Z,(A x,) = c.
+
The proof is essentially similar to that of Theorem 1.1.5 and left to Problem 1.4.1. Unfortunately, Prop. 1.4.1 only covers the situation when the point of convergence is a point of continuity of the function f or a point at which the onesided limits off exist. It thus applies to continuous functions or to those of bounded variation. This by no means solves the problem for Lebesgue spaces Lq,. In fact, we are then interested in
62
APPROXIMATION BY SINGULAR INTEGRALS
theorems which assert pointwise convergence almost everywhere of ID(Ax) towards f as p f po, a much more reasonable question since alteringfon a set of measure zero does not alter the singular integral. In this respect we have
Proposition 1.4.2. Let f E Liz and the kernel {xo(x)}of the integral ID(Ax) be an even, absolutely continuous approximate identity satisfying (1.1.7)and
(1.4.1) Then at each point x for which
(1.4.2)
JOh
+ u) + f ( x  u)  2f(x)]
[f ( x
du = o(h)
(htO+),
thus for almost all x, we have
(1.4.3)
lim
ID(Ax)
=f(x).
D+Po
Proof. Since xp is even, we have by (1.1.1) for any 0 < 8 < r
say. Setting
L~(x+ r )
(1.4.5)
+ f(x  t )  2f(~)]dr,
then, according to (1.4.2),to each E > 0 there exists a 8 > 0 such that IG(u)l s EU for all 0 < u 5 8. We now fix this 6 and estimate Iland la,respectively. By integration by parts we obtain
(1.4.6)
2~11= C(8)x0(8)
Sd 0
G(u)xXu) du,
and therefore by (1.1.5)and (1.4.1)
since one further integration by parts gives
Regarding la,we have
(1.4.7)
Val
5
IXP(U)l
which in view of (1.1.7)tends to zero as p almost all x satisfy (1.4.2).
(2
llflll + If(x)l)
po. This proves (1.4.3)since by Prop. 0.3.1
+
It is useful to observe that if xL(x) is of constant sign on (0,T ) and if X ~ ( Tis) bounded for p E A, then, in view of
is a consequence of (1.1 S). For example, by Problem 1.2.18 the Abelcondition (1.4.1) Poisson kernel {pl(x)} is absolutely continuous such that p:(x) I0 on [ O , r ] and
63
SINGULAR INTEGRALS OF PERIODIC FUNCTIONS
p,(n) = (1  r)/(1 + r) for each 0 < r < 1. Since this kernel forms an even approximate identity satisfying (1.1.7) we obtain for the singular integral Pr(f; x) of AbelPoisson Corollary 1.4.3. Iff
E
Lg,, 1
_
0. Consequently, Lip (X,,; a)Mdenotes the set of those f E Lip (X,,; a) for which w(X,,;f; 8) I MS" for all S > 0. Definition 1.5.6. A functionf E X,, satisfies a generalized Lipschitz condition of order a, a > 0, in notation f~ Lip* (Xa,; a), if w*(X,,;f; S) = O(6") as S + O+. I f w*(X,,;f; 8) = o(Sa), we write f E lip* (X,,; a). For r E N, a > 0 the class *Wktn is defined as the set of thosefE WL,, for which +(r) E Lip* (X,,; a), where E AC& with +(I) E X,, is such that +(x) = f ( x ) (a.e.).
+
Without loss of generality we may restrict the order a in Def. 1.5.5 to 0 < a I1 and in Def. 1.5.6 to 0 < a I 2, respectively, since f~ lip (X,,; 1) as well as f E lip* (Xzn;2) implies that f is a constant. This is Lemma 1.5.2(iv) and 1.5.4(iv), respectively. Moreover, it immediately follows from the definitions that f E Lip (X,,; a ) implies f E Lip* (X,,; a) for 0 < a I 1. On the other hand, also the converse of the latter assertion is valid for 0 < a < 1. This is the result of Theorem 2.4.2. Thus the classes of functionsf E X,, which satisfy a Lipschitz or a generalized Lipschitz condition of order a are equivalent for 0 < a < 1. But iffe Lip* (Xzn;l), then it is not necessarily true thatfE Lip (X,,; 1). This follows by the example of Problem l.S.l(iii). Again it is an immediate consequence of the definition that for f E Lip* (X,,; a) there exists a constant M* such that w*(Xa,,f; 6) IM*S" for all 6 > 0. In this case we writefE Lip* (Xa,; &.,thus emphasizing the dependence upon the constant M*. Obviously, f E Lip (X,,; 0 1 ) ~implies f E Lip* (Xa,; a ) a M . 1.5.2 Direct Approximation Theorems
We shall now return to the problem formulated in the beginning of this section. At first we shall consider singular integrals with (real) kernels which are even and positive. Moreover, it seems natural to assume that the integrals in question converge to f i n X,,norm, i.e.,
Since by Prop. 1.3.10 the convergence (1.51) holds if and only if lim (1  $(I)) = 0, (1.5.2) 000
69
SINGULAR INTEGRALS OF PERIODIC FUNCTIONS
our considerations may be based on (1.5.2). Thus we expect that the discussion of the rate of convergence of ( 1 3.1) will be reduced, more or less, to a discussion of the rate of convergence of (1 5 2 ) . In this section we shall only prove some direct approximation theorems, i.e., theorems which assert a certain order of magnitude of the approximation (1.5.1) provided the function f to be approximated satisfies certain structural properties. Before proceeding to a basic lemma, we note that, since the kernel {x,,(x)}is even, the complex Fourier coefficients x;(k) of (1.2.11) are real. In fact,
1"xp(u)
x:(k) = 1
(1.5.3)
T
o
cos ku du
(k E
a,
and since the kernel is positive,
Ix
(1 5 4 )
Moreover, x;(O) = 1 for all
pE
1 " m I s 1 2rr
x,(u)du
=
( k E H, p E A).
1
A by (1.1.1).
Lemma 1.5.7. Let {x,,(x)}be an even andpositive kernel. Then
( 1 S.6)
1"
= o
u'xp(u)du I
5
(1  x ; ( l ) ) L J4 
1X
X)
( j = 3,4).
Proof. To prove (1.5.5) f o r j = 2, we have by Problem 1.2.4(ii) and (1.5.3)
F o r j = 1 the last result implies in view of Holder's inequality
The proof of (1 5 6 ) runs along the same lines and is left to the reader. Theorem 1.5.8. Let f and positive. Then (1 57)
E
Xan,and the kernel {x,,(x)} ofthe singular integral I,,(fi x) be even
IIIAfi
0)
f(o)llx*, = o ( w * ~ X z " ; f id 1  X ; G )
Proof. Since the kernel is even, we have
and hence by the H6lderMinkowski inequality
1)
(P
Po).
+
70
APPROXIMATION BY SINGULAR INTEGRALS
Therefore it follows by Lemmata 1.5.4@), 1.5.7 that for each h > 0
This implies (1.5.7) by setting h = (1  ~;(l))"~. Let us apply the last result to the singular integral J,,(f; x) of Jackson as defined in Problem 1.3.9. Herej;(l) = 1  (3/(2na l)), and therefore
+
1 jc(l) = O(na)
(1.5.10)
(n + 00).
Corollary 1.5.9. I f f € Xan, then
11 Jdf;
0)
f
(0)
I x,,
= O(w*Ohn if; n '1)
(n + 00).
Thus, i f f E Lip* (Xan; a) for some 0 c a I 2, then
(1.5.11)
I Uf; f I x,, 0)
(0)
= O(n  9
(n + 03).
Thus, for the singular integral of Jackson the best possible order of approximation at this stage is O ( ~ I  ~ more ); refined properties upon f such as f " E Lip* (Xa,; a) do not imply better approximation. This depends neither on our methods of proof nor on the particular example, but turns out to be characteristic for positive polynomial operators. We shall return to this question in Sec. 1.7. Method of Test Functions It was shown in Prop. 1.3.9 that in case of positive singular integrals the test conditions (1.3.17) imply the convergence (1.3.7) for every f E Xan.Now the question naturally arises whether it is possible to strengthen (1.3.7) to a result concerning an order of approximation if the test conditions (1.3.17) include an order of approximation. The following proposition gives an affirmative answer.
1.5.3
Proposition 1.5.10. Let the kernel {xp(x)} of the singular integral I,cf, x) be positive. If at some point xo
Zp(cosu ; xo) = cos xo  &(xo),
lim &(x0) = 0,
PPO
(1.5.12) Zp(sinu ; xo) = sin x,,
 yp(xo),
lim yp(xo) = 0,
PPO
71
SINGULAR INTEGRALS OF PERIODIC FUNCTIONS
Proof. As in the proof of Theorem 1.5.8 we obtain, for any A > 0,
u2xxp(u) du)lla]. But as in the proof of Lemma 1.5.7
In
I!
du u2xxp(u)
4
I
(1  cos (xo  u)),yp(xo u) du
=n a {1 
cos XJ,(COS
u ; x,)
 sin xoZD(sinu ; xo))
2
II
0)
 f(0) IIx,,
fi
IW ( x a n ;
 1)
[1 +
An
d ~ , ( x o )cos xo
+ yp(xo) sin xo]
9
+
which implies (1.5.13) by taking A' = d/B,(xo) cos xo y,,(xo) sin xo. As an example we consider the singular integral (1.2.25) of FejCr. Here we have 1 o,(cos u ; x) = cos x  + cos x ,
a,(sin u ; x )
=
1 sin x  sin x. +
Hence (1.5.13) gives for every f E X,, IIudf;
0)
f(~)lIx~,,= O(W(Xan;f; nl'a))
(n + m);
see also Problem 1.5.10. If the kernel {x,(x)} satisfying the assumptions of Prop. 1.5.10 is also even, then by (1.5.3) I,(cos u; x) = cos x x;(l) = cos x  (1  X;(l)) cos x , (1.5.14) I,(sin u ; x) = sin x x;(l) = sin x  (1  x;(l)) sin x , and therefore (1.5.13) implies
IIlAA f(O)IIx,, 0)
=
O(~(X2n;fid 1
 X;(~I))
(P
+
Po);
thus we essentially obtain again (1.5.7) (see also Problem 1.5.8). There is an interesting interpretation of Theorem 1.5.8 in connection with the method of test functions. By (1.5.11) we know that there are singular integrals Z,(A x ) such that for everyfg Lip* (Xan; a), 0 < a I2, we have ( 1.5.15)
IIUA  f(O)IJXzn= O ( p  9 0)
(P
f
P 0).
I
72
APPROXIMATION BY SINGULAR INTEGRALS
The problem that arises is to characterize such integrals. In this connection the method of test functions calls for a set of functions and for a condition such that if every function of the set satisfies the condition, then (1.5.15) holds for every f E Lip* (X2n;a). Many important results may be interpreted in this way. Here we have Proposition 1.5.11. Let the kernel {xp(x)}of the singular integral 10(Ax ) be even and positive. r f
1
( 1.5.16)
 X;(l)
=
(P
O(p2)
+
Po),
then (1.515) holds for every f E Lip* (X2,; a), 0 < a I 2. In other words, the test set for the assertion that f E Lip* (X,,; a ) implies (1.5.15) consists of the function elx only, and the test condition is (1.5.1 7)
IIIO(eiu;  erollxan= O(P2)
(P
0)
+. Po)*
The proof follows immediately from (1.5.7) and the fact that (1.5.18)
llIP(eiu;
0)
 elO1lXan = 1  x;(l).
We mention another important interpretation of Theorem 1.5.8. A sequence of bounded linear operators Tncf;x) which are polynomial of degree n such that for every f E X,, (1.5.19)
IITdf;
0)
 f(0)llXDn
=
(n
0(~*0(2,;A nl))
f
00)
is called a Zygmund approximation sequence. We have Proposition 1.5.12. Let the kernel (xn(x)}of the singular integral In(Ax) be ecen, continuous, and positive, and suppose that In(Ax ) is a polynomial operator of degree n. A necessary and suficient condition that {I,,(Ax)} be a Zygmund approximation sequence is that 1  xC(1) = O(na)
(I. 5.20)
(n
f
a).
Proof. We first of all note that in view of Problem 1.2.15 xn(x)is an even and positive polynomial of degree n. According to (1.5.18), the necessity of (1.5.20) follows immediately since eix E Lip* (X,,; 2). On the other hand, (1.5.7) implies the sufficiency. For an example of a Zygmund approximation sequence we refer to Sec. 1.6.1. 1.5.4 Asymptotic Properties
We begin with a general theorem for positive singular integrals. Theorem 1.5.13. Let the kernel {xO(x)} of the singular integral IpV, x ) be positive and [(x)a function such that (1.5.21)
[(x)
E Can,
[(xJ
=
0, [(x)> 0, x #
xO,
Suppose that for h E L,", the limit (1 5 2 2 )
lim h(x)/{(x)= L X'XO
T
I
X , XO
I T.
SINGULAR INTEGRALS OF PERIODIC FUNCTIONS
73
exists and isfinite. Then (1.5.22) implies (1 5 2 3 )
lim l0(h;xo)/Io(5;xo) = L
0PO
if and only fi (1.5.24)
lim ao(8)/I0(5;xo) = 0 000
holds for each 0 < 6 I
rr,
where
(1.5.25) Proof. Necessity. We consider the particular function h(x) = 5"~). Then, according
to (1.5.21), lirn h(x)/l;(x)= lim [(x) = 0. xxo
xxo
Moreover, for this function
J
1
l0(h;xg) 2 2rr
5"XO
 U>Xo(U,d~2
~sa0(8),
dslulsn
where ms
E
inf,.
lul 5 x
5z(xo u ) > 0. It follows that Io(h;x0)/10(5;xo) 1 m , [ ~ , ( ~ ) / 1 0XO)l, (5;
and thus ( I 5 2 3 ) implies (1.5.24). Sufficiency. According to (1.5.22), given any E > 0 there is a 6 > 0 such that Ih(xo  u)  L5(xo  u)l < e[(xo  u ) for all IuI I 6. Since Zo(h ; xo)  LIp(5; xo) =
!2rr (
1+ 1
lul sd
)[h(xo u )
 L5(xo  u)]xD(u)du = 11 + ZZ,
d s IuI sn
say, we therefore obtain
Illl
2 .1,(5; xo),
1121
5 IIh(0)  L 5 ( o ) I I
mao(S),
which establishes (1 5 2 3 ) . Before proceeding to the next result we give the following
Definition 1.5.14. Let the function f be defined in a neighbourhood of the point xo. If lim,,,, h'[f(xo + h) + f ( x o  h)  2f(xo)]exists and isfinite, this limit is called the second Riemann derivative o f f at x,: in notation: f["(x0). For the general definition of a Riemann derivative as well as other generalized derivatives we refer to Sec. 5.1.4 (see also Problem 1.5.15).
74
APPROXIMATION BY SINGULAR INTEGRALS
Theorem 1.5.15. Let the kernel {xO(x)} of the singular integral Z,,cf; x ) be even and positive. Zfthe functionsf, g E L,“, possess a second Riemann derivative at xo, then (provided the denumerators are direrent from zero) (1.5.26) $and only if (1.5.27)
Proof. Necessity. If we set gk(X) = 1  cos kx, k zp(8k;
0) = 5
=
1,2, then according to (1.5.3)
jn (1  cos ku)xp(u) du J
= 1  x;(k).
Therefore, if (1.5.26) holds, we have
Sufficiency. If we set 2h(u) = f ( x o
+ u ) + f ( x o  u )  2f(x0), then by (1.5.8)
ZP(A xo)  f ( x o ) = 2;;
1‘ n
h(u)Xp(u)du = ZP(h;0).
Since
1 = lim f ( X 0 lim uog,(u) u + o
+ u ) + f ( x o  u )  2f(xo) = f [ 2 ] ( x o ) , 2(1  cos u )
and since [ ( x ) = gl(x) obviously satisfies (1.5.21) for xo = 0, we obtain in virtue of Theorem 1.5.13 that
if and only if for each 0 c 6 < 7r
But this is true since by (1.5.27) 1
27r(l
1
 cos 6)s d s l u l s n (1  cos U ) ~ X ~du( U )
1  2x;;(1) < (1  cos l 6)s {
Thus the proof is complete.
+ 2x;(2)} + 
75
SINGULAR INTEGRALS OF PERIODIC FUNCTIONS
We remark that every singular integral which satisfies the assumptions of Theorem 1.5.15 admits the asymptotic expansion
(I. 5.28)
ZO(A
xo)

f(X0)
= (1
 x;(~)~f[”~xo~ + o(l  x;(l))
(P
+ Po).
To give a first example we consider the singular integral W,(f; x) of Weierstrass introduced in Problem 1.3.10. Since [S& ; t)]^ = exp {  tk’}, condition (1 527) is satisfied, and we obtain
Corollary 1.5.16. I f f E L,“, has an ordinary second derivative at xo,then for the integral of Weierstrass (1 5.29)
W U xo)  f(X0)
= tf”(X0)
+ o(t)
(t + O+).
For further examples we refer to Problem 1.5.11. Unfortunately, there are very important singular integrals which do not possess the property (1.5.27). For instance we have for the FejCr kernel {Fn(x)}of (1.2.24)
Ft(1)
=
1
1 n+l
3
FQ(2)
=
1
1  F;(2) lim I  ~ ; ( l ) = 2s
2 n + 1’
hm
and for the AbelPoisson kernel {p,(x)} of (I .2.37)
Thus Theorem 1.5.15 does not apply. But in comparison with Theorem 1.5.13 this only means that for those particular singular integrals [ ( x ) = gl(x) = 2 sin2 (x/2) is not a suitable choice for the function [(x). As is shown in Problem 1.5.12, [(x) = lsin (x/2)1is now a suitable one. Finally we mention that we may also apply the method of test functions in order to obtain general asymptotic expansions for positive singular integrals. Indeed, we have in continuation of Prop, 1.3.9 and 1.5.10
Proposition 1.5.17. Let the kernel {xo(x)} of the singular integral If at some point xo the representations ZO(C0S u ; xo) = cos xo
row,x ) be positive.
+ rIl(X0) V(P) + O(V(P)), lim ~ ( p = ) 0,
(1.5.30)
0PO
Zp(sin u ; xo) = sin xo
+
172(xo)
V(P)
+ O(P)(P)),
hold, then for every f E C,, having a second derivative at xo
(1.5.31) lim 0PO
[&(Axo)  f(XO)l/V(P)
= {12(xo) cos xo
 rIdx0) sin XOlf’(X0)
 {rII(XO) cos xo + rl,(XO) provided for each 0 < 8 < s (1.5.32)
U
lim (l/p(p)) 0PO
sina xp(u)du dL IUI SJZ
=
0.
sin ~OlfYXO),
76
APPROXIMATION BY SINGULAR INTEGRALS
For the proof we refer t o Problem 1.5.13. If the kernel {xp(x)} is also even, then  ,y;(l))I, ~ ~ (= x )cosx, ~,(x) = sinx as follows by (1.5.14), and (1.5.32) reduces to (1.5.27). This again gives the expansion (1.5.28).
v(p) = (1
Problems 1. (i) Prove Lemma 1.5.2. (Hint: As to (ii) show that for any S1, 8, > 0 wo(zn;f; 6 1 8,) 5 wO(2,;f; 6,) w 0 L ; f; 821, and thus w(X,,;f; n6) Inw(X2,;f; 6) for any n E N. Now choose n E P such that n IX c n 1; see also NATANSON [81, p. 75 ff 1%TIMAN[2, p. 96 ff]. As to (iv) use (ii) to show that w(X,,;f; 6,)/6, I2w(X2,;f; S1)/6, for any 6, < S2, and thus limdto w(Xzn;f; 6)/6 > 0 unless f is a constant (a.e.)) (ii) Prove Lemma 1.5.4. (iii) Show that the function f(x) = sin x log [sin XI belongs to the class Lip* (Xz,; 1) but not to the class Lip (X,,; 1). 2. Let f be defined in a neighbourhood of a point x E R. For (sufficiently small) h E R the first onesided difference off at x with respect to increment h is defined by A; f(x) = f(x h)  f(x), and the higher differences by A:f(x) = AiALl f(x), r E N. (i) Show (by induction) that
+
+
+
+
A:f(x)
=
$ ( l)’k(L)f(x + kh).
IC=0
(ii) Iff has a bounded rth derivative in a neighbourhood of x , then (compare with Problem 1.1.7)
(iii) Show (by induction) that for n E N n1
n1
2
A;hf(x) =
ki0
* * a
2
A&f(x
kr=O
3. For f E Xznthe rth modulus of continuity (smoothness) is defined for 6 2 0 by w r 0 L ; f ; 8) = llALf(~>lIx~..
,:yzd
Show that the following properties hold for w,(X,,;f; 6): (i) wr(X,,;f; S) is a monotonely increasing function of 6, 6 2 0. (ii) w,(X,,;f; 6) 5 2’’w,(Xz,;f; 6) for any j E N with j < r. In particular, one has lim,,,, wr(X,n;f; 6) = 0. (iii) wr(Xan;f; ha) 5 (1 X)*w,(X,,;f; 6) for any X > 0. (Hint: Use Problem 1.5.2(iii) to show that w,(X,,;f; n6) Inrw,(X2,;f; 6) for any n E N) (iv) Iff E WL1, and E AC;il with +(I) E X,, is such that +(x) = f(x) (a.e.), then wjtr(X2n;f; 6) 5 6‘o~,(X~,~; #,); 6) for any j E N. In particular, i f f E Wi,,, then w(Xzn;f; 6) I6 Ild’llxn,. (Hint: Use Problem 1.5.2(ii)) (v) w,(X,,;f; S,)/& I 2‘w,(XZ,;f; S,)/Si for any 6, < 6., Thus, unless f is constant (a.e.) lim,,,, w,(Xzn;f ; S)/8 > 0. For the proofs of these and further properties of w,(X,,;f; 6) see also TIMAN[2, p. 102 ff].
+
+
4. For a > 0 the Lipschitz class Lip, (X2,;a) is defined as the set of those functions f E X2, for which wr(X,,;f; 6) = O(6“) as 6 + O + . The class Lip, (X2n;a)M,consists of those J”E Lip, (XZ,; a) for which wr(X2,;f; 6) 5 M,6“ for all 6 > 0, M , being a
given constant.
77
SINGULAR INTEGRALS OF PERIODIC FUNCTIONS
(i) Show that if f E Lip, (XZ,; a) for some j E N, then f~ Lip, (X,,; a) for every r E N withj < r. (ii) I f f € Lip, (Xzn;a),then there exists a constant M , such that w,(X,,;f; 6) IM,6“ for all 6 > 0. (iii) L e t f c X2,. g E La,. Show that w,(XZn;f* g; 6) I llglllwr(Xz,; f;6). In particular, f E Lip, (XZ,; a) implies f * g e Lip, (Xzn;a). 5. Show that for the second integral means A%(f;x) o f f € X,, IIAW;
0)
 f(o)IIx,,
(h > 0).
5 w*o(zn;f; I?)
+ +
(Hint: Use the fact that AE(f; x) = h  2 J: Lf(x u) f ( x  u)](h  u) d u ) 6. LetfE X,,. Show that a necessary and sufficient condition forfto belong to Lip (Xzn;a) is that the Fejer means o,(f; x) belong to Lip (X,,; a),uniformly for n E N. Extend to general singular integrals. (Hint: As to the necessity use Ilu,(f; o)llxan IIlfllxa,, for the sufficiency Cor. 1.4.7 and Fatou’s lemma) 7. Let f~ X,, and the kernel {x,,(x)} of the singular integral I D ( f ;x) be even and positive. (i) I f f € ACzn withf’ E XZn,show that
llw;  f(o)Ilxan
=
0)
 X;(l)w~Xzn;f’;
I f f E AC:, with f” E X,,, then llID(f;  f ( o ) I l X a , , (1.5.7) and Problem 1.5.3(iv)) (ii) I f f € ACznwith f’ E XZn.show that for each X > 0 0)
d1  x 3 ) ) ) (P ’Po). = 0(1  ,y;(l)). (Hint: Use
(Hint: Use (1.5.8) and partial integration, and then proceed as in the proof of Theorem 1.5.8) (iii) I f f € AC:, withf”
E
Xan,show that
8. Let f~ Xznand the kernel {x,,(x)} of the singular integral Show that for each A 7 0
ID(Lx ) be even and positive.
Thus in particular llf, , ( f;  f ( o ) / l x a , = O(w(X,,;f; d1  xO(1))). If even, $(1) is to be replaced by Re(X;(l)) = [xp131)/2. 9. (i) Show that for the integral means Ah V , x) o f f € X2, one has
{x&)} is not
‘1  f ( o ) I I X a ,
(h+O+).
0)
IIAh(f;
=
o(w*(xZn;f;
h))
In particular, f E Lip* (X,,; a) implies IIAh(f; 0)  f ( o ) I I x a n = O(h“). Formulate and prove counterparts for the rth integral means A;(f; x). (ii) Show that for the singular integral W,(f;x) of Weierstrass off E Xznone has
I1 wt~f; f(0)IIxan 0)
= o(w*(XZn;fi
d;))
( t +O+).
In particular, f~ Lip* (X,,; a) implies (1 W t ( f ;0)  f(o)Ilxan = O ( P ) . 10. (i) Show that for the singular integral o,V, x ) of Fejkr off E X,, one has Ilon(f;
0)
 f(O)IIxan
(n + a).
= O(w*(X~n;f;n”’))
In particular, f E Lip* (X,,; a) implies llu,,(f;0)  f ( O ) i l x , ,
= 0(na’2).
78
APPROXIMATION BY SINGULAR INTEGRALS
(ii) Show that for the singular integral Pr(f;x) of AbelPoisson off
E
Xan one has (r +. 1 ).
In particular,fE Lip* (Xzn; a) implies IIPr(f; 0 )  f(o)Ilx2. = O((1  r Y a ) . (Hint: Use Theorem 1.5.8; for improvements of the latter estimates see Cor. 1.6.5, Theorem 2.5.2) 11. Let f E L& have a second Riemann derivative at xo. Show that (n a), (i) Jn(f; X o )  f(Xo) = (3/2)nafra1(X~) o(n’) (h+O+), (ii) Ah(f; xo)  f(xo) = (1/24)haff[a1(x~) o(h7 Jn(f;x ) and Ah(f;x) being the singular integrals of Jackson and RiemannLebesgue, respectively. (Hint: Use Theorem 1.5.15, thus (1.5.28); to evaluate the second Fourier coefficient of the kernel, in the Jackson case see PETROV [l], MATSUOKA [3], for the integral means see Problem 1.2.22) 12. Let f L L have a righthand derivativef+(xo) and a lefthand derivativefL(xo) at XO. Show that (i) lim W o g n)[un(f; xo)  f(xo)l = ( 1 / a ) K (xo)  fr.(xo)l,

+ +
n
m
(ii) lim ((1 rl
 r ) ]log (1  r)I)’[Z‘rM
xo)
 f(x0)l
=
(l/a)Lf:(xo)  f(xo)l.
(Hint: Use Theorem 1.5.13 with [(x) = lsin (x/2)1, see also NIKOLSKI~ [3], NATANSON [81, p. 1631, KOROVKIN [5, p. 1161 for (i) and MAMEDOV [9, p. 941 for (ii); f : ( x o ) is defined through f:(xo) = limh,o+ Lf(xo h)  f ( x O ) ] / h . For further asymptotic expansions see Cor. 9.2.9) 13. (i) Let f E C,, have an ordinary second derivative at xo. Show that f(x) = f(xo) f’(xo) sin (x  xo) 2f”(x0) sina ((x  x0)/2) q(x) sina ((x  x0)/2), where 71 is bounded with lim,~,, ~ ( x = ) 0. (Hint: Use L’Hospital’s rule, see also NATANSON [81, p. 2121) (ii) Prove Prop. 1.5.17. (Hint: Insert the expansion of (i) into Z,(f; xo)  f(xo),see also MAMEDOV [2]) 14. Let the kernel {xD(x)}of the singular integral Z,(f; x ) be even and positive and suppose rn(x,; 1) = 0, where the ath (absolute) moment mk,;a) of ,yo is defined that lim,,,, through (1.6.9). (i) For f e X,, show that \lZ, Lf; 0)  f ( o ) I I X 2 , I2w(X2,;f; rn(xD; 1)). (Hint: See also Prop. 1.6.3) (ii) For f E Xi, show that IIu,(f; 0)  f(o)llxan S 2w(Xan;f; a(1 2 log n)/4n). (Hint: Use Lemma 1.6.4) (iii) Suppose that xD(u)du = o(rn(.yD; 2)), p po, for each 0 c 6 < T.I f f € Can has an ordinary second derivative at xo, show that ZAf; X O )  f(x01 = (m(.y,,;2)/21f”(x0) + o(rn(x, ;2)) (p Po). Apply this t o the singular integral of Jackson so as to obtain again the assertion of Problem 1.5.1 l(i) (withfLal replaced byf”). (Hint: Use the expansion
+
+
+
+
+
fi
+
+

f(x) = f b o ) + f’(xo)(x  xo) + f”(xo)(x  xo)a/2 + q(x)(x XoY, where q is bounded with limz,zo q(x) = 0, cf. Problem 1,5.13(i); see also NATANSON [41) 15. (i) Let the function f be defined in a neighbourhood of the point xo. If lim h’lf(xo
h0
+ (h/2))  f ( x o  (h/2))1
exists and is finite, this limit is called the first Riemann derivative off a t xo; in notation: f[l1(xo). Show that if the ordinary first derivative f’(xo) exists, so does f[ll(x~)and f [ l l ( x o ) = ~ ( x o ) .
79
SINGULAR INTEGRALS OF PERIODIC FUNCTIONS
(ii) Show that if the ordinary second derivative f”(xo) exists, so does f[21(xo),and f[”’(xo) = f”(x0). 16. Let f E L,”,, and the kernel {xp(x)}of the singular integral I J f ; x) be even and positive. (i) Suppose (see (1.6.9)) that lim,,,, m(x,; 1) = 0 and J: uxp(u) du = o(m(x,,; l)), po, for each 0 < S < 7r. Show that if the onesided derivativesf;(xo), fl(xo) p exist at xo (cf. Problem 1.5.12), then 44fi XO)  ~ ( x o=) (m(xo ; 1)/2)Lf; (xo)  K(x0)l + o ( m b ; 1)) (P PO). (Hint: Use (1.5.8) and the expansion f ( X 0 + u) + f(x0  u)  2f(xo) = Lf;(xo)  f:(xo)lu + q(u)u, [l]) where q is bounded with lim,,o+ q(u) = 0; see also TABERSKI (ii) Establish again the assertions of Problem 1.5.12. (Hint: Compare also with Lemmata 1.6.4, 2.5.1) (iii) Suppose that lim,,,, rn(xp;2) = 0 and J: u2xxp(u)du = o(m(Xp;2)), p + po, for each 0 < 6 < rr. Show that if the second Riemann derivativef[21(xo)exists at xo, then lim IUf, xo)  f(xo)l/m(x,; 2) = fCz1(x0>/2.

,’Po
17. Let the kernel {x,(x)} of the singular integral &,(Ax) be even and positive and let m E N. Then for each f E X,, with period 27rlm (Hint: See also DE VORE[2])
1.6 Further Direct Approximation Theorems, Nikolskii Constants
We continue the investigations of the preceding section in order to consider direct x). First we shall introduce the approximation theorems for the singular integral lp(fi singular integral of FejCrKorovkin and apply the results so far obtained. In discussing the integral of FejCr, it will be observed that the corresponding results are unsatisfactory and have to be completed by further direct theorems. The section concludes with a few selected results concerning best constants in asymptotic expansions for Lipschitz classes.
1.6.1 Singular Integral of FejkrKorovkin The singular integral of FejkrKorovkin is defined for f E Xanby
(1.6.1)
Kn(f; x ) =
1 1’f(x  u)k,(u) du 27r n
with kernel (kn(x)}given by (jE Z)
(1.6.2) kn(x) =
2 sin2 (7r/(n + 2)) cos ((n + 2)x/2) a ,x# [cos (7r/(n + 2))  cos x n+2
+
+ 2112
*
1
(n
,x=
7r +
7r +
+2h + 2371
80
APPROXIMATION BY SINGULAR INTEGRALS
and discrete parameter n E P, n + to. For each n E P, k,(x) is an even, positive trigonometric polynomial of degree n which may be represented as n
( I .6.3)
k,(x)
=
1
+2 C
k=l
&(k) cos kx
with &factor given for 1 5 k 5 n by (cf. Problem 1.6.5) (1.6.4) e,(k) 
1
2(n
+ 2) sin (n/(n + 2))
(n  k
+ l n  (n  k + + 3) sin kn+ 2
Thus kf(1) 3 e,(l) = cos (n/(n + 2)), and we note that in view of Problem 1.6.4 this value of the first coefficient already determines the kernel (1.6.2) uniquely. Since {k,(x)} is a positive kernel and limntmkZ(1) = 1, it follows by Prop. 1.3.10 that KnW, x) converges in X,,norm toffor everyfE Xzn. In order to apply the results of the last section, we observe that 1  kf(1) = O(n,) for n +to. Thus the integral of FejtrKorovkin has properties that are analogous to those of the integral J,(f; x) of Jackson (compare (I 510)). However, the degree of Jncf;x) is (2n  2) whereas the degree of K,,(f; x) is at most n. As an immediate consequence of Theorem 1.5.8, in particular of (1.5.9), we note Corollary 1.6.1. I f f E X2,, then
II&(f; Thus, iff (1.6.5)
E
0)
f(o)/Ixp,,
5 18~*0(2n;S;f ~  ’ ) .
Lip* (X2,; a ) for some 0 < a 5 2, then
IIUf;0)
f(0)
IIx,,
=
O(n  ”)
(n + m).
The kernel {k,(x)} of FejCrKorovkin also satisfies (1 5 2 7 ) . Indeed, (1.6.6)
1  kf(2) 1  kf(1)
=
2(n + 1) ( 1 n+2
+
Therefore by Theorem 1.5.1 5 Corollary 1.6.2. I f f E L& has an ordinary second derivative at xo, then
(1.6.7)
(n + a).
1.6.2 Further Direct Approximation Theorems
In connection with Cor. 1.6.1 it is important that, in a certain sense, the results are best possible. In fact, if for examplefis such that the approximation (1.6.5) holds, then, for 0 < a < 2, f belongs to Lip* (Xzn;a) as will follow in Sec. 2.4 (see Problem 2.4.6(i)). Thus for the integral of FejCrKorovkin the order of approximation given by (1.6.5) cannot be improved for functionsf€ Lip* (X2n;a), 0 < a < 2.
81
SINGULAR INTEGRALS OF PERIODIC FUNCTIONS
For the singular integral of Fejtr, the situation is quite different. Indeed, it follows by Theorem 1.5.8 that f E Lip* (Xzn;a), 0 < a < 1, implies IIa,,(f; 0 )  f(o)IIx,, = O(n"'2)(see Problem 1.5.10).But this estimate is by no means best possible. For, as will be seen in the next corollary, f e Lip* (Xzn;a), 0 < a < 1, even implies IIun(f; 0)  f(0)IIXzn = O(n"), which again is best possible in the sense that the converse is also true (see Theorem 2.4.7). To improve the order of approximation of Problem 1.5.10 we need the following result.
Proposition 1.6.3. Suppose that the kernel {xp(x)}of the singular integral Ipcf;x) is even. Then for every f E Lip* (Xzn;a), 0 < a I 2,
(1.6.8)
IIIAS; 0 )  f(o)IIx,, = O(m(Xp; 4 ) where the ath absolute moment m k p ;a) of xp is defined b y
(P
+
Po),
The proof is rather simple. Indeed, it follows as in the proof of Theorem 1.5.8 that
Thus in view of Problem 1.5.4(ii) I I I P W 0)
 f(O)llX,,
I (MZ/W
Ion
U"lXP(U)l
du,
which already establishes (1.6.8). Let us emphasize that Prop. 1.6.3 is valid for arbitrary kernels; the kernels need not be positive (or even, see Problem 1.6.7). To apply (1.6.8) to particular singular integrals the moments (1.6.9) have t o be estimated. It is to be expected that in this procedure the particular kernels must be estimated more refinedly. For the FejCr kernel a sharper version of Lemma 1.5.7 is valid.
Lemma 1.6.4. Let {F,,(x)}be the Fejkr kernel. Then (0 < (0
0,
(1.7.2)
n. m
then (1.7.3)
lim inf nail Vn(f; 0) n m
 f(0)IIXan= 0
implies that f is constant (a.e.).
Proof. First we note that for each f
E Xzn,Vn(f; x) is a polynomial of degree m(n) at most, given by (1.2.40). Moreover, by Problem 1.2.2(i) one has for each k E Z
Therefore for Ikl 5 m(n)
'1' 2rr
[Vn(f;
X)
f ( x ) ]
dx
=
(O,(k)  l)f"(k),
and hence
In view of the hypotheses there exists a subsequence {n,}, lim,+ ,,, n, = 00, such that on the one hand lim, n7l B,,(k)  1 I = c k > 0 for each k E Z, k # 0, and on the other hand lirn,.,,, n7II Un,(f; 0)  f(o)II1 = 0. B y (1.7.4) this implies cklf^(k)I = 0, and thus f "(k) = 0 for each k E E, k # 0. Hence it follows by Cor. 1.2.7 that f is constant (a.e.). For an application we return to the FejCr means un(f; x). By (1.2.43) the FejCr kernel is of the form (1.7.1) with m(n) = n and &(k) = 1  (Ikl/(n 1)). Moreover, it satisfies (1.7.2) with cc = 1. Therefore
+
Corollary 1.7.2. I f f E X,, and (1.7.5)
lim infn IIun(f; nm
0)
f(0)IIXan
= 07
then f is constant (a.e.). In other words, in the spaces X,, the FejCr means un(f; x ) never give an approximation to f with error o(nl) unlessfis constant. This answers the question posed at the
88
APPROXIMATION BY SINGULAR INTEGRALS
beginning. Furthermore, most examples of polynomial summation processes V,(f; x) of the Fourier series off do possess the property (1.7.2) and thus admit an assertion analogous to Cor. 1.7.2 (cf. the Problems of this section). Hence, if one sums the Fourier series off E Xznby such processes, then on the one hand this produces convergence to everyf and even approximation within a certain order which corresponds to the smoothness off. But on the other hand, the order of approximation cannot be improved beyond the critical order O(n.), n + 00, where a is given by (1.7.2). If this holds, the summation process V,V, x ) is said to be saturated, the relevant value a being the saturation index. We shall return to such questions in Chapter 12, give the x). formal definitions and deal with general singular integrals IOU, Here we mention a further interesting phenomenon. Let us suppose that C,(x) of (1.7.1) with m(n) E n is a positive function for each n E N. In view of Problem 1.6.4 IS,(l)l Icos (r/(n 2)), and therefore the sequence {n211e,(l)  lI} does not tend to zero as n + 00. Since U,(cos u; x ) = O,,(l) cos x and correspondingly for sin x, this implies (compare (1.5.14), (1.5.18))
+
Proposition 1.7.3. Let the kernel {C,,(x)}of the singular integral U,V, x ) be positive and given by (1.7.1) with m(n) = n. Then the sequences {n211U,(cos u ; 0)  cos oIIXan) and {n211U,(sin u ;  sin 011 xa,} do not tend to zero us n + 00. 0)
Thus, even iff is an infinitely often differentiable function (as cos x is), the approximation off by positive singular integrals cannot in general be better than O(n2), n+m.
Moreover, we have in view of Problem 1.6.6 that if C,(x) of (1.7.1) with m(n) = n is a positive function for each n E N, then lle,(k)l Icos (r/([n/k]+ 2)) for each 1 Ik In. Therefore lim inf,, oo n21le,(k)  1I > 0 for each k E Z, k # 0. This gives Proposition 1.7.4.
If under the hypothesis of Prop. 1.7.3 lim inf nail V,(f;0) n+ w
 f(o)I)Xzn= 0
for some f E Xan, then f is constant (a.e.).
Problems 1. Let& XZ, and JnV,x) be the singular integral of Jackson. Show thatfis constant (a.e.), provided lim inf,,, naIIJ,(f;  f(o)Ilxan = 0. 2. Let f E Xzn and Kn(f; x ) be the singular integral of FejCrKorovkin. Show that f is constant (a.e.), provided lim inf,,, nzIIKnCf;0 )  f(o)Ilxan = 0. 3. LetfE Xznand Bn(f; x ) be the singular integral of Rogosinski. Show thatfis constant (a.e.), provided lim inf,,, naIIBn(f;  f(o)Ilxan = 0. 4. Let f e X2, and the kernel {C,(x)} of the singular integral U,(f;x ) be given by (1.7.1). If there exists an a > 0 and 1 E N such that for each fixed k E H, (k(> I, condition (1.7.2) is satisfied, show that (1.7.3) implies thatfis a polynomial of degree I at most. 5. Let {xI)(x)}be the kernel of the singular integral ZJA x), and let ~ ( p be ) any positive function on A. 0)
0)
89
SINGULAR INTEGRALS OF PERIODIC FUNCTIONS
(i) Show that for each k E Z, k # 0, the following assertions are equivalent : (a) One of the following two sequences does not tend to zero as p + po: IIZ,(sin k u ;  sin kolIx2J. {p(p) IlZ,(cos k u ; 0 )  cos kollx,,J, (b) The sequence {p(p) IX;(k)  1I } does not tend to zero as p Po. (ii) If, furthermore, the kernel is even, show that for each k E H, k # 0, the following assertions are equivalent : (a) p(p) Ix;(k)  1 I does not tend to zero as p + po. (b) d p ) IlI,(cos ku;  cos ko /IXsn does not tend to zero as p + po. (c) p(p) IlZ,(sin ku;  sin kollxl, does not tend to zero as p + po. (Hint: See also BUTZERNESSELSCHERER [l]) 0)
f
0)
0)
1.8 Notes and Remarks Sec. 1.1. An early and important paper on singular integrals is that of LEBESGUE [2] (1909); an earlier but less general theory was given by D u BOISREYMOND and DINI. The work of HAHN[l] and HOBSON[I] is also basic. Most of our results are to be found in HARDYROGOSINSKI [l, pp. 5367], and indeed in books on Fourier series such as ZYGMUND [71, p. 84 ff I, EDWARDS [II, p. 59 ff]. For Theorem 1.1.5, Prop. 1.1.6, 1.1.7 see also HOFFMAN [I, pp. 1720]. A sequence {~,,(x)}:=~ definingan approximate identity (or unit) is sometimcs referred to in the literature as a (Dirac) delta sequence. For the term ‘approximate identity’ compare also Prop. 4.1.4. For further general comments on singular integrals see the remarks to Sec. 3.1. The term ‘singular’ was already used by LEBESGUE [2], HOBSON [2, Chapter 71 and DE LA V A L L ~POUSSIN E [3]; it is also used by TITCHMARSH [6, p. 30 ff], the Russian school, in particular by NATANSON [6, pp. 309359; 81, and by HILLEPHILLIPS [I]. However, it is not standard and many authors have no specific name for it. Still others, especially the Chicago school of ZYGMUND, mean something very different by the word singular (see CALDER~NZYGMUND [11, C A L D E R ~ [3]; N the corresponding onedimensional situation is treated in Part 111 and leads to conjugate functions and Hilbert transforms). In our concept a (convolution) integral is singular in the sense that the limit off* ,yo for p + po deliversf; the individualf* x, may be very smooth. Therefore it would be more correct to call the whole set {fo(f; x)} of integrals (1.1.3) a singular integral, in correspondence with the definition of a kernel. Singular integrals need not necessarily be of convolution type. Indeed, there is a general theory dealing with integrals of type s:f(u)x,,(x, u) du for which the reader is referred to HOBSON[2, p. 422 ff], NATANSON [6, p. 309 ff]. Strong and weak derivatives are considered in books on functional analysis, compare LJUSTERNIKSOBOLEW [ l , p. 192 ff], HILLEPHILLIPS [l, p. 58 ff]. Just as the usual pointwise derivative, they are defined successively by means of the derivatives of lower order (in contrast to the generalized derivatives considered in the pointwise sense in Sec. 5.1.4 and in the weak and strong topology in Sec. 10.1). For further comments, in particular for the connection with distributional derivatives, see the remarks to Sec. 3.1. Our results concerning derivatives of convolutions are standard topics in Fourier analysis, see e.g. EDWARDS [ l I , p. 55 ff], LOOMIS[3, p. 165 f]; they followed immediately since we assumed strong differentiability. The same is true for relation ( I .I .20) for which one may consult EDWARDS [lI, p. 601 or NATANSON [81, p. 2161. for example. The corresponding pointwise assertions are less obvious, see Sec. 1.4. Sec. 1.2. The material here constitutes the basic results of the classical theory of trigono
metric and Fourier series to be found in any book on the subject. Our treatment is brief as
90
APPROXIMATION BY SINGULAR INTEGRALS
ours is not intended as a text on Fourier series per se. We prefer to view Fourier coefficients from the standpoint of transform theory; this matter is dealt with systematically in Chapter 4. The standard references are also cited at the beginning of Sec. 4.4; see also the relevant chapters of texts on real variable theory, e.g. ASPLUNDBUNGART [I, Chapter 101. In this line, the reader will note that the pointwise convergence theory of Fourier series has been relegated to the Problemsection. But this theory is covered in every book on Fourier series; let us just mention the excellent accounts in BARI[l], ZYGMUND [7]. For an elementary proof of an explicit formula for the numbers L, of Prop. 1.2.3, due to FEAR,see CARLITZ [l]. In connection with the fact that the Dirichlet kernel does not constitute an approximate identity, KOREVAAR [ l , p. 3231 analysed its properties and considered ‘delta sequences of Dirichlet type’ from a general viewpoint. The Weierstrass theorem has been the object of much study in recent years. M. H. STONE in 1937 treated very general approximation theorems concerning closed subalgebras of the Banach algebra (with pointwise operations) of continuous functions on any compact Hausdorff space, For theorems of StoneWeierstrasstype compare HEWITTSTROMBER~ [l, p, 90 ff], EDWARDS [3, Section 4.101. The literature abounds with names for what we have called a closed system (Prop. 1.2.6); some speak of complete, fundamental or spanning systems. Concerning summability, a number of books do not work with general &factors but with specific examples, such as Fejtr or Abel, or with rowfinite factors, thus with triangular matrices (see BARI[HI, p. 2 ff]). In most examples of interest (including those of this section) B,(k) = 8(k/p), B(x) being a definite function of (one) real variable x. The reasons for this become apparent in view of the results of Sec. 3.1.2, 5.1.5. A fundamental theorem on linear methods of summation of sequences, the terms of which are general complex numbers, is due to 0. TOEPLITZ (see HARDY [2], ZELLER [l] and the literature cited there): Given a matrix A = (ajk);k=O, A associates with each sequence {sk}?O a sequence {U,}?=O, defined by u! = 2 b oa j k f k , provided that these series converge. If uj+ s, the sequence {sk} is said to be Asummable to s; the uj are called the Ameans. The theorem states that, for every convergent sequence {sk},s, + s implies u,,,s for n + Q) (thus A is regular) if and only if A is a Toeplitz matrix. Such matrices satisfy the three conditions:(i)lim,,,ajk=O(k= 0, 1, ...),( ii)Iimj+, 2?=0ajk= l,(iii)supj2:,”,0la,kl < co. We are usually concerned with sequences s k = 2rsoc, defined by partial sums of infinite series. The choices ajk = 0’ I)’ for 0 Ik Ij , = 0 f or j c k and a j k = (1  r,)rF, 0 < rj < 1, rj + 1 , then lead to Ceshro and Abel summability of the series, respectively. In this general setting, a series summable Ceshro is summable Abel to the same sum, but not conversely. Thus many results involving Abel summability follow from corresponding theorems that treat Ceshro summability. Nevertheless, an independent study of the former is of interest, particularly when dealing with Fourier series of functions f. This is true, not only because such series may be Abel summable under weaker conditions on f than are necessary to guarantee their Ceshro summability, but also because the Abel means have special properties related to the theory of harmonic and analytic functions that are not enjoyed by the Ceshro means. However, in contrast to the above Toeplitz approach, we commenced with the means (1.2.30). Thus we do not deal with Ameans of the partial sums of the Fourier series [ l , p. 53 ff]), but insert the factors B,(k) directly into the series (HARDYROGOSINSKI (ZYGMUND [71, p. 84 ff]). For an abstract approach to the AbelPoisson integral see RUDIN[4, p. 1091. Concerning Problem 1.2.7(iv), although the series (h0/2) + hk cos kx converges to a sum f(x) for all x # 2nj, j E Z, it is not necessarily the Fourier series off (see Problem 6.3.2(iii)). But if the sequence {hk} is furthermore assumed to be convex on P, then it follows that f E Lin, and the above series is the Fourier series off, see Sec. 6.3.2. Sec. 1.3. For Prop. 1.3.1, 1.3.2 see GOES[l] and the literature cited there. As the operators in question are of convolution type, it was rather easy to determine their norm, Prop. 0.8.8,
+
zzl
SINGULAR INTEGRALS OF PERIODIC FUNCTIONS
91
0.8.9 on bounded linear functionals being applicable. Similar assertions are also available for more general kernels. If x(x, u) is a continuous function of both variables x, u in the square a 5 x, u Ib, and an operator T of C[a, b] into itself is defined by (Tf)(x) = f,f(u)x(x, u) du, then 1I T I( = sup, b J: Ix(x, u) I du; compare KANTOROVICHAKILOV [1, p. 1081. Prop. 1.3.3 seems to have been first stated explicitly by ORLICZ[l]. In the Russian literature the integral of Rogosinski is usually referred to as that of Bernstein, see NATANSON [81, p. 2171. For Theorem 1.3.5 see also NATANSON [IIII, p. 87 ff], KANTOROVICHAKILOV [I, p. 255 ff]. Prop. 1.3.6 may be considered as a weak version of the famous theorem of DU BOISREYMOND (1872) as stated in Problem 1.3.5(i). This result has been extended in several directions. Thus there exist continuous functions whose Fourier series diverge on sets that are uncountable, of the secondcategory, and everywhere dense (see EDWARDS [lI, p. 155 ff], RUDIN[4, p. 101 ff]). In 1966 CARLESON [l] solved a problem posed by LUSINin 1915. It states that the Fourier series of any function f E L& converges pointwise almost everywhere to f. This result was extended by HUNT[l] to all LS, with p > 1. In 1926 KOLMOGOROV showed that there exist integrable functions whose Fourier series diverge everywhere. For a proof see ZYGMUND [71, p. 310 ff] or BARI[lI, p. 455 ff]. For a discussion from a general point of view, see KATZNELSON [l, p. 55 ff]. Other extensions are connected with the names of FABER, BERMAN,MARCINKIEWICZ, HARSILADSE and LOZINSKI~ (see Problems 2.4.132.4.17). Compare CHENEY [l, p. 210 ff], LORENTZ[3, p. 96 f]. For historical remarks on the integral of Jackson of Problem 1.3.9 and its many generalizations see e.g. GORLICHSTARK [2]. For remarks on the integral of Weierstrass of Problem 1.3.10 see BUTZERG~RLICH [2, p. 340, p. 366 f]. For comments to Sec. 1.3.3 see the Notes and Remarks to Sec. 1.5. Sec. 1.4. Much of the material of this section was influenced by HARDYROGOSINSKI [l,
p. 58 ff] and BOCHNERCHANDRASEKHARAN [l, p. 13 ff]; see also the comments to Sec. 3.2. For Prop. 1.4.1 compare ZYGMUND [71, p. 861. Cor. 1.4.3, 1.4.7 are standard, the latter often called the theorem of FejerLebesgue. Prop. 1.4.8 for the particular kernel of AbelPoisson (then known as a theorem of Fatou) is to be found in ZYGMUND [71, p. 1001, HOFFMAN [ I , p. 34 f]. Their proof looks simpler since they divide the central difference of p(x) by sin x in which case {O,(x)} constitutes an approximate identity. We divided by sin (x/2), there also being a singularity at P (compare (9.2.30)). Prop. 1.4.10 also does not seem to be stated explicitly for general kernels; for the (C, a) means it is given in ZYGMUND [71, p. 105 f]. Problem 1.4.2, due to Fatou, may also be stated for nontangential limits ;see BARI[lI, p. 152 ff], PRIVALOV [l]. At this stage it would be possible to study dominated convergence of singular integrals. An integral Z,(f; x) with f c X,, converges dominatedly tofas p po if limp,poZ,(k x) = f(x) a.e. and there exists some g E Li, such that suppeAIZo(f; x)l 5 g(x). We have seen (see Problems 1.2.10, 1.2.1 1) that i f f c BV,,, is 2~periodic,then the partial sums &(f; x) converge boundedly tof(x) as n 00. More generally, if {xp(x)}is a kernel which is majorized by {xz(x)} satisfying (1.1.5) and (1.4.1) with constants M* and M:, respectively, then sup IZP(f;X)l I( M * 2M,I)Ocf,x).


+
DEA
sx"
The latter function, known as the HardyLittlewood majorant off, is defined by
SCr; x)
(  n Ix In). s u ~ If(u)l du 511 t x LetfE L$, 1 c p < co. Then OCr; x) E Lg,; in particular, the integral u,(f; x) of Fejkr converges dominatedly to f. Moreover, llsupnal lcr,cf, ,)I 1, IApllfl\p for some constant A,. This material receives excellent coverage in ZYGMUND [71, p. 29 ff, 154 ff]; compare also K. L. PHILLIPS[ll. Sec. 1.5. At first some comments to Sec. 1.3.3. For continuous functions, the analog of Theorem 1.3.7 for approximating sums and its proof is due to BOHMAN[l]; his paper
=
O 1, then in view of Prop. 1.1.15, 1.1.16
[(Kn  V'+I'(x) = [(Kn
 V'+'l(xh
and therefore we may continue as above with 36 &(Xan;f) 5 ; ]I(&  I)[(Kn
 O'a+'lll~2n
By a repeated application of Prop. 1.1.15,1.1.16, Cor. 1.6.1 and Problem 1.5.3 we have 36 r  l En(Xa,;f) 5 II(Kn 
);(
Now, (2.2.2) may be obtained from (2.2.1) by the following device. Recalling that AE(f; x) denotes the second integral mean off (see Problem 1.13, we write f(x) = &(f; x ) g(x). In view of Problem 1.1.7 it follows that f E Wk,, implies &(f; x ) E W&inaand
+
[A%(s;o)]"+yx) = h  a [ p ( x
+ h) +
p ( X
 h)  2+'"(x)].
Therefore II[A%(f;~ ) ] ( ~ + ~ ) ( ~I) l h'U*(X,,; lx~,
4"'; h).
On the other hand, g E Wk,,, and since by Prop. 1.1.15, 1.1.16, and Problem 1.5.5
II[&(4;
0)
 +(~)I(~'/lx,,, = llA%(+(r);  +(rY~)llxa, 5 W*(Xan; 0)
h)t
we have by Problem 2.1.2 and by (2.2.1) &(Xan;.f) I E n o L z ; I
0))
+ &O(a,;
e)rtahau*(xan; 4"); h)
g)
+ E)'u*(xan;
Finally, setting h = l/n, n E N, E,,(XZn;f) I((36)"(36a + l))n'w*(X,,; and the proof is complete. As an immediate consequence we state
n'),
+@);
+(');
h).
99
THEOREMS OF JACKSON AND BERNSTEIN
Corollary2.2.4. Iff E *W;, for n + m.
for some r
E
N and0 < a I 2, then E,(X,,;
f) = O(nra)
Problems 1. Let r E N and f E Wka,, i.e., let there exist 4 E AC&' with #(x) (a.e.). Show that there is some constant C such that
I$(~) E
X2, such that f(x) =
En(Xzn;f) ICnr~(X2,; +(r); n'). In particular, i f f € Wk&, 0 < a I 1, then E,(X,,;f)= O(n..).
2. Let f E C,, be rtimes differentiable such that ftr) E L,",. Show that E,(C2,;f)I (36/n)rllf(r)llm. Moreover, if f t r ) eC,, is such that f ( r ) ~ Lip* (Can;a) for some 0 < a I 2, then En(C2,;f) = O(n..). 3. Let f E Xzn.I f f € lip (Xzn;a) for some 0 < a I 1, show that En(X,,;f)= ~ ( n  ~ ) . 4. ForfE W& show that limn+mnYE,(X2,;f) = 0 for every y > 0.
2.3 Theorems of Bernstein It is the purpose of this section to establish results converse t o those of the last section. In Cor. 2.2.4 it was shown that iffis rtimes differentiable andf(')E Lip (Czn;a), then E,(C,,;f) = O(n+.). The question then arises as to whether conversely the boundedness of the sequence nr+"En(Cz,;f)implies the existence of f:r) such that f r ) E Lip (&; a). As we shall see, this is indeed the case; in other words, E,(C,,;f) does not tend to zero more rapidly than O(nra)in general, thus the estimate of Problem 2.2.1 cannot be improved for 0 c a < 1. We begin by establishing the inequality of Bernstein on the order of growth of the derivatives of (trigonometric) polynomials. This inequality is fundamental for what is to follow.
Theorem 2.3.1. (2.3.1)
If t , is a polynomial of degree n, then I I ~ I x ~ 5, 2n IItnIIX2,.
Proof. We write (cf. Problem lS2.l4(iii)) tn(x) =
27
jn
t.(u)[ 1
+2
k=l
cos k(x
 u)] du.
Differentiation gives tn(x
+ u ) [2 k:2l k sin ku
To the term in brackets we may add the sum 2 k sin (2n  k)u (all of its terms have degree greater than n) which does not change the value of the integral since 1, E T,. We then obtain
fn(x + u) 2n sin nu Fa,@) du,
APPROXIMATION BY SINGULAR INTEGRALS
100
{F,(x)} being FejCr's kernel. Hence by the HolderMinkowski inequality
and the proof is complete. Corollary 2.3.2. I f f , E T,, then for every r E N
I tf' IIx ~ 5, (2nY IItn I1Xan
(2.3.2)
We remark that by a more refined but nevertheless elementary argument one can show that the factor 2 in inequality (2.3.1) is superfluous. On the other hand, the resulting inequality IItkIIxzn I n IItnIIXz,cannot be improved as the example tn(x) = sin nx shows. Theorem 2.3.3. If; for some a > 0, f polynomial of degree n such that (2.3.3)
E X,,
can be approximated for each n E N by a
En(Xan;f) = O(n9
( n + m),
then 4Xan;f;
O(W, O < a < l 8) = O(6 (log 61 1, cc=l
{
O(S)Y
( 6 + O+).
a >1
Proof. If t: denotes the polynomial of best approximation to f, then Ilf  t: I Bn" for some constant B > 0. We write Ua(x) = t;a(x), V,(x) = t&(x)  t &  ~ ( x ) , n = 3,4,. . ., U,(x) being a polynomial of degree 2". Then for n = 3 , 4 , , . . IIUnllx,,
5 IIt& fllx,,
and since (1 Uallx,, 5 B2aa + n = 2,3, ...
Cnlllx,,, I ( 1
zE=a
+ 2")~2"",
we may choose a constant C such that for all
1I UnIIX,,
(2.3.4) Now,
l f( xur,
+ [If
C2""
uk(x) = t:n(x) converges in X,,norm t o f a s n + coy thus
(2.3.5)
Since U, (a polynomial) is arbitrarily often differentiable, we have
uk(x + h) 
uk(X)
=
lohuL(X +
U)
dU.
101
THEOREMS OF JACKSON AND BERNSTEIN
Moreover, according to Bernstein's inequality
m k=m+l
2 2k""' + 1  2" m
< 2C6 
k=Z
Supposing 0 < 6 < 3, we now choose m 2 2 according to
2m1 I116 c 2".
(2.3.6)
Then with C1 = 2l*C/(l  2  9 we have
We now consider the different cases. (i) I f 0 < a < 1, then and since 2"+l I416,
+
C16. But 2"+' I 416 implies (ii) If a = 1, then w(X,,;f; 6) I 2C6(m + 1) m + 1 I2 + (]log 6)/Iog2), giving w(X,,;f; 6) = O(6 /log 61 1. (iii) If a > 1, then
Thereby the proof is complete. Next we prove a theorem which establishes the existence of derivatives of a function f E X,, if the sequence {E,,(Xzn;f ) } of its best approximations tends to zero sufficiently rapidly.
Theorem 2.3.4. If, for some r E N and a > 0, f n E N by a polynomial of degree n such that then f E W;Cz,. Moreover, if+
w(Xz,;
E AC',; l,
+(');
6) =
+(')
E
E
X,, can be approximated for each
Xzn, is such that
4 = f i n X2,,
O 0, f E X,, can be approximated for each n E N by a polynomial of degree n such that E,,(X,,; f ) = O(n'"), then f E Wk,,. Moreover, if+ E AC',; l, +(r) E X,,, is such that q5 = f i n X,,, then w*(X,,;
p;6)
=
I
O(V, O(S2110g Sl), 0(W,
O 0,f E X2, can be approximated for each n E N by a polynomial of degree n such that E,(X,,; f) = o(n.), n + co, show that 4X2,;f;
8) =
r8”’ llog 811,
m,
O < a < l a=l a > 1.
2. Prove Theorem 2.3.6. 3. If, for some r E N and a > 0,f E X,, can be approximated for each n E N by a polynomial of degree n such that E,(X,,;f) = o(n’.), n + a, show that f E W:,. Moreover, if # E ACL; l, #(‘) E X2,, is such that #(x) = f(x) (a.e.), then O(W,
w*(X,,;
{
p;6) = o ( P llog Sl), O(W,
O < a < 2 a = 2 a > 2.
(Hint: See also NATANSON [81, p. 1091, BUTZERNESSEL [l])
2.4 Various Applications
Let us first summarize some of the results so far obtained in this chapter, and for the convenience of the reader state them in form of theorems.
Theorem 2.4.1. L e t f E X2,. (i) For 0 < a < 1, En(X2,;f) = O(n.) e f Lip ~(X2,; a), (ii)
En(Xzn;f)= O(nl) O
~
Lip* E (X2,; 1).
To prove (i), the fact that f E Lip (Xzn;a) implies En(X2,;f) = O(n.) follows by Cor. 2.2.2, since in view of Problem 1 . 5 . 4 f ~Lip (X2,; a) yieldsf€ Lip* (X2,; a). The converse is given by Theorem 2.3.3. Similarly, the equivalence of the statements of (ii) follows from Cor. 2.2.2 and Theorem 2.3.5. In part (ii) of this theorem one cannot in general replace the class Lip* (X2n;1) by Lip (X2,; 1); indeed, i f f E Lip (X2,; I), then E,(Xzn;f) = O(nl), but conversely, given the latter estimate, it does not necessarily follow that f E Lip (X2,; 1). This is verified by the example of Problem 2.4.2. The question remains as to whether for functions f~ Lip (X2,; 1) the quantity En(X2,;f) converges more rapidly to zero than the estimate E,(X,,;f)= O(nl) permits. By means of the examplef(x) = Isin XI (cf. Problem 2.4.12(i))it follows that the relation limn+mnEn(X2,;f)= 0 does not hold for everyfE Lip (X2,; 1). Thus we are faced with the fact that the class Lip (Xz,; 1) is a true subclass of the class of function f for which E,,(X2,;f) = O(n’), the latter being in turn contained in that class of functionsffor which w(X,,;f; 6) = O(S llog Sl), i.e., Lip OGz; 1) c {fe x2 zIE nO G n; f ) = O(nl)> c
{ f X~2 ~ ~ 4 X 2 n 8); A= O(S llog Sl)>*
And part (ii) of Theorem 2.4.1 shows that the ‘intermediate’ class of functions is characterized as the class Lip* (X2,; 1). There seems to be no (cf. Sec. 2.6) simple characterization of the class Lip (Xz,; 1) expressed directly in terms of the order of best
THEOREMS OF JACKSON AND BERNSTEIN
105
approximation. For very different characterizations of the class Lip (X,,; 1) we refer to Sec. 10.2. The results of this chapter may be used to prove certain nontrivial relations between Lipschitz classes. As an example we state
Theorem 2.4.2. Let f E X,,. (i) For 0
w(X,,;f; 6)
a),
0(6 /log 61). Proof. If f E Lip (X,,; a), then obviously f E Lip* (X,,; a). Conversely, if f E Lip* (&,; a), then En(X2,;f)= O(n@')by Cor. 2.2.2 which impliesf E Lip (X,,; a) by Theorem 2.3.3 if 0 < a < 1. This proves (i). Regarding (ii), iff E Lip* (X2,; l), then E,,(X,,;f) = O(nl) by Cor. 2.2.2, and thus w(X,,;f; 6) = 0(6 [log 61) by Theorem 2.3.3. In particular, f E Lip* (X,,; 1) implies f E Lip (X,,; a) for every O 00. (Hint: Use Theorem 1.5.15, thus (1.5.28)) 8. (i) Let f e Can be continuously differentiable. Show that V;(f;x) =f'(x) uniformly for all x . (Hint: Use Prop. 1.1.15) (ii) LetfE L,",, and suppose thatfis differentiable at xo,Show that limn+mV,'(f; x o ) = f'(xo). (Hint: Use (1.1.18) and proceed as in the proof of Prop. 1.4.8; see also NATANSON 181, p. 2151) 9. (i) Show that A*(u,; 2) = 2n' o(nl). Thus, for the Nikolskii constant, N*(2) = 2. (Hint: Use Prop. 1.6.6) (ii) Show that for each 0 < a I1
+
A(v,,; a) = {2'n1/ar((l
+ a)/2)}n"Ia + ~ ( n  " ~ )
(n + a).
(Hint: NATANSON [2], for a = 1 see also [81, p. 2101) (iii) Show that the same formula as in (ii) for A(u,,; a) holds for A*(o,; a) for every 0 < a I2. (Hint: TABERSKI [l]) 10. Show that the geometric mean of a positive cosinepolynomial C,,(x) = ho + A, cos kx, not identically zero, is given by exp {L 2n
s"
n
log Cn(u) du} = Ihn(0)la= hE(O),
where h,(z) is an algebraic polynomial in z having real coefficients such that C,(x) = (h,,(eix)(a. (Hint: Apply Problem 1.6.3, see P~LYASZEGO [III, p. 841) 1 I . If C,,(x) of the previous Problem has geometric mean equal to one, show that: (i) C,,(x) 5 2an;equality holds if and only if Cn(x) = 2ancosan((x  x0)/2), thus for the kernel of de La Vall6e Poussin (apart from a constant). (ii) The arithmetic mean of C,,(x) is given by ho = (1/2n) Cn(u)du I(a,"); when does equality hold in this case? (iii) Ak s 2(,,?!k),k = 1,2,. . ., n ; when does equality hold in this case? 12. Let W t ( f ;x) be the singular integral of Weierstrass off E Xan.Show that (compare Problem 1.5.9(ii))
E,
for 0 < a < 2,f
f
(Xzn;a) * II Wt(f;0 ) Lip* (Xan;2) 3 I W d f ;0 )
E Lip* E
 f(o)IIx2,
 f(o)IIxpn
= O(ta12), t t
= O(0,
 +.+, +
0 0
116
APPROXIMATION BY SINGULAR INTEGRALS
2.6 Notes and Remarks The presentation in this chapter is brief since this is not intended to be a text on approximation theoryper se. The theorems of Sec. 2.12.4 are standard and do not represent the best possible results currently known: only that degree of generality is considered that is relevant to its later use in the text. The most useful general references here are LORENTZ[3] and TIMAN[2], where the leitmotiv is actually that of the order of approximation by polynomials of best approximation. One may also consult the classical treatises of ACHIESER [2] and NATANSON [8]. The results of Sec. 2.5 are not standard in texts. Sec. 2.1. The existence theorem for best approximations (Theorem 2.1.1) was first estab
lished in the doctoral dissertation of KIRCHBERGER [l] in 1902 and also by BOREL[l] in 1905. P. L. CHEBYCHEV was perhaps one of the first (1857) to concern himself with polynomials of best approximation and their characterization, using the mathematical techniques of those days. The concept E n ( X a n ; f is ) best treated from the point of view of the theory of normed linear spaces. If X is a normed linear space, and g,, . . .,gnare n linearly independent given vectors in X , the best approximation of YE X by linear combinations (‘polynomials’) 2Ezlakgk (arc being scalars) is E n 0 = minak.lsksn Ilf  2lt=1 akgkII. Then the exisfence of polynomials of best approximation tofis a simple consequence of the fact that a continuous function defined on a compact set achieves its infimum. But the question when such a polynomial of best approximation is unique is much more involved. A general reference to the unicity problem is CHENEY [l, Chapter 1; 31 and the literature cited there. Thus a sufficient condition which assures uniqueness is the strict convexity of X. X is called strictly convex if IlfiII 5 r, l[flla I r imply llfi + fall < 2r unlessfl = fa. Although the spaces Lbn, 1 c p < a,are strictly convex, this is not true for p = 1 nor for the space C2,. These spaces must therefore be treated by their own methods. An important problem here is the characterization of the polynomial of best approximation. One such result in case { g l , . .,gn}is a system of elements of Cansatisfying the Haar condition (often termed a Chebychev system) is the BorelYoung or Chebychev alternation theorem in case of uniform approximation. The system { g l , . .,gn}satisfies the Haar condition if each nontrivial polynomial 2EG1 akgk has at most n  1 distinct roots (in any interval of length 2 ~ )In . this respect, Haar’s unicity theorem states that the polynomial 1 of best approximation to f E Can in the uniform norm is unique for all choices off if and only if { g l , . . .,g,,} forms a Chebychev system. The unicity problem in the case of best approximation in La,norm is solved by an important theorem of JACKSON: If { g l , . .,gn} is a Chebychev system in Can, then each f E Can possesses a unique polynomial of best approximation in La,norm, see CHENEY [ I , p. 218 ffl. Other references: HAVINSON [l], KRIPKERIVLIN[l], PTAK[l] and RIVLINSHAPIRO[l]. For the unicity and characterization problem in the setting of arbitrary Banach spaces we refer to PHELPS [l], GARKAVI [l] and to the many papers and monograph of SINGER[14]; see also AUMANN [ 11 as well as BUTZERGORLICHSCHERER [ 11. The theory of nonlinear approximation, in particular on the existence, characterization and unicity of nonlinear best approximations, has been the subject of mathematical investigations only during the last few years. Here we must mention the names of MOTZKIN [2], RICE[III], MEINARDUS [l, pp. 1311881, BROSOWSKI [l], as well as the literature cited in the last text. However, the special case of rational approximation is a much older discipline, see e.g. CHENEY [l, Chapter 51.
.
.
.
Sec. 2.2. Theorem 2.2.1 (in C,,space) for the ordinary modulus of continuity was proved by JACKSON [l] in 1911 in his dissertation (written under E. LANDAU at Gottingen). These investigations were stimulated by earlier work of LEBESGUE [l] and DE LA VALLBEPOUSSIN [ l ] in 1908. It seems that KOROVKIN [5] was the first to use the polynomials K n ( f , x ) explicitly in proving the first Jackson theorem (JACKSON himself had employed the polynomials Jn(f; x)). The proof of Theorem 2.2.3, the second Jackson theorem, is based upon
THEOREMS OF JACKSON AND BERNSTEIN
117
an interesting nonstandard method of NATANSON [5] which fits in well with our approach based upon the theory of convolution integrals. The proof of (2.2.2) is taken from ZYGMLJND [3; 71, p. 1171. The fact that one can replace the ordinary modulus of continuity by the generalized one (cf. Theorem 2.2.1 and the estimate (2.2.2)) was first noted by ZYGMLJND 121 in 1945. Best approximation in Lg,space, including the Jackson theorems, was first studied systematically by QUADE[I] (see also the text of TIMAN[2]). For various improvements of the Jackson theorems, determination of the best possible constants occurring, and various connected results see e.g. the references in CHENEY [ 1 , p. 2301. For more recent papers see SHISHA111, ROULIER[l], DE VORE[l], LORENTZZELLER [l]. Sec. 2.3. The theorems here (in Cz,space) are from the doctoral dissertation (1912) of BERNSTEIN [l]. The proof of Bernstein's inequality (2.3.1) given is due to F. RIESZ[l]; for the refined estimate Iltillxz,5 nlltn((xan compare NATANSON [81, p. 90 f]. The method of proof of Theorems 2.3.3, 2.3.4 is particularly interesting. To appreciate BERNSTEIN'S argument (cf. H. S. SHAPIRO[I, p. 781) it is worth while to note what a straightforward approach would yield. If E,(C,,;f) = O(na), then the polynomials t: (of best approximation t o f ) are uniformly bounded, and thus by Bernstein's inequality IItn*'IICln = O(n). Therefore it follows for h > 0 that
+
+
+
Ilf(0 + h)  f(0)IIczn 5 Ilr*(o h)  tR*(o)llcz, 2~,(C,,;f) IC(nh n") = O(ha'(l+a)) if we choose n around / t  l ' ( l + a ) . This shows fe Lip (C,,; a/(1 + a))rather than Lip(&; a). The reason for this imperfect result is that the estimate Ilfn*'llcz, = O(n) is too crude. The extra information that {t;} converges with a certain speed must be exploited to produce a sharper estimate for t:'. In fact, we obtained Iltn*'(ICzn = O(nl'). The Bernstein method has been adapted to other situations, too, in particular to various singular integrals, see Sec. 2.5, 3.5, 13.3. Theorem 2.3.5 is due to ZYGMUND [21(1945). We owe to Zygmund the discovery of the fact thatfE Lip* ( X z n ; 1) if and only if En(Xzn;f)= O(nl). He was also the first to recognize the importance of the function class Lip* (&,; 1) in various branches of Fourier analysis. The Bernstein theorems in Lg,space were first proved by QUADE[l]; see also TIMAN[2, p. 334 f]. These results have been developed further by many authors. There is an elegant generalization by DE LA VALLBEPOUSSIN [3, pp. 5358] (extended to higher moduli of continuity by BUTZERNESSEL [l]) as well as an extension of a different type by S. B. STEEKINwhich has given rise to a large number of investigations on the subject. See e.g. LORENTZ [3, pp. 5863], TIMAN [2, pp. 344491. For generalizations of the theorems of Jackson and Bernstein as well as of certain results of M. ZAMANSKY and S. B. STEEKIN(and their converses) to arbitrary Banach spaces (in the setting of the theory of intermediate spaces) the reader is referred to BLJTZERSCHERER [141. Sec. 2.4. The applications presented here are standard, see e.g. NATANSON [8],GOLOMB [I]. Theorem 2.4.3 is proved in LEBESGUE 121. Condition lim,,o+ w*(Cz,;f; 6) )log 61 = 0 is known as the generalized DiniLipschitz condition. For Theorem 2.4.8 see ROGOSINSKI [l, 21, BERNSTEIN [3] or NATANSON [81, p. 217 ff]; for part (iii) of Theorem 2.4.9 see ZYGMUND [3]. Problem 2.4.11 is due to DE LA V A L L ~POUSSIN E [3, p. 341. Concerning Problem 2.4.14(ii), CHENEYHOBBYMORRISSCHLJRERWLJLBERT [1, 21, CHEWYPRICE [I] have recently shown that the Fourier series projection is the only minimal projection of CZ, onto Tn. Lip(Czn; 1) cannot be characterized in terms of En(C2,; f);see SCHERER 111. Let R, be the operator which assigns to eachfE C,, its polynomial of best approximation, thus R,f = t:(Czn;f). For each n E N,R, is an operator of C,, onto T, with the property that Rntn = r, for all r, E T,. Hence RZ = Rn. Moreover, limn.+mIIRnf  f\lcan = 0 by Weierstrass' theorem. The operator R, is in fact continuous but not linear; see, for example, CHENEY[I, p. 2101. It is therefore natural to ask whether a linear operator can exist having the above properties. In view of Sec. 1.7 this operator cannot be positive. Indeed, the result of HARSILADSELOZINSKI~ of Problems 2.4.152.4.16 shows that n o such operator does
118
APPROXIMATION BY SINGULAR INTEGRALS
exist. In L:,space the situation is quite similar. However, in L;,space, 1 < p < 00, there does exist a sequence of bounded linear operators of L;, onto T,, namely the partial sums of the Fourier series off, whose order of approximation to a given f is indeed that of the best approximation E,(L%,;f);see Prop. 9.3.8. Sec. 2.5. Concerning Theorem 2.5.2 see ANGHELUTZA [ 11, also SALEMZYGMUND [I],
NATANSON [3], TABERSKI [l], or BUTZERBERENS [ l , p. 1221. Theorem 2.5.4 was first established for the AbelPoisson integral on the unit nsphere for continuous functions by Du PLESSIS [2]. In the form given the result was rediscovered by BERENS [ l ] in his doctoral thesis. The method of proof of the theorem is that of BERNSTEIN (cf. Theorem 2.3.3). Prop. 2.5.6 is due to ZYGMUND [3]. The integral (25.11) was introduced by DE LA VALLBEPOUSSIN [2] in 1908. Further papers on the subject are NATANSON [2], BUTZER [l], TABERSKI [l], MATSUOKA [l]; see also NATANSON [81, p. 206 ff]. Prop. 2.5.8 is due to NATANSON [ 11. For Lemma 2.5.9 see BERENS [3, p. 571; the reader may note the important rale played by the identity (2.5.19) in the proof. Once such identities (cf. the treatment in BUTZERPAWELKE [l]) have been established for a general class of singular integrals, a complete approximation theory may be built up for such a class. One identity of this type is given by the semigroup property (cf. Sec. 13.4.2). The proof of Theorem 2.5.10 in this form does not seem to be given elsewhere; see also BERENS [3, p. 56 ff]. For Prop. 2.5.12 see BUTZER [l]. There is also an abstract approach to the material of this section. Indeed, results of the type given by Cor. 2.5.5, 2.5.1 1 may be established for a general class of approximation processes on Banach spaces. Such a process (cf. Def. 12.0.1) is defined by a family of commutative, bounded linear operators { T J D , oon a Banach space X to itself which approximate the identity strongly as p co, and satisfy a Jacksontype inequality f
(2.6.1) I I T D f  fllx as well as a Bernsteintype inequality
5 ~lP'llfllY
cfe Y)
(2.6.2) I l T D f I l Y 5 C2P"llfllx (fE )o for a suitable Banach subspace Y of X and exponent u > 0. In particular, 'polynomial' operators may be studied. In this respect, we refer to BUTZERSCHERER [2, 3, 51 as well as to their monograph [l, p. 73 ff]. These results are also given in the setting of the theory of intermediate spaces.
3 Singular Integrals on the Line Group
3.0 Introduction The material of this chapter is in many points a straightforward adaptation of the periodic counterparts of Chapter 1. This is particularly true for Sec. 3.1 which corresponds to Sec. 1.1, 1.3, thus treating convergence in the norms of the spaces X(R). Special emphasis is placed upon the study of singular integrals of FejCr’s type. In Sec. 3.1.2 to each approximate identity on the real line a periodic approximate identity is associated via (3. I .28), (3.1.55), respectively. Important examples of singular integrals such as those of Fejtr, GaussWeierstrass, and CauchyPoisson are introduced. Sec. 3.2 deals with pointwise convergence of convolution integrals, the results correspond to those of Sec. 1.4. Sec. 3.3 is concerned with nonperiodic counterparts of Sec. 1.5, thus with questions on the order of approximation by positive singular integrals on the real line. The method of test functions is touched upon and certain asymptotic expansions are given. Furthermore, Nikolskii constants for periodic singular integrals of FejCr’s type with respect to Lipschitz classes are determined; these complete the results of Sec. 1.6.3 in the fractional case. Sec. 3.4 deals with direct approximation theorems for singular integrals, the kernels of which need not necessarily be positive. In case the order of approximation is O ( P  ~ )0, < CL < 2, the results correspond to those of Sec. 1.6. For applications of these concerning higher order approximation we refer to Sec. 6.4 where certain periodic counterparts are also formulated. In Sec. 3.5 inverse approximation theorems for singular integrals of FejCr’s type are given. The proofs follow by a direct adaptation of Bernstein’s idea, already familiar from Sec. 2.3, 2.5. In Sec. 3.6 some aspects concerning shape preserving properties of approximation processes are discussed. For a certain class of functions f i t is shown that the approximation off by the GaussWeierstrass integral is monotone if and only iff is convex. The concept of variation diminishing kernels is introduced. The main result here is that a kernel is variation diminishing if and only if it is totally positive. For counterparts of Sec. 1.2 concerning the classical theory of Fourier series the reader is referred to Chapter 5.
APPROXIMATION BY SINGULAR INTEGRALS
120
3.1 NormConvergence
3.1.1 Definitions and Fundamental Properties Just as for periodic singular integrals in Sec. 1.1 we begin with Definition 3.1.1. Let p be a positive parameter tending to infinity. A set of functions { ~ ( x p)} ; will be called a nonperiodic kernel or a kernel on the real line if ~ ( 0 p) ; E L' for each p > 0 and
x(u; p) du = z/z;;
(3.1.1)
A kernel { ~ ( x p)} ; will be said to be real, bounded, continuous or absolutely continuous if x(x; p) is a real, bounded, continuous or absolutely continuous function of x for each p > 0. A realkernel{X(x;p)} isevenorpositive ifx(x; p) = x(x; p) or,y(x; p ) 2 0 a.e. for each p > 0. We shall usually just speak of a kernel, whether it is periodic or not. The distinction will be apparent either from the context or the different notations { ~ ~ ( xand ) } { ~ ( x ;p)}. Instead of (3.1.1) sometimes the condition (3.1.2)
is used in the literature. But there is no loss of generality since we may always normalize a kernel by (3.1.1). The normalization (3.1.1), which is different to that of ( l . l . l ) , is rather convenient as will be shown in Chapter 5. Definition 3.1.2. For f
E
X(W) the convolution 1
rw
(3.1.3)
defines a singular integral generated by the kernel { ~ ( x p)}. ; The singular integral is said to be positive or continuous if the kernel is positive or continuous. As an immediate consequence of Prop. 0.2.2 we state Proposition 3.1.3. Let f E X(R) and { ~ ( x p)} ; be a kernel. For each p > 0, it follows that Z(f; p) E X(W) and 0 ;
(3.1.4)
IIIW ";P)llX(Oa) 5 IIx(0; P)lllllf
IIXW).
With [Z(p)fl(x) = Z(f; x; p), the integral (3.1.3) defines a bounded linear transformation I(p) of X(R) into itselffor each p > 0. In order to produce convergence of the integral Z(f; x; p) towards f as p + co we introduce the notion of an approximate identity. Definition 3.1.4. A kernel { ~ ( x p)} ; is called an approximate identity (on the real line) if there is some constant M > 0 with (3.1.5)
IIx(0;
P)II1 5 M
(P > 01,
SINGULAR INTEGRALS ON THE LINE GROUP
121
(3.1.6) Apart from (3.1.6) we sometimes use ( 8 > 0).
(3.1.7)
In contrast to the periodic case (see (1.1.6) and (1.1.7)) condition (3.1.7) does not imply (3.1.6). Another new feature is that in most of the examples the dependence of the kernel upon the parameter p > 0 takes the special form x(x; p ) = px(px). For this case it is easy to see that Lemma 3.1.5. {px(px)}dejnes an approximate identity for every x E NL'. For such kernels, which are said to be kernels of Fejir's type, the singular integral (3.1.3) is denoted by (3.1.8)
J ( f i x ; p) = Y d2a
Srn f ( x m
U)X(fU)dU
and called a singular integral of Fejir's type. Since every x E N L 1 generates a kernel (and even an approximate identity) via {px(px)},we shall abbreviate the notation and also call x itself a kernel. Concerning convergence of singular integrals we have Theorem 3.1.6. I f the kernel { ~ ( xp)}; of the singular integral (3.1.3) is an approximate identity, then (3.1.9)
lim
P. m
for every f
E
IIm";PI f(o)llxm)
=0
X(R). I f f E L", then for each h E L' r m
(3.1.10)
lim Pm
J
m
[I(f;x ; p )  f(x)]h(x)dx
= 0.
The proof is similar to that of the corresponding results on periodic functions and left to the reader. The same is true for the following Proposition 3.1.7. Let the kernel { ~ ( xp)}; of the singular integral (3.1.3) satisfy (3.1.5). Zflimo.+mIIZ(h; p)  h(o)llxcw)= 0 for all elements h of a set A c X(R) which is dense in X(R), then for every f E X(R) (3.1.11) lim I I U P ) f(o)Ilx(R, = 0. 0 ;
pm
0 ;
The problem now is to find a suitable dense subset A of X(R). To this end we have Proposition 3.1.8. For each a > 0 the functions of the form p(x)exp { ax2}, where p(x) is any algebraic polynomial, are dense in the spaces Coand Lp, 1 Ip < co. For a proof one may consult Problems 3.1.183.1.20. In order to specialize the last proposition to a more convenient form we introduce the Hermite polynomials
122
APPROXIMATION BY SINGULAR INTEGRALS
d" h,(x) = ( 1)" eYa {eXa} dx"
(3.1.1 2)
(nE
and the Hermite functions
Hn(x) = h,(x) exa'2
(3.1.13)
the elementary properties of which are left to Problem 3.1.3. In terms of the Hermite functions, Prop. 3.1.8 now reads (see Problem 3.1.3(iv))
Corollary 3.1.9. The Hermite functions form a fundamental set in the spaces Co and L P , 1 I p < co, i.e., the linear manifold generated by the Hermite functions is dense in these spaces. 3.1.2 Singular Integral of Fejer Let us consider the singular integral of Fejkr (3.1.14) with parameter p > 0. Putting (3.1.15 )
F is an even, positive function belonging to Con L', thus to We may write the integral (3.1.14) as
LP
for every 1 5: p I co.
(3.1.1 6) which thus defines a singular integral of type (3.1.8). Indeed, F belongs to NL' since
j y mu  sin2 ~ u du = r (compare Problem 5.2.8). According to Prop. 0.2.1, 3.1.3, Lemma 3.1.5, and Theorem 3.1.6 we have
Corollary 3.1.10. For f E X(W) the singular integral of FejPr exists for all x E R and p > 0 and defines a function in X(W) n C satisfying (3.1.17) (3.1.18)
IIG lim
o. m
IIdf;
O;
f)IIX(W)
5
Ilf IIX(R)
(P
> 01,
";P ) f(o)llx(W) = 0.
There is a close connection between the integrals of Fejdr introduced in (1.2.25) and (3.1.14). In fact, a wellknown result of the theory of meromorphic functions states that the series (3.1.19) converges absolutely and uniformly on every compact set of the complex plane which does not contain any of the points k r , k E Z (see Problem 3.1.13). It follows that (3.1.20)
sin2 ( n + 1)x sin2 x
sin2 (n + 1)x
(n E P)
SINGULAR INTEGRALS ON THE LINE GROUP
123
converges absolutely and uniformly on the real line. Therefore, beginning with the integral (1.2.25) for somefE Can,we obtain by the periodicity off sin2 ((n
1
+ l)u/2) du
k=m
the last integral converging sincefis bounded. Thus in terms of (3.1.14) un(f;x) = u ( f ; x ; n
(3.1.21)
+ 1)
(n E P).
This relation shows that the periodic integral (1.2.25) of FejCr can be classified under the integral u(f; x; p) of (3.1.14), the continuous parameter p > 0 being replaced by the discrete (n 1). But in the form (3.1.14), the periodicity off does not enter explicitly, and so, in studying the convergence of u(f; x ; p ) towards f as p + 00, the version (3.1.14) will also give a result for nonperiodic functions as well. Furthermore, (3.1.14) has the technical advantage that the analytical dependence on the parameter, in contrast to (1.2.25), is a very simple one. Indeed, the kernel of the integral (3.1.14) is generated by the function F ( x ) of one variable through the simple scale change
+
PF(PX).
In the following we shall see that there are many situations in which a possible 'reallineform' of a given periodic singular integral is more convenient. It is therefore very useful to know that with every kernel on the real line one may associate a periodic kernel. To show this, let g E L1. Setting (3.1.22)
g*(x) =
dZ
m
2
g(x
k=m
+ ZkT),
it follows that g* belongs to Lin. Indeed, by Prop. 0.3.3
and since the last term is finite, the series (3.1.22) converges absolutely for almost all E (7, 7r). Thus the resulting sum g*(x) exists almost everywhere, and since it is independent of the order of the terms of the series, it is periodic. Moreover, it follows by (3.1.23) that
x
Furthermore, i f f € X2', then (3.1.25)
'1' 2m
d~ 1
m
n
f ( x  u)g*(u)du = 
mf(x
Apart from these facts we shall need the following
 u)g(u) du
(a.e.).
124
APPROXIMATION BY SINGULAR INTEGRALS
Proposition 3.1.11. Let g E L1 n BV. Then g* is a 2nperiodic function of BVloo the series (3.1.22) being absolutely and uniformly convergent. Furthermore, (3.1.25) holds for all x, and thus in particular for every f E X,, (3.1.26) Proof. Setting
1
gk(x)
=
we have for n < m Since g E L1, the series 2; more
5
1
x+P(ktl)n
x t Pkn
g(u) du,
 1 gk(x)l converges uniformly on every finite interval. FurtherID
and thus
+
Ig(x + 2kr)1 5 Igk(x)l ek(X)s where ek(x) is the total variation of g over [x 2km, x 2(k 1)7r], Since g E BV, the series 2 ; ,,. ek(x) converges uniformly on every finite interval. Therefore d z 2;=  .,, g(x 2kr) converges absolutely and uniformly on every finite interval. Since its sumg*(x) is already known to be 2rperiodic, it follows that the series (3.1.22) is absolutely and uniformly convergent on the whole real line. Moreover; for any finite partition x = xo < xl < xa < . . c xn = r of the interval [ T,r]
+
+
+
+
.
Thus g +is of bounded variation on [r,T], in fact g* E BV1,, and (3.1.27) llg*lIn"2n 5 llgllnv. In particular, g* is a bounded function. Therefore the convolution (f*g*)(x) exists everywhere by Prop. 0.4.1, and (3.1.25) holds for all x in view of the uniform convergence of the series (3.1.22).
We proceed with the case that g is a kernel of FejCr's type (for general kernels see Problem 3.1.1 1). Thus, let x E NL' and set
x m
(3.1.28)
Then :x belongs to
m
=
%a 2 PX(P(X k=  m
3 2 W )
(P
'0).
Lin and
(3.1.29)
for every p > 0. As an immediate consequence we note that for every p > 0 and h E [w Ilx: 0 and deJines a function in X(W) n C" satisfying (3.1.40) (Y > Oh IIP(f; ";Y)IIX(W)5 II f IIX(R) (3.1.41) lim IIP(f; 0 ; Y )  f ( O ) I I X ( R ) = 0. u 0 + Similarly as for the singular integral of GaussWeierstrass the integral (3.1.38) also plays an important r61e in the theory of partial differential equations. As may easily be verified, it is a solution of Dirichlet's problem for the upper halfplane with prescribed boundary value f E X(W), thus of
(3.1.42)
For further details see Sec. 7.2.2.
SINGULAR INTEGRALS ON THE LINE GROUP
127
There is a close connection between the singular integral (1.2.36) of AbelPoisson, which is a solution of Dirichlet's problem for the unit disc, and the singular integral (3.1.38) of CauchyPoisson. Indeed, setting r = e#, the kernels {p,(x)} of (1.2.37) and {pp(px)} of (3.1.39) are again related via (3.1.28) according to (see Sec. 5.1.5) (3.1.43)
As we have seen in Sec. 1.2, the singular integral (1.2.36) of AbelPoisson for realvalued f may be regarded as the real part of a function F(z) holomorphic in the unit disc. The analog holds for the singular integral P ( f ; ;&of CauchyPoisson, too. Indeed, for any realvalued f E X(R) the function H(z) = 
(3.1.44)
7r
Im   z f(u)  u du
(z = x
+ iy)
= {(x, y ) X E R,y > 0} and H ( z ) = P ( f ; x; r>+ iQ(f;x; r) (x E R, Y > 01,
is holomorphic in the upper halfplane Ram+ (3.1.45)
where P ( f ; x; y ) is the integral (3.1.38) and Q ( f ; x; y ) is given by (3.1.46)
As in the periodic situation of (1.2.52) we call Q ( f ; x; y), also denoted by P  ( f ; x; y), the conjugate of the singular integral of CauchyPoisson. If we set
p(x) = d G x ( 1
(3.1.47)
+
x2)1,
then p"(x) is called the conjugatefunction (or Hilbert transform) of p(x). {pp"(px)} is not a kernel in the sense of Def. 3.1.1 since p" 6 L'. For further information about Q ( f ; x; y) we refer to Sec. 8.1, 8.2. There the definition (cf. (8.2.10)) of the conjugate I " ( f ; x; p) of a general singular integral Z(f; x; p ) will be given, followed by a detailed discussion of its properties. Problems 1. (i) Prove Theorem 3.1.6 for approximate identities satisfying the weaker condition (3.1.2) instead of (3.1.1). (ii) If the kernel is of Fejkr's type, prove Theorem 3.1.6 with the aid of Lebesgue's
dominated convergence theorem. (Hint: Use the estimate compare with the proof of Prop. 3.2.1; see also RUDIN[4, p. 1861) (iii) Let { be a sequence of functions in X(W) which are bounded and continuous in X(W)norm, uniformly with respect to n E N. If { ~ ( xp)} ; is an approximate llI(A;0 ; p)  fn(o)IIx(~) = 0 uniformly for n E N. identity, show that (iv) Let f be bounded and continuous, but not necessarily uniformly continuous. If { ~ ( x p)} ; is an approximate identity, show that Z(f; x; p ) = f ( x ) uniformly
fn}cml
on each bounded interval.
128
APPROXIMATION BY SINQULAR INTEGRALS
2. (i) Let &x) be defined through [(x) = exp { l / ( x a  1 ) ) for 1x1 < 1, = 0 for 1x1 2 1. Show that 5 belongs to Cro. (ii) Show that Cz0 is dense in Coo,that is, every function in Coocan be approximated uniformly by functions in C& (Hint: For any f E Coo consider the singular integralJ(f; x ; p ) of Fej6r's type with (positive) kernel x(x) = 5(x)/115111,5 being given as in (i). Show that J c f ; x ; p) E CG for each p > 0 and Jdf; x ; p) = f ( x ) uniformly for all x ; see also ZEMANIAN [ l , p. 31) (iii) Show that Cgo is dense in Co and Lp, 1 5 p < 03. (Hint: Use (ii) and Prop. 0.3.4) (iv) Let f E X(R) and suppose r:mf(u)+(u) du = 0 for every 4 E Cgo. Show that Ilfllxg, = 0, i.e.,f(x) = 0 (a.e.). (Hint: By (ii) one has j:mf(u)+(u) du = 0 for every 4 E Coo.By Prop. 0.3.4, rZrnf(u)+(u) du = 0 for every # E L' if X(R) = C, for every (b E Co if X(R) = L', and for every E Lp' if X(R) = Lp, 1 < p .c 03. Now use (3.1.33, for example) 3. Let h,(x) be the nth Hermite polynomial (3.1.12) and H,(x) the nth Hermite function (3.1.13). (i) Show that h,(x) = 2xhn'(x)  h:,(x) for each n E N. As obviously ho(x) = 1 , it therefore follows that hn is an algebraic polynomial of degree n of the form h,(x) = 2"x" 2;: akxk.In particular, hl(x) = 2x, hz(x) = 4x2  2 , . . . (ii) Show that for each n E N h:(x) = 2nhn1(~), hn+l(x)  2xhn(x) + 2nhn1(~)= 0, hfi(x)  2xh:(x) 2nhn(x) = 0. (iii) Show that Hi(x) = (xa  2n  l)Hn(x)for each n E P. (iv) Show that the Hermite functions form an orthogonal sequence on R, i.e., J?m Hn(x)Hm(x)dx= 0 for n # m. (Hint: Use (iii) and partial integration) The Hermite functions and thus the Hermite polynomials are linearly independent. (v) Show that I? Hz(x) dx = 2n JZm H,21(x) dx. (Hint: Partial integration and the first relation of (ii).) Therefore JZ H?(x) dx = 2/;2"n!. (vi) Show that N , , + ~ ( X=) XHn(X)  Hi(x) for each n E P. This recursion formula, together with Ho(x) = ex''a, determines the sequence of Hermite functions uniquely.
+
+
+
(Hint: For these elementary properties of Hermite polynomials and functions one may consult SZ.NAGY [S, p. 334 fl, HEWITI~TROMBERG [ l , p. 243 ff]) The following three Problems deal with counterparts to Sec. 1.1.2. Thus, if r is a natural number, we introduce the classes E C I fEC'} (3.1.48) W&a) = c f Lp ~ 1 f = 4 a.e., E AC;;', $ck) E LP, 0 Ik Ir} ( 1 I p < 03).
("
+
For p = 1 this is equivalently expressed by W[i = { f L'~ I f = 4 a.e., 4 E AC'l}. Analogously, the definitions of an rth strong derivative Df'f and an rth weak derivative DS'f of f~ X(W) are quite obvious: In Def. 1.1.9 and 1.1.11 one only has t o replace Xanby X(R), BVzn by BV, and L$ by Lp'. Sometimes, we consider functions x belonging to W: n Wli; in this case, we abbreviate the notation and write x E WtnLi. 4. (i) Show that f E W ~ Wimplies , the existence of the rth strong derivative Dp'f of f, which in turn implies the existence of the rth weak derivative Dy'foff. If X(W) = C, then f(')(x) = ( D f ' f ) ( x )= (D$)f)(x)for all x E R; if X(R) = LP, 1 Ip < 03, then 4(r)(x)= (D:'f)(x) = (D$)f)(x) a.e., 4 E ACIG' with #(k) E Lp, 0 Ik Ir, being such thatf(x) = +(x>a.e. (for the converse of the above statements the reader is referred to Prop. 10.5.3, Problem 10.6.16). (ii) Show that x E WinL1implies y, E W ~ for P every p > 1 .
SINGULAR INTEGRALS ON THE LINE GROUP
5.
129
(i) Let f E X(R) and x E NL’ be the kernel of the singular integral J ( f ; x ; p). Show that if x E W&1, thenJCf; x ; p) is an rtimes continuously differentiable function of x. In particular, for every x E R and p > 0 (3.1.49)
thus J ( f ; x ; p ) E WE for every f E X(R) and p > 0. (ii) Let the kernel x E NL’ of the singular integral J ( f ; x ; p) be bounded. Iff E W&R), show that J ( f ; x ; p) is an rtimes continuously differentiable function of x , and for every x E R and p > 0 (3.1.50)
+
[JW,
0 ;
p)](’)(x) = J ( D f ’ f ;x ; p) = J(+(r);x ; p), +(lC) E X(R), 0 Ik I r, is such that +(x) = f ( x ) (a.e.).
where E AC;&l with Moreover, (3.1.51)
lim
pm
II[J(f;
0 ;
p)I(Yo)
 (Df’f)(o)IIx(~)= 0.
(iii) State and prove counterparts for the (general) singular integral Zcf; x ; p) with kernel { x ( x ; p)}. (iv) As an application of (i) show that the Weierstrass integral W ( f ;x ; t) belongs to C“ for every f E X(R) and t > 0. Give applications to further examples. 6. Let f E W&(R, and g E L’. Show that the rth strong derivative (in X(R)) off * g exists and that D p ) l f * g ] = (Dp’f*g ) . (Prop. 10.5.3, Problem 10.6.16 indeed give f * g E W;(R,, in other words, if f E W;CR)and g E L’, then also f * g E W;(W,.) State and prove further counterparts to Problems 1.1.8, 1.1.9. 7. The first integral means (or moving averages) off E X(R) are defined as for periodic functions in (1.1.15). However, in accordance with our different notation for periodic and nonperiodic singular integrals we now employ the notation A ( f ; x ; h), thus
AW, x ; h) = h  1
jXt(”” f(u) du
=
h’
f(x
x  (hl2)
 u) du.
Show that A(f; x ; h) is a singular integral of Fejir’s type with even and positive kernel x ( x ) = ~ / % K [  ~ , ~ , and ~ ~ ~parameter ~ ( X ) p = h’, h  j 0 Hence for every f E X(R)
+.
lim IIA(f;
hOt
0 ;
h)  f(o)Ilx(nu = 0.
For further properties see the comments connected with (1.1.15). 8.
(i) Let { ~ ( xp)} ; be a kernel and the iteratedsingular integral Z2((f;x ; p) off E X(R) be defined by
Prove counterparts of Problem 1.1.4 for Z2(k x ; p). (ii) The rth integral means Ar(f; x ; h) off E X(R) are defined as for periodic functions h); x ; h). State and prove counterparts to Problems by Ar(f; x ; h) = A(Ar’(f; 0 ;
1.1.5, 1.1.7.
(iii) Since the kernel of the rth integral means has compact support, Ar(f; x ; h) is welldefined for every f E Lioc. For those functions f show that Ar(f; x ; h) E C;gl for each h > 0 and Iimh+ot A Y f ; x ; h) = f ( x ) a.e. (see also Prop. 0.3.1). If f E CLocr then limh,o+ Ar(f; x ; h) = f ( x ) for every x E R. 9. (i) Analogously to (3.1.3) we may assign to each function p E BV a singular integral by (3.1.52) 9F.A.
X(X

U ; p)
44~).
130
APPROXIMATION BY SINGULAR INTEGRALS
Show that if { x ( x ; p ) } is a kernel, then Z(dp; x ; p ) L'~ for each p > 0 and IIWP;0 ; p)lll 5 Ilx(0; p)IIIIlpllev. Moreover, if { x ( x ;p ) } is an approximate identity, then for every h E C (3.1.53)
lim
h(x)Z(dp; x ; p) dx =
0m
(Hint: Compare with Prop. 1.1.7) (ii) If the kernel is of FejCr's type, the notation will be
Show that if x E NL', then IIJ(dp; p)II1 9 llxlllllpllsv for all p > 0, and (3.1.53) is valid as well. 10. (i) Show that if g E L' is even (odd), then also g* as defined by (3.1.22) is even (odd). (ii) Let g E L1, and suppose that the ath absolute moment m ( g ; a) of g exists for some a > 0 (cf. (3.3.6)). Show that m ( g * ; a) c 00 (cf. (1.6.9)) and 0 ;
In particular, if x E NL1 satisfies m k ;a) < 00, then m(Xp*;a) I~  ~ r n (a) x ;for all xp* being given by (3.1.28). On this basis, examine the estimates of Lemma 1.6.4 once more. (iii) Let x E L1 and xp*,be defined through (3.1.28). Show that limp+" Ilxp*IILin = IIxIIL1. (iv) If x E NL1 is positive, show that {xp*(x)}as given through (3.1.28) is a positive (periodic) approximate identity. ; be a kernel (on the real line) and set 11. Let { ~ ( xp)} p > 0,
m
xp(x) = .\/%
(3.1.55)
2
x(x
k=m
+ 2kn; p).
Show that {xp(x)}defines a periodic kernel with parameter set A = (0, co) and p o = 00. llxPIIL;,5 Ilx(0; p)llLl for each p > 0. Furthermore, if { ~ ( xp );} is an approximate identity on the real line, then {x,,(x)} is a periodic approximate identity. 12. Let p E BV and define a function p* through
In particular,
p*(x) = .\/%
(3.1.56)
a
2
k i m
Show that p*
E
[p(x
+ 2kn)  p(2km)l.
BV2,,. Iff E Can,then
(Hint: See also KATZNELSON [I, p. 1341, BERENSGORLICH [I]) 13. Prove (3.1.19). (Hint: HILLE[41, p. 2611) 14. The singular integral of Jacksonde La Vallbe Poussin is defined through (3.1.58) with kernel N(x) = (3/.\/&)(~/2)~sin4 (x/2). Show that N E NL1 (compare with Problem 5.2.8). Thus {pN(px)} is an even, positive, continuous kernel of FejCr's type. Show that IIN(f; 0 ; p)IIx(~)IIlf Ilxm and limp+ IINf; 0 ; p )  f(o)IIx(w) = 0 for POUSSIN [3, p. 451, ACHIESER [2, p. 138, p. 3751, every f E X(R) (see also DE LA VALLBE BUTZERGBRLICH [2, p. 3781). 15. (i) Show that 3 2;  m ( z + kn)* = (2 + cos 22) s i r 4 z. (Hint: Differentiate (3.1.19))
SINGULAR INTEGRALS ON THE LINE GROUP
131
(ii) Replacing the continuous parameter p in (3.1.58) by the discrete n E N, show that the periodic singular integral N,*cf; x) of Jacksonde La Vallk Poussin as associated with NCf; x; n) through (3.1.28), (3.1.30) is given by
+
with kernel {N,*(x)}, N:(x) = (2n3)'(2 cos x)[sin (nx/2)/sin (x/2)I4. Show that N,*(x) is an even, positive trigonometric polynomial of degree 2n  1. Compare with the (original) singular integral of Jackson as defined in Problem 1.3.9. For the coefficients of the polynomial N,*(x) see Problem 5.1.2(v). Show that limnm IIN,*(f;  f(0)llxan = O for every f E XZn. 16. Let v E NBV. Show that for everyfe X(R) 0)
(Hint: Since Y E NBV, it follows by the HolderMinkowski inequality that
NOWuse the continuity off in X(R)norm and Lebesgue's dominated convergence theorem; see also the proof of Prop. 3.2.1) 17. Let the kernel { ~ ( xp)} ; of the singular integral I(f; x; p) be positive. Suppose that at some point xo E R lim I(ua;xo; p) = xt. lim Z(u; xo; p) = xo, o+
P+
"
Show that limo+" IIZ(f; 0 ; p )  f ( o ) I I X ( ~ ) = 0 for everyfE X(R). Apply to the singular integral of GaussWeierstrass. (Hint: Although x and xa do not belong to X(R), the hypothesis assumes that these two functions belong to the domain of definition of the singular integral under consideration, in other words, that the second moment m(x(0; p); 2) of x(x; p) (cf. (3.3.6)) exists for each p > 0. Hence proceed as in the proofs of Theorem 1.3.7, Prop. 1.3.9 and use the identity 1 x:  2xoZ(u; xo; p) I(u2; xo; p) = W u ; p) du, d 2 a OD which tends to zero as p +00. Thus { ~ ( xp)} ; satisfies (3.1.6), and the assertion follows by Theorem 3.1.6; see also Prop. 3.3.1) 18. (i) Show that the function enax, n E N, a > 0, can be uniformly approximated on 0 Ix < co by functions of the form p(x) e"#, where p(x) is an algebraic polynomial. (Hint: Suppose a = 1 and proceed via mathematical induction; see STONE[ l , p. 721, also Natanson [8II, p. 1551) (ii) For each a > 0, the functions of the form p(x) eax, wherep(x) is any algebraic polynomial, are dense in the spaces Co[O,co) and Lp(O, co), 1 Ip < 03. (Hint: Consider first the case Co[O,a),the space of all continuous functions f on [0, a) with limx,, f(x) = 0. Introducing a new variable t = eaX, the function 7 ( t ) = f((  1 /a)log t), ~(0) = 0 is continuous on 0 5 f I 1, so that the (ordinary) Weierstrass approximation theorem (cf. Problem 1.2.12) applies. Now use (i). In LP(0, co) the discussion may be reduced to functions f E Coo[O,00) (compare Prop. 0.3.4), and then use the result for Co[O, 0 0 ) ; see also STONE[l, p. 741 where the results are also derived from the general StoneWeierstrass approximation theorem) 19. The Laguerre polynomials are defined by (n E P) (3.1.60) In(x) = (l/n!) ex (d/dx)"[x" e#]
+
1"
132
APPROXIMATION BY SINGULAR INTEGRALS
and the Laguerre fimctions by L,,(x) = Z,,(x) e  x J a . (i) Show that f,,(x) = zg=o( l)k(;)xk//k!. Thus l,,(x) is an algebraic polynomial of degree n. In particular, fo(x)= 1, ll(x) =  x 1. (ii) Show that j," e"l,,(x)l,,,(x)dx = 0 for n # m, = 1 for n = m. Thus the Laguerre functions form an orthonormal system on [0,03).The Laguerre functions and polynomials are linearly independent. (iii) As an application of the previous Problem show that the Laguerre functions form a fundamental set in the spaces Co[O,00) and Lp(O, a),1 I p < 00. (Hint: Among others, one may consult TRICOMI [l, p. 212 ff], HELMBERG [l, p. 55 ff]) 20. Prove Prop. 3.1.8. (Hint: For a proof by an application of the general StoneWeierstrass approximation theorem see STONE[I, p. 791; this also contains a variant which makes appropriate use of the (ordinary) Weierstrass approximation theorem for functions continuous on a compact set of the plane. For reductions t o (generalized) Laguerre functions one may also consult NATANSON [81I, p. 1581, TRICOMI [l, p. 2391, compare also HELMBERG [ 1, p. 55 f f ] ) (3.1.61)
+
3.2 Pointwise Convergence In this section we establish results for singular integrals on the real line which we have already shown for periodic singular integrals in Sec. 1.4. In what follows we shall restrict our discussion to singular integrals of Fejdr's type. Analogous results for the general singular integral (3.1.3) are stated in Problem 3.2.1.
Proposition 3.2.1. Let f E Lm and x E NLl. (i) At every point xo of continuity o f f , limo+m J c f ; xo; p ) = f ( x o ) . (ii) r f f is continuous on (a  v, b 7 ) f o r some r] > 0, a < b, a, b E R, then Jw x ; p) = f ( x ) uniformly on [a, b]. (iii) Ifx is even and xo is such that lim,,o+ If(xo h) f ( x o  h)] = 2f(x0), then limo+m J ( f ; xo ; p) = f(x0).
+
+ +
Proof. We shall only prove (iii). We have J ( f i xo; p )
f(X0)
=
Srn 1 d2lr
0
[fh + p'u) + f(x0  p'u)  2f(xo)lx(u) du.
According to the hypotheses, the integrand is dominated by 4 (1 f Ilrnlx(u)Iand converges to zero pointwise for almost all u as p + 00. Hence the conclusion follows by Lebesgue's dominated convergence theorem. We now proceed with convergence almost everywhere. The first result is concerned with convergence at Dpoints (compare (0.3.3)).
Proposition 3.2.2. Let f E LP, 1 5 p I co, and x E NL1 be an even, positive function, monotonely decreasing on [0, co). Then at each point x for which
133
SINGULAR INTEGRALS ON THE LINE GROUP
thus for almost all x,one has
(3.2.2)
lim J c f ; x;p) = f(x). D+m
Proof. Since for each x > 0
and since the lefthand side tends to zero as x f a,one first of all obtains that lim xx(x) = 0.
(3.2.3)
xm
Setting
(3.2.4)
=
IOU [f(x + 0 + f(x
0
 2f(x)l dt,
then, according to (3.2.1),given any e > 0 there is a 6 > 0 such that IG(u)l 5 all 0 < u I8. Therefore, in view of
eu
for
+
12,
(3.2.5) J ( f ; x;P )  f(x) 
& (jod + /)f(x +
u)
+ f(x
 u )  2f(X)lX(fU) du = I1
we have by partial integration
x(u) du. Furthermore, for p say. Then 1; I2 If(x)l j,"d
whereas for 1 < p < co la' 5
{L4%
m 6
If(x
+ u ) + f(x
{!42;;
I""' {(pG)x(pS)}l'p
0. Similarly as in Sec. 1.5 we conclude for any
h>O
But for the second moment of the kernel we have (see Problem 3.1.17)
1
"
u2x(u;P ) du = Axe; P )  2xoS(xo; PI,
4%
which implies (3.3.2) by taking A' = d / r ( x o ;p )  2x0P(x0;p). According to Prop. 0.1.9 the assumption (3.3.1) implies that the singular integral Z(f; x; p ) converges to f in the metric of the space X(W) for everyf E X(W). In Sec. 1.3 and 1.5 we already remarked that the test functions which 'test' the approximation for the full class X(W) need not necessarily belong to X(R). Prop. 3.3.1 serves as an illustration to this fact. Of course, we then only obtain a sufficient condition for approximation in contrast to Theorem 1.3.7 which also gives necessary conditions. Before applying Prop. 3.3.1 to the singular integral of GaussWeierstrass we observe that if the kernel { ~ ( x p)} ; is also even, then Z(u; x ; p) = x , i.e., b(x; p) = 0. As is easily seen, we then have an estimate in terms of the generalized modulus of continuity given by
IIZ(f; : P )  f
(3.3.3)
O
(0)
IIX(R) = O(W*~X(W;f; l/y(xo ;P)))
(P
+
a).
Since the kernel of the integral W ( f ; x ; t) is even and since W ( u 2 ;x ; t ) = xa + 2t, we therefore obtain Corollary 3.3.2. For the singular integral W ( f ; x ; t ) of Gauss Weierstrass one has for every f E X(W)
IWM
(3.3.4) Zffurthermoref
(3.3.5)
E
O
;  f (0) IIX(R) = O(w*(X(R);f;
t1'2))
(t + O + ) .
Lip* (X(W); a )for some 0 < a I2, then
II W ( f ;
O;
t )  f(O)IIX(R)
=
O( t"'2)
( t + O + ) .
APPROXIMATION BY SINGULAR INTEGRALS
138
We conclude with a short discussion concerning asymptotic expansions of singular integrals on the real line. The methods of proof of Sec. 1.5.4 may be carried over to obtain analogous theorems for singular integrals Z(f; x ; p) of type (3.1.3), keeping in mind that Fourier coefficients are replaced by moments (compare Problem 3.3.4). Here we shall give a simple result concerning expansions for singular integrals of Fejtr’s type. If a is a positive number, the ath absolute moment of x is defined by (3.3.6) Proposition 3.3.3. Let x E NL1 be an even andpositivefunction for which m(x; 2) exists as afinite number. I f the second derivativef” o f f E L” exists at some point xo E R, then (3.3.7)
lim p 2 [ J C f ; xo; P) f(xo)l = (m(x; 2)/21f”(xo).
p.m
Proof. Since the kernel is even, we have (3.3.8) J ( f ; X ; p)  f ( x ) =
!” 1/28
0
[f(x + u)
+ f ( x  u )  2f(X)lX(PU)h.
Obviously, the hypothesis implies that f is continuous at xo. Therefore the quotient Hif(xo)/u2takes the indeterminate form O/O as u approaches zero. To eliminate this indeterminacy, the existence o f f ” at xo includes the existence off’ in a neighbourhood of xo. Therefore L’Hospital’s rule may be applied, and it follows that
the latter relation being valid by definition off”(xo). Hence f(X0
+
+Ax0
 u )  2f(xo) = U T ( X 0 )
+ U 2T ( U ) ,
being an essentially bounded function satisfying limu+o~ ( u = ) 0. Therefore by (3.3.8)
r]
P  ~ ( ~ G 2)/21f”(xo) (; + P2Rp, > 0, let 8 > 0 be such that Ir](u)I Ie for 0 Iu I8. Then =
say. Given E
the latter integral tending to zero as p f 00. Thereby relation (3.3.7) is established. Applying this result to the singular integral of GaussWeierstrass we obtain Corollary 3.3.4. I f the second derivative o f f E L“ exists at some point xo, then for the singular integral of Gauss Weierstrass there holds the asymptotic expansion (3.3.9)
SINGULAR INTEGRALS ON THE LINE GROUP
139
Let us finally return to Nikolskil constants for periodic singular integrals as introduced in Sec. 1.6.3. It is the aim here to determine these in the fractional case for the rather general class of singular integrals of FejCr's type.
Proposition 3.3.5. Let the periodic approximate identity {xp*(x)}be generated via (3.1.28) by x E NL1, and let Z,*(f; x) be the corresponding (periodic) singular integral (3.1.30). Suppose that x is even, positive and that the ath absolute moment m&; a ) exists as afinite number for some 0 < a 5 2. Then (3.3.10)
lim paA*h:; a ) = m(X; a). 0m
Thus the Nikolskii constant of the integral I,*(f; x ) for the class Lip* (Can; .)a is equal to the moment m(X; a ) of x.
Proof. Let f each p > 0
E
With J ( f ; x ; p ) of (3.1.8) it follows by (3.1.31) that for
Lip* (C28;
for all x E R. Therefore by (3.3.8)
I l Z 3 f i  f(o)Ilc,, 0)
=
IIJcf;
P)
 f1/1/;;.
to+
6. Establish Problem 1.6.12 as an application of Prop. 3.3.5; likewise for Problem 1.6.14 in case 0 < a < 1. (Hint: Use (3.1.36), (3.1.43)) 7. (i) Let the periodic approximate identity {xz(x)}be generated via (3.1.28) by x E NL1, and let Zp*(f; x ) be the corresponding (periodic) singular integral (3.1.30). Suppose that x is positive and that there exist constants fl, c with 0 < fl s 1, c > 0 such that x(x) = O(lxl'8) for 1x1 + co and
Show that &,*; p) = cp8 log p + O ( P  ~ )p, + co. Thus the NikolskiI constant of the integrals I,*(f; x) for the class Lip(C2,; fl)l is given by the constant c.
APPROXIMATION BY SINGULAR INTEGRALS
142
Bl and
(Hint: By the periodicity off, w(C2,;f;6) 5 m6 for every f E Lip(&; 6 > 0, and thus by (3.1.31) (compare also Problem 3.4.2)
llmf;
0)
 f(0)Ilc2,
5 ( p  w m
s”, lulBx(4
j3) 1 (l/d%)
On the other hand, &p*;
dl4
+ (m”i./23jIUI x ( 4 h. 2 P
12 sin (u/2p)lBx(u)du, and thus
E,
[lu/pIs  12 sin (u/2p)16]x(u)du; see also NEWEL [4]) where R(p) = (l/d%) (ii) Show that the Fejer and AbelPoisson kernels satisfy the assumptions of (i) with /3 = 1, c = 2/m. Therefore an application of (i) again delivers Problems 1.6.1 1 (i), 1.6.14 (case (Y = 1).
3.4
Further Direct Approximation Theorems
In this and the following section only singular integrals of FejCr’s type are considered. Results for the integral (3.1.3) are left to the reader. However, in contrast to the preceding section the kernel need not be positive.
Proposition 3.4.1. Let f E X(R) and x E NL’ be an even function for which the second absolute moment exists as a finite number. Then for the corresponding singular integral J ( f ; x ; p) one has
I J ( f ; ; PI  f (.I IIX(R)
(3.4.1)
O
=
ObJ*O((W ;f; P  ‘1)
(P
+
a).
Proof. It readily follows by (3.3.8), the HolderMinkowski inequality and Problem 3.3.1 that
$go + m
IIJ(f;O;
P)
f(o)Ilx(R) 5
Ilf(0
5 w*(X(R);f; p1)
u)
+f(o
 u )  2f(O)IIX(t3),,IX(PU)l du
Irn +
2 (1
1
dZ
0
pu)alX(pu)l du
{ dL 5 (1 + uYlx(u)l du} w * O w ) ; f ; p  9 , Lo
=
0
which proves the assertion. As an example, we treat the singular integral of Picard (cf. Problem 3.2.5).
Corollary 3.4.2. For the singular integral Cacf; x ; p) of Picard one hasfor everyf IICdf; ”; P )  f ( O ) I l X ( R ) = O(w*~X(W;f; P  3
Iffurthermorefe Lip* (X(R); a ) for some 0 < (3.4.2)
I Gdf;
0
;P )  f
(0)
a I2,
E
X(R)
(P
+
(P
+ 00).
00).
then
I1x(R) = O(P “)
143
SINGULAR INTEGRALS ON THE LINE GROUP
If the second moment of the kernel does not exist, but the first, a corresponding direct approximation theorem holds. But even this weaker hypothesis is not satisfied by many particular singular integrals. For instance, the Fejtr kernel (3.1.15) does not possess a first moment. Here the counterpart of Prop. 1.6.3 may be of interest.
Proposition 3.4.3. Let x E NL' be an even function for which the ath absolute moment m(X;a ) exists as afinite number for some 0 < a 5 2. Then for every f E Lip* (X(R); a) (3.4.3)
IIJcf; ";P )  f(0)IlXCR)
Indeed, iff
= O(P 9
(P
+
a).
Lip* (X(R); a)M.,then
E
5 (~/vz)M*
jOm ualx(pu)l du
=
( ~ * / 2 ) m &a)pa, ;
establishing (3.4.3). As an immediate consequence we state
Let
IIx@p,
=
0, 1,. . ., r, is such that +(x) = f ( x ) (a.e.). In 5 (2/r!)m&;r ) ll+(')lIx(R)P'.
Proof. We have
With 4 and R(x, u) as specified in this Lemma, we obtain by partial integration (3.4.1 0)
f ( x  u)  f ( x ) =
I
2 (+(k)(x)/k!)(u)k+ ~ ( xu), k1
(a.e.).
144
APPROXIMATION BY SINGULAR INTEGRALS
Thus (3.4.6) follows. Moreover, by an elementary substitution and the HolderMinkowski inequality
which implies that (3.4.12)
IIR(., u)llxca, 5 2
ll+~r~llx~R~l~lr/~~~
Substitution into the estimate .
IIJ(f;
(3.4.13)
0 ;
p)
 f(O)IlX(R)
m
I( p / d % )
IIR(0;
~ ) l l X ( R , l X ( ~ U ) ldu
delivers (3.4.8).
Proposition 3.4.6. (i) Let r E N and 0 I a I 1. x E Mr exists, then f E Wa&) implies
If the (r + a)th absolute moment of
(3.4.14) IIJcr; ";P )  f ( o ) I l x ~ ) = O(p('+')) (P 00). (ii) Let r = 2r, s E N, and 0 < a I2. If the (2s a)th absolute moment of the even kernel x E M' exists, then f E *W;i&, implies (3.4.14). (iii) Let r E N and 0 < a I2. I f x E Mrta, then for f E W;CcR, +
+
IIJ(fi
0;
p>
f(o>llx~)
I (2/r!)[m(x;r
+ 2) + m(x; r ) ~ p  ~ w * (P); ~ ( ~p11, );
+ being defned as in Lemma 3.4.5. In particular, f
*W;i&, implies (3.4.14). Proof. (i) SincefE W;&, there exists a constant E such that Il+(')(o  u)  +(r)(~)Ijx(R) B IuI" for all u E R. Therefore by (3.4.11) \ \ R e , u)llx(a I(lulr/(r  1)!)B
lutl"(1
E

f)rl
5
dt I( E / r ! ) (ul'?.
This implies by (3.4.13) that
(ii) Since x is an even function, it follows by (3.4.6) that J ( f ; x ; p)
 f(x)
= (p/d%)
An elementary calculation yields R(x, u)
+ R(x,  u )
=
Thus, if r = 2s, s E N,then IIR(0, u) + R(0, u)llx(P, I
1'
'41 (r  l ) !
1
14' (r  l)!
0
JOm
[+'"(x
ll$(')(o
M x , u)
+ R(x,
 u)lx(pu)
du.
+ ut)  +'"(x)
+ (l)'#Yx  ut)  (  l)'~jW(x)](l 
+ ut) +
#')(o
so '
ry1
dr.
 ut)  2+(')(o)Ilx(~,(l ty' dt.
Now the hypothesisfE *W?(&,yields the existence of a constant B* such that IIR(0, u)
+ N o , u)IIx(R)
I(lul'/(r  l)!)E*
which implies (3.4.14) as in (i).
lutl"(1
 V  l d t I(B*/r!)IuI*+",
145
SINGULAR INTEGRALS ON THE LINE GROUP
(iii) The proof follows by (3.4.8) as for (2.2.2). Indeed, if A 2 ( f ;x; h) denote the second integral means off(see Problem 3.1.8), we writef(x) = A 2 ( f ;x; h) g(x). Againfrz WLW) implies A2(f; x ; h) E WLT& and (j[A2(f; h)]('+2'(0)(IX(R) I h2w*(X(R); +(I); h). On the other hand, g E W;OW,,and since by Problem 3.1.5(ii) and the counterpart of Problem 1.5.5 [/[A2(+; h)  +(o)l'"(~)Ilxca = llA2(+(f);0 ; h)  + T O ) I I X ( R ) I w*o((R); +(I); h), Minkowski's inequality and (3.4.8) give
+
0 ;
0 ;
 f ( 0 ) I I X t W ) I IIJ(A2(f; 11); p)  A 2 ( f ; h)IlX(W, + IIm; p)  8(0)IlX(R) 2 I7 m(x; r + 2) l l [ ~ ~ ( +0;; h)](r+2)(o)IIX(~)p(f+2)+  m(x; r)w*(X(W); +(I); h)p'. (r + 2). r!
IIJ(f;
0;
p)
0 ;
0;
0 ;
0;
2
This implies the assertion by setting h = p'.
For applications of the preceding results we refer to Sec. 6.4. Problems 1. Formulate and prove the results of this section so as to obtain estimates in terms of the first modulus of continuity w(X(W);f; 6) (instead of w * ) , for Lip (X(R); a)classes (instead of Lip*), etc. 2. (i) Show that for every x E NL' and f E Lip (X(R); a)M
I J(f;
0;
p)
 f(0)IIxm
I
Mp"
lul"lx(4l du + 2
IlfllX(R,
J
Ix(u)l du,
Iulz p a
where a is a positive number which one is free to choose suitably (compare also H. S. SHAP~RO [I, p. 261). (ii) As an application prove that for the singular integral of Fejkr f E Lip (X(R); 1) implies Ilu(f;0 ; p)  f(o)IIx(m) = O(p' log p), p co. Show by an example that this estimate cannot in general be improved for the class Lip (X(R); 1). 3. Let f~ W$$, and x E N L ' be an even, positive function for which m(x; 2r + 1) < co. Show that f
11 J ( f ;
 k2 M x ; 2k)/fJ2"2k) !)d'2k'(o) IIX(P) m l I
0
;p)

f(0)
I[(m(x; 2r)
+ m(x; 2r + 1))/(2r)!]p*'w(X(R);
+car); p  l ) , where is given as in Lemma 3.4.5. Extend to kernels which are not necessarily even and positive. (Hint: See also BUTZER[6]) 4. Let x E NL' be such that m(x; 1) c co. I f f € L" is differentiable a t xo, show that
+
(Hint: Use Lebesgue's dominated convergence theorem and (see also H. S. SHAPIRO [I, P. 391)
5.
(i) Let x E NL1 be such that m(x; r) < co for some r E N. Show that for and p + 00
~OF.A.
YE W&(m,
146
APPROXIMATION BY SINGULAR INTEGRALS
where # is given as in Lemma 3.4.5 and y(x; k ) = (l/d%) J?m ukx(u)du. (Hint: Apply (3.4.10) and show that the lefthand side of the assertion may be estimated by (PWG)( + I I N o , u)Ilxtalx(pu)l du.
1 1)
IuISd
lulad
in the mean for the first, (3.4.12) and the Using (3.4.11)and the continuity of existence of m(x; r) for the second integral, the assertion follows) (ii) Let x E NL1be an even function for which rnk;2) < 00. Show that for f E W& lim IIpWf;
om
0;
p)
 f(0)l  b ( X ;
2)/2)#(2Yo)llxt~~ = 0.
Compare with Prop. 3.3.3. (iii) Show that for the singular integral of GaussWeierstrass o f f € W2tR, lim Iltl[ W ( f ;0 ; t )  f@)]  #t2)(o)llx(~)
= 0.
:0+
j,“~  ~ a f f ( du x ~exists ) as a Lebesgue integral at some point xo. Show that for the singular integral of CauchyPoisson
6. (i) Let f~ Lm and suppose that
lim P ( f ; xo; Y )  f ( x 0 ) = 1. Y’O+
Y
y
f(x0
+ u) + f(x0  u)  2f(x0) du.
= o
l4a
In case of the FejCr integral show that p[u(f; xo;p)  f(xo)] tends to the same limit as p + 00. (Hint: Use (3.3.8)and Lebesgue’s dominated convergence theorem for P ( f ; x; y ) , the RiemannLebesgue lemma (Prop. 5.1.2) for u(f; x; p); see also H. S. SHAPIRO [l, p. 40 f]) (ii) Let f~ X(R). Show that for the FejCr integral
[l], GOLINSK~ [2]) (Hint: EFIMOV[l], see also ZAMANSKY
3.5 Inverse Approximation Theorems Bernstein’s Theorem 2.3.3, particularly its method of proof, was the starting point for a series of investigations on the inference of structural properties of a function from a given order of approximation. In the present section we discuss one of these generalizations by considering the approximation o f f € X(W) by singular integrals J ( f ; x; p ) of Fejkr’s type.
Theorem 3.5.1. Let the kernel x E NL1 of the singular integral J ( f ; x ; p) belong to W& L1. If,for some CY > 0,f E X(W) can be approximated by J ( f ; x; p) such that (3.5.1)
IJM
O;
P)

f(O)IlXCR,
= 0bY
(P
+
a),
then (3.5.2)
w*(X(W);f; 8)
=
o xo, for example,
it follows that
SINGULAR INTEGRALS ON THE LINE GROUP
149
On the other hand, U,”)is an absolutely continuous function for each j = 0, 1,. . ., r  1. Therefore and letting l+ 00, the righthand side converges to [ g ,  l ( x )  g,l(xo)] for almost all x E R. Together with (3.5.11) this shows that R,&) = g, l ( x ) a.e., and sincego(x) = f(x) a.e. by (3.5.41, the assertion f E W:, is established (compare (2.3.8)). If E AC;&.l with +(’) E X(R) for each j = 0, 1, . . ., r is such that +(x) = f ( x ) (a.e.), I/+(r)  X Z 2U g ) I l x ( ~=) 0 by the preceding arguments. Thus II+(r)IIx(~) I then X = ZII UPIlxc~) and l +(r)@ + h) + + ( y o  h)  2+(‘)(0)IlX(R) 00
+ + U p ( 0  h)  2Ug)(o)Ilx(w)+ 4 2 IIUf)IlX(R). Sincex E Wt+naL1,Uf) E Weand llUf)(o + h) + Ug)(o  h)  ~U~)(O)IIX(R) IhaIIUf+a)Il~(~) for every h E R and k = 2, 3, . . .. Moreover, as in (3.5.10) 5
f
kP
/lUg)(o h)
ll~ft2)ll
X(R)
k=m+l
< 2 r t a + 1 IIX ~ ( r t a ) 1112k(aa). 
Combining the results, one has for every integer m 2 2 w*(X(R);+(f); 8) Ic 82
{
2 2uaa) + k  m2+ l k=2 m
2“”}’
where C is a constant. By considering different cases as in the proofs of Sec. 2.3, (3.5.9) follows. For many of the particular singular integrals so far considered it is easily seen that the kernels are of class C;, all derivatives being absolutely integrable. Therefore the previous results may be applied for any positive integer r. Corollary 3.5.3. Let W ( f ;x; t ) be the singular integral of GaussWeierstrass of f E X(W (i) For 0 < a < 2, f E Lip* (X(R); a) e 11 W ( f ; (ii)
feLip*O((W;2)*
0 ;
t )  f(o)IIx(w)
=
O(ta”),
t +O f ,
I I W ( f ; o ; t )f(o>Ilx(*, = W), t + O + .
The proof follows by Cor. 3.3.2 and Theorem 3.5.1. The inverse implication of (ii) is valid as well. But for this purpose other methods of proof must be employed which will be presented in Chapters 12, 13. There it is shown that an order of approximation beyond the critical value O(t) is impossible for nontrivial functionsf. Indeed, Prop. 12.3.2, 12.4.3, 13.2.1, and Problem 13.2.1 assert that 11 W ( f ;0 ; t )  f ( 0 ) IIxm) = o(t) as t +O + necessarily implies f to be constant (which must be zero in Lpspaces). Corollary 3.5.4. Let u(f; x; p) be the singular integral of Fejdr o f f E X(R). (i) For 0 < a < 1, llu(f;
0 ;
p)
 f(o)IIxcR,
 f E Lip (X(W; a), = 0bl) =.fE Lip* (X(W; 1)= O(p9
I I a 0 ; P)  f ( O ) I I X ~ , (ii) The proof is given by Cor. 3.4.4 and Problem 3.5.1. The result in (ii) is not best possible as is to be shown in Sec. 12.4.1, 13.2.5, where the exact characterization of the class of functions {f E X(R) I IIu(f; p)  f(o)Ilxm, = O ( p  l ) } will be given. For approximation by Fejkr’s integral the value a = 1 is the critical one. For Lp, 1 I p I 2, 0 ;
150
APPROXIMATION BY SINGULAR INTEGRALS
this will be a result of Sec. 12.3.1, where it is shown that IIo(f; 0 ; p) implies Ilfll, = 0.
 f(o)IIp
= o(p')
Problems 1. Let the kernel x E NL1 of the singular integral J ( f ; x ; p) belong to WI, nL1for some r E M. If, for some a 5 0, f E X ( R ) can be approximated byJ(f, x; p ) such that (3.5.1) holds, show that w,O((W;f; 6) =
{:;
o(srl h 3 811,
O < a < r a = r a > r
(6
+
O+).
2. Let the kernel x E NL1 of the singular integral J ( f ; x; p) belong to W'&lL1 for some r e N. If, for some a > 0, f s X ( R ) can be approximated by J c f ; x ; p) such that (3.5.8) holds, then f E W:(R). If 4 is as in Theorem 3.5.2, show that w(X(R); #r); 6) admits estimates as in the previous Problem for r = 1. 3. Let f~ X(R). Show that (i) for O < a c 1, f E Lip* (X(W); a) 9f E Lip (X(R); a), (ii) f e Lip* (X([w); 1) => w ( X ( R ) ; f ; 6 ) = 0(6 ]log 81). 4. Let NCf; x ; p) be the singular integral of Jacksonde La Vallke Poussin of f~ X(R).
Show that (i) for 0 < a < 2, f E Lip* ( X ( R ) ; a) * IIN(f; 0 ; p)  f(O)IIx(R) = O W " ) , p+ 03, (ii) f E Lip* ( X ( R ) ; 2) =. IIN(f; 0 ; p)  f(0)IIxm = O@a), p 03. (Hint: Apply Prop. 3.4.1, Theorem 3.5.1. For the inverse implication of (ii) see Problems 12.3.4, 12.3.5, and Sec. 10.6, 13.2) 5. Let Pcf; x; y ) be the singular integral of CauchyPoisson off E X(R). Show that (i) for 0 < a c 1, IIP(f; 0 ; Y ) f(o)IIx(~) = 0(ya) =fE Lip O W ) ; 4, +

(ii)
IIP(f; 0 ; Y )
 f@)llX(W) = O(Y) * f E Lip* ( X ( W ; 1).
3.6 Shape Preserving Properties
In this section we will give a brief account concerned with the following general problem: Assume that the graph of a function f has a certain shape; for example, assumefto be monotone or convex. What can be said about the approximation by a certain process 7 One result is that the approximation of a convex function f by the GaussWeierstrass integral is monotone. Furthermore, one asks for those approximation processes which assume the shape of the graph offas well. This problem will be discussed in terms of the variation diminishing property. Roughly speaking, an approximation process is called variation diminishing if the approximators do not oscillate more often about any straight line than the function to be approximated. The result is that a singular integral is variation diminishing if and only if the kernel is totally positive. 3.6.1
Singular Integral of GaussWeierstrass
The discussion of the problems of this subsection will be restricted to the representative example of the singular integral Wcf;x; t ) of GaussWeierstrass as defined in
SINGULAR INTEGRALS ON THE LINE GROUP
151
Sec. 3.1.3. Let W denote the class of all functionsf€ C,,,which grow to infinity in such a way that for each Q > 0 (3.6.1)
lim eaxa
=
Ixlrm
0.
Obviously, every algebraic polynomial belongs to W. For everyfE W and t > 0 the integral W ( f ;x; t ) is welldefined as a function of class Cloo(see also Problem 3.6.1). In particular, (3.6.2)
W(1;x; t ) = 1,
W ( u ; x ;t ) = x.
A realvalued functionfis said to be conuex on R if d i f ( x ) 2 0 for all x E R, h > 0 (compare Sec. 6.3.1). Note that a function convex on R cannot be bounded unless it is a constant.
Proposition 3.6.1. If the function f E W is conuex on R, then Wcf; x; t ) is convex on R for each t > 0. Indeed, one has
1
fliwcf;x; t ) = 
d i f ( x  u) eua'4tdu, d4pt  m so that the assertion follows since the kernal is positive.
Proposition 3.6.2. I f f € W is conuex, then W ( f ;x; t ) 2 f ( x ) f o r all x E R, t > 0. Proof. Since f is continuous and convex on R, there exists a righthand derivative f;(x) and a lefthand derivativefl(x) (cf. Problem 1.5.12 for the definition) for all x E R; these derivatives are monotonely increasing, andf;(x) = fL(x) for almost all x (compare (6.3.3), Problem 6.3.5). Let xo be fixed and choose c such that (3.6.3)
f'(x0)
=
{fL(x,) < c < f;(x,)
iff'(x,) exist, otherwise.
In view of the monotonicity of the derivatives one has (3.6.4)
c I f ; ( x ) for x > xo, c 2 f : ( x ) for x < xo.
For the first degree polynomialp(x) = c(x  x,) p(x) and thus
(3.6.5)
+ f(x,), (3.6.2) implies W ( p ;x ; i) =
W ( p ;xo ; t ) = f ( x 0 ) .
By (3.6.4) it follows that (cf. (6.3.3))
f ( x )  f ( x o ) =  /*'ffL(u)
du Ic(x  x,)
for x < x,,
X
and thereforef(x) 2 p ( x ) for all x E 08. Since the GaussWeierstrass kernel is positive,
APPROXIMATION BY SINGULAR INTEGRALS
152
this implies Wcf;x; t) 2 W ( p ;x; t ) , and thus Wcf; xo; t) 2 f ( x o ) by (3.6.5). Since xo is arbitrary, the assertion follows. Lemma 3.6.3. Let f E W. Then for each x E R and every fired pair 0 < 8 < r] (3.6.6)
Proof. Condition (3.6.1) implies that for each a > 0 there exists a constant c such that If(x + 4 1 5 c exp {2au2}for all u E R. Let a be fixed and choose t < 1/8a. Then
On the other hand,
Upon collecting the results, it follows that
<  c1 
 d a ) 14;
where c1 is a suitableconstant. Since the righthand side tends to zero as t +.O + ,this proves one of the assertions (3.6.6), the proof for the other being similar. With the aid of the previous lemma one proves by the usual technique (compare also Problem 3.6. I). Lemma 3.6.4. I f f
E
W, then lirnt,o+ W ( f ;x ; t) = f ( x ) for every x E R.
We may now turn to the converse of Prop. 3.6.2. Proposition 3.6.5. Let f E W. If W ( f ;x; t ) 2 f(x) for all x E R, t > 0,thenf is convex.
Proof. Assume that f is not convex. Then there exist points x1 < xa < x3 and a first degree (algebraic) polynomial p(x) which intersects f ( x ) in x1 and x3 but for which f ( x 2 ) > p(x2) (see Problem 6.3.5(v)). Defining the function g by g(x) = f ( x )  p(x), then g(xl) = g(x3) = 0 and g(x2) > 0. Let M = max,,,,,,, g(x) and y = max { x E [xl, x,] I g(x)
=
M},
z =
min {x E [xl,x3] I g(x)
If the first degree polynomial q ( x ) is defined by q(x) 4(Y) =f ( Y ) , q(x) 2 f ( x ) for x E [xl,&Ir Furthermore, let
= p(x)
f(x))
+
M}.
+ M , then
4 6 ) > f ( x ) for x E [XI,z)
for x E [xl,(xl + 21/21 U [(Y + x3)/2, 711= Y  xi, 6 , = Y  (xi ~ ) / 2 , qa = x3  Y , 8 2 = (x3 + mo = min (q(x)
=
u ( A %I.
 Y.
153
SINGULAR INTEGRALS ON THE LINE GROUP
Then
v1 > 6,
> 0, 712 > 6, > 0 and
4(x)  f ( x ) 2 0
d4  f ( x ) 2 mo
for x E [Y  81, Y + > 0 for x E [Y  71,Y 
821,
811
u [Y
+
82,
Y
+ 721.
This implies that [q(y  u)  f ( y  u)]
du 2
 nz
+ / “ ) [ q ( y  u)  f ( y  u)] eua/4tdu 61
2 mo{Jda eUa/4tdu +
[
eUa/4t}.
W2
Since q(x)  f ( x ) E W, it follows by Lemma 3.6.3 that there exists to such that for all O 0 for IuI > uo. Let x E R be fixed. Then
W ( f ;x;
1,) 
W ( f ;x ; t z )
= (4771’2
154
APPROXIMATION BY SINGULAR INTEGRALS
Setting u1 = x  uo, ua = x + uo, one has g(x  u) I 0 for u1 Iu 5 ua and g(x  u) > 0 for u < ul, u > ua. On the other hand, let p(u) be the (algebraic) polynomial of first degree which intersects f ( u ) in u1 and u2. Then for
h(u) = one has (3.6.7) Sincef is convex, it follows (cf. Problem 6.3.5(v)) that h(u) 5 0 for u1 I u S uz and h(u) 2 0 for u < ul, u > ua. Therefore h(u)g(x  u) 2 0 for all u E R, and thus (h * g)(x) 2 0. But
1
m
(47r)ll2
m
p(u)g(x  u) du = W(p; x ; t l )  W ( p ;x ; f a ) = p(x)  p(x) = 0.
This implies by (3.6.7) that
S_mmm h(u)g(x  u) du
= (uz
 ul)
lmm
f(u)g(x  u) du.
Hence J?m f(u)g(x  u) du 2 0 which proves the assertion, Proposition 3.6.7. Let f then f is convex.
E W.
If W ( f ;x ; tl) 1 W (f ; x ; fa) for allx E R and0 < t z < t l ,
Proof. This is an immediate consequence of Prop. 3.6.5; for letting t 2 tend to zero, the assumption implies by Lemma 3.6.4 that W(f ; x ; tl) 2 f ( x ) for all x E R and t l > 0. The results of this subsection may be summarized in the following Theorem 3.6.8. Let f E W, and W ( f ;x ; t ) be the singular integral of Gauss Weierstrass. (i) A necessary and suficient condition for f to be conuex on R is that W ( f ;x ; t ) 2 f (x)for all x E R, r > 0. (ii) A necessary and suficient conditionfor f to be conuex on R is that W ( f ;x ; t l ) 2 W ( f ;x ; ta)for all x E R and 0 < t2 < tl. 3.6.2 Variation Diminishing Kernels In this subsection it is always assumed that f belongs to the space C and that the kernel x E NL' of the singular integral J ( f ; x ; p) is continuous. The number u ( f ) of sign changes off ( x )on R is defined in the following manner : If x1 < x2 < . < x , is any finite increasing sequence of reals, let v(f(x,)) denote the number of sign changes in the finite sequence {f (x,)}. Then u ( f ) is defined by u ( f ) = sup u(f(x,)), the supremum being formed for all ordered finite sets {xr}.The singular integral J c f ; x ; p) is said to be variation diminishing if for every f E C and p 0 (3.6.8) @Cf; f ) ) 5 ocf).
=
0;
SINGULAR INTEGRALS ON THE LINE GROUP
155
In this case one also says that the kernel x is variation diminishing. The problem now is to characterize such kernels x. At first the elementary
Proposition 3.6.9. Let the continuous kernel x E NL1 be variation diminishing. I f f is monotone on R, then JCr; x ; p) is a monotone function of x for each p > 0. Proof. For any real
a
E
C
consider the relation
J ( f ; x ; p)  a = 
dG
m
If(x
 u)  a]x(pu) du.
Sincefis monotone,f(x)  a changes sign at most once. By the variation diminishing property of the kernel the same is true for J ( f ; x ; p)  a. Since this holds for every real a, J c f ; x ; p) is a monotone function of x for each p > 0. A realvalued, continuous kernel x E NL1 is said to be totally positive if for any < u, one has n E N and sets of reals x1 < x2 < . < x,, u1 < u2
0. Every totally positive function is positive (take n = 1 in (3.6.9)). Sometimes a totally positive kernel is called Pdlyafiequency; a positive kernel x E NL' is then calledfrequency. To prove the main result, we need some facts concerning variation diminishing properties of finite matrix transformations. Let A = (ajk) be a real (m, n)matrix. The linear transformation n
(3.6.10)
Yj
=
2 ajkxk
( j =1,2,...,
k1
m),
or the matrix A, is said to be variation diminishing if u(y,) I v(&) for any finite sequence of reals xlr. . .,xn. The matrix A is called totally positive if all minors of A of any order are positive?. If rank A denotes the rank of A which is supposed throughout to be different
from zero, then Lemma 3.6.10. If the matrix A is rotalIy posiriue, then v(y,) values of X I , . . .,Xn.
Irank
A
 1 for
all real

Proof. Consider first the particular cases that rank A = 1 and rank A = m 1. Let rank A = 1. Since A is totally positive, all elements of A are positive. Since rank A = 1, one of the linear forms (3.6.10) does not vanish identically and all the others differ from that one only by a positive factor. Thus v(y,) = 0 = rank A  1. Let rank A = m  1. Certainly, since y is an rndimensional vector, one has 10,) 5 m  1, and it is to be shown that equality is impossible. To this end, assume that there are XI,. . ., X n such that u(yJ = m  1 = rank A. The matrix A has at least one nonzero minor B of order m  1. Let Al, Aa, . . ., A,,, be the m minors of A of order m  1 which may be constructed from the m  1 columns of B. Obviously, B is one of these minors. With a property of rank one has 2El ( l)'+lA,y, = 0. As A is totally positive, all minors A, are positive; as v(yJ = m 1, the sequence {y,} alternates in sign. Therefore 2yXlly,l A, = 0. But this is impossible since no y , vanishes and since at least one A, is different from zero. Thus u(y,) Im  2 = rank A  1.

t
Recall that a quantity c is called positive if c 3 0.
156
APPROXMATION BY SINGULAR INTEGRALS
The proof of Lemma 3.6.10 now follows by induction. Suppose that the assertion is valid in case rank A = 1,2,. . ., 1  1, and let rank A = 1. One may assume that 1 < m ; for, if 1 = NI, then trivially v(y,) Im  1 = rank A  1. Suppose that the assertion is not valid for rank A = 1, i.e., there exist XI, . . ., X n such that v(y,) 2 1. Then one may select 1 1 linear forms of (3.6.10) such that u(y,,, . . . , y j , + J = 1. Let us denote this transformation by
+
n
y: =
2
k1
( j = 1 , 2 ,..., 1 + 1 ) .
aTkxk
Setting A* = (&, then certainly A* is again totally positive and rank A* Irank A = 1. However, if rank A* < 1, then o(y:) = 1 > rank A*, which is impossible by the induction hypothesis. On the other hand, if rank A* = 1, then the second particular case applies (with A replaced by A* and m by 1 1) which would imply 1 = u(y:) 5 rankA*  1 = 1  1, again a contradiction. Therefore u(yj) c 1 so that u(yJ I1  1 = rank A  1. This completes the proof.
+
Lemma 3.6.11. If the matrix A is totally positive, then A is variation diminishing. Proof. It is sufficient to prove the assertion in case u(y,) = m  1. For, if v(y,) < m  1, one may select m* = v(yJ + 1 elements of the sequence yl,. . .,ym and denote them by y:, . . ., such that v(y7) = v(yJ = m*  1. Then v(y:) Iv ( x k ) would imply o(y,) I &k), and thus the assertion. Therefore one may assume that the x l , . . ., x,, are such that v(y,) = rn  1, and one has to show that necessarily v ( x k ) 2 m  1. By Lemma 3.6.10 and the assumption it follows that m  1 = u(y,) IrankA  1. Therefore rank A = m, and in particular n 2 in. The proof now follows by induction on n. Suppose n = rn. Then no x k vanishes; for, if X k = 0 for some k, one would have a new system with m  1 columns, in contradiction to rank A = m. Since A is totally positive and the y j alternate in sign, it follows from det ( a j k ) # 0 and Cramer’s rule that the XI, alternate in sign as well. Therefore v ( x k ) = n  1 = m  1, and the assertion is shown for n = m. Now suppose that the assertion has been established for n = m, m + 1,. . ., 1  1, and let n = 1. Then there are two cases: (i) The x k alternate in sign; then v(&) = n  1 > rn  1, and the assertion is shown. (ii) The X k do not alternate in sign. But then either x , = O f o r s o m e s o f l ~ s ~ n o r x , x , + ~ > O f o r s o m e s~os f~ 0, then there exists X > 0 such that x , + ~ = Ax,. Setting for k < s Xk f o r k IS x: = x k + l f o r k s, u?k = + k + l fork = s for k > s, aj k + l one has v(xZ) = v ( x k ) and
,
n
7, =
2
k 1
n1 ajkxk
=
2
k1
a?k$
(j =
1, 2,.
. ., m).
Since the matrix A* = (a:k) is again totally positive but has only n  1 = 1  1 columns, the induction hypothesis applies, giving the assertion for n = 1. This completes the proof.
On the basis of the preceding lemma it is possible to prove the following
Proposition 3.6.12. Let the continuous kernel x E NL1 be totally positive. Then the singular integral Jcf; x ; p ) is variation diminishing.
157
SINGULAR INTEGRALS ON THE LINE GROUP
Proof. One may assume u ( f ) c co and u(Jcf; 0 ; p)) > 0, for otherwise the assertion (3.6.8) holds trivially. Let xo < x1 < . < x, be such that the numbers

J c f ; XO;PI, J d f ;
XI;
PI, 
 , J c f ; X m ; p)
alternate in sign, i.e., u(Jcf; x,; p)) = m. In order to show that m I u ( f ) , one has for every x E R
Obviously, the convergence is uniform for the m + 1 values xo, . . .,x,. Therefore one may choose a, b such that the function g(x), defined by
(3.6.1 1)
+
will also alternate in sign over the m 1 points xo, . x, (note that J ( f ; x; p) E C for eachfE C, p > 0 by Prop. 3.1.3). The interval [a, b ] is then subdivided into n equal parts by a = uo < u1 < . . . < u, = b so that the length of each subinterval is given by u k  u k  1 = ( b  a)/n. Considering the corresponding Riemann sum of the integral (3.6.1 l), the numbers g(x,), j = 0,1, . . .,m, are approximable by a ,
n
(3.6.12) for n+a. Hence one may choose n so large that also the numbers K,,, K ~ ..., , K, alternate in sign, i.e., u(K,) = m. Since x is totally positive, all minors of the matrix A = k ( p x ,  p u k ) ) of the transformation (3.6.12) are positive. Therefore Lemma 3.6.1 1 applies, giving u(K,) I u ( f ( u k ) ) . Since v c f ( u k ) ) I ucf) by definition and u(K,) = u(Jcf; x,; p)) by construction, it follows that u(Jcf; x,; p)) I u r n . Noting that the finite sequence of reals x, was arbitrary, this indeed implies u(J(f; p)) I u r n for each f E C and p > 0, and the proof is complete. 0 ;
Theorem 3.6.13. Let x E NL' be continuous. The corresponding singular integral Jcf; x; p) is variation diminishing fi and only if the kernel x is (up to the sign) totally positive. That total positivity is sufficient for the variation diminishing property is given by the previous proposition; for the necessity the reader may consult the literature cited in Sec. 3.7. Let us again consider the integral W ( f ;x; t ) of GaussWeierstrass. By Problem 3.6.5 the kernel w(x) = (1/2/2) exp { x2/4} is totally positive. Prop. 3.6.12 then gives that the approximation by the GaussWeierstrass integral is variation diminishing, i.e., u(Wcf; 0 ; t ) ) I ucf) for everyfE C, t > 0. Problems 1. Let f~ W. Show that W ( f ;x; t ) E Goo for each t > 0 and limt+,,+ W ( f ;x; t ) = f ( x ) for each x E R. 2. Let f E C and suppose that f does not reduce to a constant. Show that for each x E Iw
APPROXIMATION BY SINGULAR INTEGRALS
158
there exists a monotone nullsequence {t,} such that W ( f ;x; t,) < f(x) and W ( f ;x; t,) c W ( f ;x; t j + l )for every j E N. (Hint: Use Theorem 3.6.8 and the fact that a function convex on R cannot be bounded unless it is a constant; see also ZEGLER[l]) A kernel { ~ ( xp)} ; is said to be strongly centered at the origin if for each fixed pair of values 0 c 6 c r ]
Show that if { ~ ( xp)} ; is strongly centered at the origin and satisfies ( 3 . 1 3 , then it is an approximate identity. Show that the GaussWeierstrass kernel is strongly centered at the origin. Concerning the results of Sec. 3.6.1, state and prove a similar analysis for the singular integral of Picard, defined in Problem 3.2.5. Show that the GaussWeierstrass kernel w(x) of (3.1.33) is a totally positive kernel. (Hint: Obviously, det (w(x,  uk)) = 2"la det (e4I4 et%14
Now use P ~ L Y A  S Z E[lG11, ~ p. 491, see also SCHOENBERG [4]) (i) Let xl, Xa E NL1 be two totally positive functions. Show that the convolution x1 * Xa is again totally positive. (Hint: SCHOENBERG [4]) (ii) Show that the Picard kernel Cz(X) of Problem 3.2.5 is totally positive.
3.7 Notes and Remarks Sec. 3.1. The notion of an approximate identity on the line and Theorem 3.1.6 are standard; see, e.g., BOCHNER [6, p. 1 ff; 7, p. 57 ff], TITCHMARSH [6, p. 341, DUNPORDSCHWARTZ [lI, p. 218 ff], HEWITT[ l , p. 186 ff], and H. S. SHAPIRO [I, p. 10 f]. ACHIESER [2, p. 133 ff] apparently coined the term 'kernel of Fejtr's type' and in his treatment only assumed f(x)/(l xa) E L1; in the latter respect see also BOCHNER [7, p. 138 ff]. The concept of an approximate identity is connected with Friedrich's mollifiers; see FRIEDMAN [2, p. 274 f]. The results on singular integrals on the line group play an important r61e in the summability of Fourier inversion integrals to be treated in Chapter 5. In this respect the reader is also referred to TITCHMARSH [6, p. 26 ff]. The corresponding problems for periodic singular integrals were already touched upon in connection with summability theory of Fourier series. For Prop. 3.1.8, Cor. 3.1.9 see STONE[l, p. 78 ff]. Prop. 3.1.11 is due to G. H. HARDY, see also ACHIESER[2, p. 126 ff]. For this material see especially BOCHNER [6, p. 19 f ] and ZYGMUND [71, p. 681. The fundamental conversion relation (3.1.21) seems first to have been observed by DE LA VALLBEPOUSSIN [3, p. 30 ff] (see also Problem 3.1 15); he was also the first to recognize the importance of the property of (periodic) kernels to be of FejCr's type, particularly in solving periodic problems by unrolling them onto the real axis via (3.1.31) (compare the comments to Prop. 3.3.5). For further results leading to the Poisson summation formula see Sec. 5.1.5. Concerning Problem 3.1.4, there is a close connection between weak derivatives and distributional derivatives. A distribution (or generalized function) is an element of the set D' of all continuous linear functionals over D ( = C&), the set of Schwartz' test functions. The usual notation for the operation of f E D' on E D is , thus emphasizing that the distribution facts on the function considered as a function of the independent variable x. In this terminology, the translate f(x h), h E R,
+
+
+>,
+,
+
159
SINGULAR INTEGRALS ON THE LINE GROUP
+
of f~ D’ is then defined by ( f ( x h), +(x)> = ( f ( x ) , +(x  h)). A distribution f i s called regular if it is generated by some (ordinary) function f~ Ltocvia 1. Whereas a large part of finite Fourier transform theory may be subsumed under the L:,theory, not even the definition of the Fourier transform by formula (11.3) is directly applicable to everyfE Ls. A new approach is needed; it even turns out to be quite different for 1 c p 5 2 a n d p > 2. However, the resulting Lptheory has rather more symmetry than in the case p = 1. In particular, f andf" play exactly the same rBle in L2. Nevertheless, a large part of the theory of Fourier series and finite Fourier transforms is not only intimately connected with but very similar to the theory of Fourier transforms, and it is the purpose of this presentation to develop the common features. To this end, it is often useful to think of the periodic functions as defined on the additive group of real numbers modulo 27r or on the perimeter T of the unit disc of the complex plane. The theory of the finite Fourier transform is then often referred to as the harmonic analysis associated with this circle, or the reals modulo 2r. In the same way, the Fourier transform is associated with the additive group of real numbers. Harmonic analysis can be associated with a variety of domains. In particular, let us briefly consider the transform for functions defined on the group of integers. This leads to a rather simple but illuminating theory and provides additional motivation for the preceding. Consider the integers as a measure space in which each point has measure 1, and an integrable function f defined on this measure space; that is,f is a sequence {f(k)),"= of complex numbers such that < co, or f E I' in the notation of Sec. 0.3. The Fourier transform o f f € I' is then the functionf" whose value at x is
z?= [)@fI
2 m
(11.6)
f"(x)
=
k=m
f ( k )elkx.
166
FOURIER TRANSFORMS
Since this series is absolutely and uniformly convergent, the transform f" belongs to CSn. Furthermore, termbyterm integration is possible, and thus the inversion formula
is immediate; here we do not encounter any of the difficulties met above when trying to expressf in terms off ". At this point we call attention to the duality that exists between the interval [ n, n] and H in case of formulae (ILl), (11.2)for the Fourier transform associated with the circle group, between H and [ 71, n]in case of formulae (11.6), (11.7)for the transform on I', and between R and R in formulae (11,3),(11.5). This observation is important in the study of Fourier transforms on more general groups. The analogy between Fourier transforms associated with the different groups becomes especially apparent by comparing the theory on the Hilbert spaces of squareintegrable functions. Whereas Chapter 4 is concerned with the finite Fourier transform, Chapter 5 is reserved to the Fourier transform on the real line R. The material is presented in such a fashion that the parallel results of the two theories come to light. These results are analogous in statement and often in proof. To avoid repetition, however, emphasis is often laid upon different methods of proof. Chapter 6 is concerned with a detailed treatment of representation theorems. Necessary and sufficient conditions such that a functionf(k) on Z is representable as a finite Fourier or FourierStieltjes transform are given ;this problem is also considered for functionsf(0) on the line group. Moreover, rather convenient sufficient conditions for representation are supplied as well as a short account on (classical) multiplier theory. Chapter 7 is devoted to the first and bestknown application of Fourier transform methods, namely to the solution of partial differential equations. The technique is to be further developed and refined in Parts IV and V so that profounder and also more theoretical problems can be handled.
4 Finite Fourier Transforms
4.0 Introduction
The plan of this chapter can be outlined as follows: Sec. 4.1 is concerned with a detailed treatment of the fundamental operational properties of the finite Fourier transform, including the RiemannLebesgue lemma and convolution theorem. Discussion of the inversion problem is kept to a minimum since it turns out to be the convergence problem for Fourier series. Indeed, the theory of periodic singular integrals of Chapter 1 enters in when considering summation processes. Of particular importance in later applications is Theorem 4.1.10 on the transform of derivatives off. Sec. 4.2 is devoted to the special features of the finite Fourier transform in Lg,, p > 1. Although the definition of the transform for L:,functions also applies to X2,functions, nevertheless several of its important properties are only valid under the additional assumption YE Lg,, p > 1. The Parseval equation (4.2.6) and RieszFischer theorem for p = 2 and the HausdorffYoung inequality (4.2.15) for 1 c p < 2 can, for example, be mentioned in this connection. This section also contains a few words on the harmonic analysis associated with functions in Ip for p > 1. It may be regarded as precursory to Sec. 5.2, reserved to the definition and properties of Fourier transforms on Lp,p > 1. In particular, the clear and elegant results of Sec. 4.2 may serve as models of those to be expected in Sec. 5.2. Sec. 4.3 deals with the definition and properties of the finite FourierStieltjes transform, including a detailed inversion theory. The classes V[X2,; $(k)] are introduced, and the fundamental Theorem 4.3.13 is derived in case +(k) = (ik)’. 4.1 L:,Theory
Fundamental Properties We recall that the finite Fourier transform of a functionfE L,: defined on E, whose value at k is the kth Fourier coefficient
4.1.1
(4.1. I )
is the functionf,
FOURIER TRANSFORMS
168
Some of the elementary operational properties of this transform are collected in the following Proposition 4.1.1. ForfE L:, we have
+
(i) Lf(0 h)]"(k) = e"lcf^(k) (ii) [ei'"f(o)]"(k)=f"(k j ) (iii) uJlA(k) =f^(k)
(h E R, k E Z), ( j , k E Z), (k E Z).
+
The proof follows immediately by direct substitution into (4.1.1). Regarding the finite Fourier transform as a transformation from one Banach space into another we have
Proposition 4.1.2. Iff, g E Liz and c E @, then (9 Lf+ g l 7 4 =fW +g w , [CflW = C (ii) lim f"(k) = 0,
f W
(kE Z),
Ikl+m
(iii) f"(k) = 0 on Z impIiesf(x) = 0 a.e., (iv) there are sequences of class I; which are not thefinite Fourier transform of a function f E Liz, yet the set [L:,]" = (a E ;I I u = f " , f ~ Liz} is dense in IF. In other words, the finite Fourier transform defines a onetoone bounded linear transformation of L:, into (but not onto) .;1 In the following the set of all those elements of I; which are the finite Fourier transform of somefE X,, will be denoted by [X&.
Concerning the proof, part (i) being obvious, it follows from (4.1.2)
IfWl 5 llflll
(kE Z)
that the finite Fourier transform defines a bounded linear transformation of Lin into I". Moreover, we have for any k E Z, k # 0, 2nfA(k) =  C , f ( u
+ i)eIkudu = 2
1'
n
[f(u)  f(u
+ i)]e'kudu.
Hence (4.1.3)
(kE Z, k # 01,
and thus Lemma 1.5.2(iii) implies (ii) which is known as the RiemannLebesgue lemma. Property (iii), often referred to as the uniqueness theorem of the finite Fourier transform, is already given by Cor. 1.2.7, whereas the proof of (iv) is left to Problem 4.1.3.
Theorem 4.1.3. Iff, g E Liz, then (4.1.4) [f* g l W = f^(klg^(k) ( k E Z). Proof. By Prop. 0.4.1 we have f*g E Li,, and thus by Fubini's theorem for any kEZ
FINITE FOURIER TRANSFORMS
169
This proves the conoolution theorem which in particular shows that the finite Fourier transform converts convolutions into pointwise products and actually defines an isomorphism of the commutative Banach algebra L:, (with convolution as multiplication) into the commutative Banach algebra I$ (with pointwise multiplication). Moreover, in Sec. 0.7 we already mentioned that the Banach algebra L:, has no unit element. The convolution and uniqueness theorem of the finite Fourier transform enables us to give a simple proof of this fact. We have Proposition 4.1.4. L:, is a commutative Banach algebra without unit element. But there are approximate identities, thus sets of functions xp E L:, p E A, such that for eoery f E L:, limp+,,Ilf * x p f 111 = 0. Proof. Suppose there exists e E Liz such that e * f = f f o r everyf E L:,. Then, taking f ( x ) = exp (ikx}, k E Z, it would follow that e"(k) = 1 for all k E Z. But this would be a contradiction to the RiemannLebesgue lemma which asserts lim,kl+ae"(k) = 0. Theorem 1.1.5 and the various examples of Chapter 1 then complete the proof. In Problem 1.2.13 we saw that every uniformly convergent Fourier series of a function f E C,, represents f at each x. Let f E L:, be such that f " E I1. Then  ,f"(k) eikX converges uniformly and thus represents a function fo E C,,, i.e., fo(x) =  ,f"(k) elkX for all x . Moreover, for the Fourier coefficients of fo we have f ; (k)= f"(k) for every k E E , and hencefo(x) = f ( x ) a.e. by the uniqueness theorem. Therefore (for (ii) below see the Jordan criterion of Problem 1.2.10)
z:=
z;=
Proposition 4.1.5. (i) Let f E L:, be such that f n E 1'. Then
(4.1.5)
m
f(x) =
2
f"(k) etkx a.e.
k=m
Hence f is equal a.e. to a function in Czn.Iff (ii)
Iff E BV,,,
E
C2,,then (4.1.5) holds everywhere.
is 2~periodic,then for eoery x n
(4.1.6) It follows by direct substitution that (cf. Problem 1.2.14(ii))
2 f*(k)ck ejkX n
(4.1.7)
( f * t,)(x) =
k= n
and any complex trigonometric polynomial t, with Coefficients Ck E @. for everyf E We shall now give severaI extensions of (4.1.7). Proposition 4.1.6. I f J g E L:, andg is such that g"
E
II, then for eoery x
2 f^(k)gA(k)efkx. m
(4.1.8)
( f * g)(x) =
k= a
Proof. Since g" E II, Prop. 4.1.5(i) implies g(x) = go@)a.e. with go E C,,. Concerning the convolution f * g, we may replace g by go and, sincef * goE C,, by Prop. 0.4.1, we first o f all obtain that (f * g)(x)exists for all x and f * g E C,,.
FOURIER TRANSFORMS
170
According to Theorem 4.1.3 the righthand side of (4.1.8) is the Fourier series of f * g . Since it converges absolutely and uniformly and since f * g is continuous, the equality (4.1.8) again follows for all x by Prop. 4.1.5. Let us mention that under the hypotheses of Prop. 4.1.6 (4.1.9) In particular, iff = g, then (4.1.9) shows that the Parseaal equation
II f I L%,
(4.1.10)
I l f "ll la
=
E Li, for which f" E 1'. However, the latter relations are also valid under weaker hypotheses. Here we shall show
is valid for all functionsf
LA, and g E BV,, be 2~periodic.Then
Proposition 4.1.7. Let f
E
(4.1.11)
( f * g)(x) = nlim rm
n
2n f"(k)g"(k) eikx
k=
for all x . In particular, (4.1.12)
277
1',
f(u)g(u)du = lim nm
$
k=n
f"(k)g^O.
Proof. First of all we observe that, since g is bounded, (f* g)(x) exists everywhere by Prop. 0.4.1. For the nth partial sum of the Fourier series of g, namely S,(g; x) =  n g"(k) elkx, we have by Problem 1.2.11 that, for all x and n E N, IS,(g; x)l I I / ~ I I L ; , + 211gll~v,, = My say. Hence If(x  u)S,(g; u)l IM I f ( x  u)l uniformly for all n E N. Moreover, limn+mSn(g;u ) = g(u) for all u by Jordan's criterion (cf. Prop. 4.1.5(ii)). Therefore by Lebesgue's dominated convergence theorem
'sn
277
f ( x  u)g(u)du = lim
,
nm
277
1',
f(x  u)
$
k = n
g"(k) elkudu,
which proves (4.1.1 1). On replacing g(u) by g(  u), (4.1.1 1) implies (4.1.12) by setting x = 0. For a further set of conditions which ensure (4.1.8)(4.1.10) we refer to Prop. 4.2.2. If we begin with some sequence a E I' and define its Fourier transform ~"(x)by (11.6), then a" E C,, as we saw, and according to (11.7) we obtain for the finite Fourier transform of a" that [a"]"(k) = a(  k) for every k E E. Thus, if we apply Prop. 4.1.6, we have every f E L B (4.1.13)
( f * u")(x) =
and in particular (4.1.14)
277
In n
m
2
k=m
f(u)a"(u)
a( k)f"(k)
elkX,
m
du
=
C k=m
a(k)f"(k).
We shall refer to relations such as (4.1.9), (4. c. 14) as Parseval formulae.
FINITE FOURIER TRANSFORMS
171
Inversion Theory So far, given a function f E Li,, we have definedf " as a function on Z. We shall now study the inversion problem of the finite Fourier transform, in other words, if we know that a function in I; is the finite Fourier transform of somefe Liz, we wish to determine the original functionffrom the values off" on Z. According to (4.1.1), 4.1.2
f A ( k )=
1 $ 2n
,
f(~)e~~~du,
and the formal inversion would be given (see (4.1.5)) by
i.e., the inversion problem of the finite Fourier transform is nothing but the convergence problem for Fourier series. It is clear from the results of Chapter 1 that (4.1.15) does not hold in general but must be interpreted in some generalized sense as has already been specified in Sec. 1.2and 1.4. Although we only need to refer to the relevant sections of Chapter 1, we shall, for the reader's convenience, recall some of the results in the new terminology. In Sec. 1.2 we introduced 0factors (1.2.28) and summed the series (4.1.15) in the form m
u,(f;x) = k =2 m
(4.1.16)
e,(k)fA(k) etkX.
According to (4.1.13) we have (4.1.17)
V,(s; x)
=
277
1, ,
f ( x  u)Q(u)du
which is (1.2.31), since in view of (1.2.32) and (11.6) m
(4.1.18)
C,(x)
G
1
+ 2 2 BP(k)cos kx = Q(x). k= 1
Hence, if the assumptions of Prop. 1.2.8 are satisfied, we have for everyfe X,, (4.1.19) which may be regarded as a certain type of inversion formula for the finite Fourier transform. If we are more interested in recapturing the original function f by a pointwise limit, the results of Sec. 1.4 assure that for eachfE Liz m
(4.1.20)
lim PPo
2
k=m
0,(k)f"(k) efkn= f ( x ) a.e.
in case the hypotheses of Prop. 1.4.2 or 1.4.6, for example, are satisfied by the kernel
{c(x)}. For explicit formulae of the most important examples of 0factors we refer to Problem 4.1.4.
FOURIER TRANSFORMS
172
Let us finally observe that if the restrictive hypotheses of Prop. 4.1.5 are satisfied, then of course we need not introduce convergence factors. For a further result on inversion see Problem 4.1.5. Fourier Transforms of Derivatives In what follows r always denotes a natural number.
4.1.3
Proposition 4.1.8.
iff^ AC',il,
then
(k E H).
Lf"']"(k) = (ik)'f"(k)
Proof. If r
=
1, thenfis absolutely continuous and an integration by parts gives
+
277[f]"(k) = f(u) e'kul?,, ik
1"
f(u) e'kUdu = 27rikfn(k).
I
The result for general t follows by induction. Next we consider the converse of the latter assertion. Proposition 4.1.9.
Iffor f E Xan there exists g E X,,
(4.1.21)
such that
(kE E, k # 01,
(ik)'f"(k) = g"(k)
then f E Wk,, (for the dejinition see (1.1.16)). Proof. We set G,(x) = g(x),
(4.1.22) Gr(x) =
[" [G,,(x,)  G;,(O)]
J
dxl
B
dxr[f(o)
11.
+ g(xr)~* *
Then Gr E AC',; if we can show that G, is 27;periodic. Since g is 27;periodic, we have Gl(x + 27;)  Gl(x) = g(x,) dx,  27;g"(O) = 0, and thus G1 is 2wperiodic. If we now assume Gkl for some k with 1 Ik Ir to be 2aperiodicYthen again
Jz+2n
Gk(X 4 27;) = Gk(X) 4= Gk(X)
+
j:"'
lIz
[Gkl(U) Gk,(U) du
 G$l(O)]
dU
 27;Gc1(0)
=
G&).
Thus G, E ACL;' and Gr)(x) = g(x) (a.e.). By (4.1.21) and Prop. 4.1.8 it follows that
(ikyf"(k) = $(k)
=
[G?)]"(k)
=
(ik)'G;(k)
(kE Z,k # 0).
FINITE FOURIER TRANSFORMS
This implies If  G,]"(k) = 0 for all k ness theorem (4.1.23)
=
f ( x ) = const
173
k 1, & 2, . . .,and therefore by the unique
+ Gr(x)
(a.e.),
giving f E W;Ca,. It is now convenient to introduce the following notation : Let #(k) be an arbitrary complexvalued function on Z. Then W[X,,; #(k)]is the set of all functionsf€ X,, for which there exists g E X,, such that +(k)fA(k)= gA(k)for every k E Z,i.e., (4.1.24)
W[Xa,;
#@)I
= 1.f~X a n
I ~Wlf"(k)= gA(k),g E Xde
If we now combine the results of the last two propositions, we arrive at the following characterizations of the class WLa,.
Theorem 4.1.10. L e t f E Xa,. The following statements are equiualent: (i) f E W;C2,, (ii) .fe W[Xa,; (WI, (iii) there exist constants ak E @, 0 5 k 5 r
 1, and g E Xansuch that (a.e.)
The classes W;CZn and their various representations will play a significant r81e in our later considerations. In particular, the fact that the finite Fourier transform converts differentiation to multiplication by (ik) makes the finite Fourier transform a useful tool in the study of differential equations as we shall see in Chapter 7. Problems 1. (i) Let f~ L;, Show thatf"(k) is an even (odd) function on Z if and only iff(x) is an even (odd) function on [ n, TI. (ii) Let f,f, E L;, be such that limn+m]If fnlll = 0. Show that lim,,,f;(k) = f"(k) uniformly for k E Z. (iii) Let f~ Xan.Show that I f^(k)l 5 llfllxln for all k E Z. 2. Let f~ X,, andf^(k) = 0 for Ikl > n. Show that f(x) = t,(x) (a.e.) with 1, E T,. 3. Prove Prop. 4.1.2(iv). (Hint: H~wrrr[l, p. 161 examines the example a(k) = [log k]1 for k = 2, 3 , . . ., = 0 otherwise; see also RUDIN[4, p. 1041). 4. (i) Give examples of functions f~ Ll, for which f" 4 I' (cf. Problem 4.1.5(i)). (ii) LetfE X2,. Show that the Fourier series offis Ceslro, Abel, and Gauss summable to f (almost everywhere), i.e.
In particular, all these relations are valid at each point of continuity off:
174 5.
FOURIER TRANSFORMS
(i) Let m*(x) be the 2rperiodic function which is defined on [ r, 7l] by ~ ~ K ~  I . O ] . K ~  being ~ . ~ the ~ characteristic function of the interval [ l,O]. Show that [m*]"(k) = (ik)I(etk  1) for k # 0, = 1 for k = 0. (ii) Let 0 < h < n. Show that for f E X,, f(u) du =
+
j"
f ( x  u)m*(u/h) du = (f*rn*(./h))(x) 271 ,
and [j!f(. u) du]"(k) = (ik)l(eihk  l)f"(k) for k # 0, = hf"(0) for k = 0. (iii) LetfE Liz. Show that J'tf(x u) du (as a function of x) is locally of bounded variation and
+
for all x and h. In other words, the Fourier series of a function f c L;,
may be integrated term by term (whether the Fourier series itself is convergent or not), i.e.
and hence for almost all x
(Hint: Apply Prop. 4.1.7 with g replaced by nt*(o/h) or, to be independent, use Jordan's criterion; see also HARDYROGOSINSKI [ I , p. 301, ASPLUNDBUNGART [l, p. 4361, EDWARDS [II, p. 921) 6. Show that W[XZn;(ik)'] c W[X2,; (ik)'] for everyj = 1, 2, . . ., r  1. 7. (i) Let f E Lip(X,,; a), 0 < CL I 1. Show that f " ( k ) = O(lkl"), k + m. (Hint: Use (4.1.3) and Problem 4.1.1(iii)) (ii) Show that 41f"(k)l I w*(X2,;f; r / l k l ) for every f~ X,, and k # 0. Thus, f A ( k ) = O( Ikl  a ) , k 00, for every f E Lip*(Xzn;a), 0 c a I 2. (iii) L e t f E BV,,, be 27rperiodic. Show thatf"(k) = O( lkl'), k  t m . (Hint: Compare with Problem 1.2.I I ) (iv) L e t f E AC,,. Show thatf"(k) = o( Ikl'), k +m . (Hint: Use Prop. 4.1.2(ii), 4.1.8; see also HARDYROGOSINSKI [ 1, p. 261) f
4.2 Li,Theory,p > 1
The Casep = 2 Having considered the finite Fourier transform mainly as a transform on Liz, we shall here establish further results in case the functions in question are squareintegrable, for example. It will be seen in particular that the inversion problem is completely solvable in Li,space and, as a matter of fact, solvable without the introduction of convergence factors. We recall that the definition of the Fourier transform for L?j,functions as given in (4.1 .I) also applies to X,,functions. The following proposition deals with an interesting minimal property of the partial sums S,(f; x) of the Fourier series off.
4.2.1
~roposition'4.2.1.Let f e L:,. (4.2.1)
Then for any t,
1I &(s;
0)

E T,
S(0)II2 5 I rn(o)
(with coeficients
 f(0)II2,
Ck
E C)
175
FINITE FOURIER TRANSFORMS
equality holding if and only i f ck
= f"(k),
(4.2.2)
(kl 5 n. Furthermore IIfllLaa,.
IlfA1112
The latter inequality is known as Bessel's inequality.
Proof. Using the Hilbert space notations of Sec. 0.7 we have IItn
fll%
= (tn
f, tn
f) = (tn,
tn)
 (f,tn)(f,
tn)
+ IIfll%
Moreover n
IItn
flli 
IIsn(f;
0)
f(o)Ili
=
2 k=
n
Ick
fA(k)I2,
and thus (4.2.1)is established. (4.2.2)is an immediate consequence of (4.2.3). Next we establish assertions (4.1.8)(4.1 .lo) under different hypotheses,
Proposition 4.2.2. I f . f , g E L&, then
(4.2.4) for all x, the series being absolutely and uniformly convergent. In particular,
(4.2.5) (4.2.6)
llfllLS.
=
llf"lIl~*
Proof. By Prop. 0.4.1 we obtain that f * g E C2n,and by Holder's inequality that the series in (4.2.4)converges absolutely and uniformly. Since it is the Fourier series of f* g by Theorem 4.1.3,(4.2.4)follows by Prop. 4.1.5. Relations (4.2.5)and (4.2.6), known as the generalized Parseval equation and the Parseval equation, respectively, are now easy consequences of (4.2.4). As an immediate application of (4.2.3)and (4.2.6)we have Proposition 4.2.3. I f f € Lin, then
According to Prop. 1.2.3and Theorem 1.3.5,the analog of the latter proposition in L:,space is not valid. Therefore, in order to produce convergence of the Fourier series, thus to have inversion of the finite Fourier transform on L&, we either supposed the functions to be smooth enough (see e.g. Prop. 4.1.5)or we introduced a summation
176
FOURlER TRANSFORMS
process. But for p = 2, the nth partial sum of the Fourier series always converges to the original function in the mean of order 2, no convergence factor being needed. Although the Dirichlet kernel {Dn(x))is not an approximate identity for L:,space, it behaves like one for Li,space, and even for L;,space, p > 1, as we shall see in Sec. 9.3.3. Up to the present we began with a functionfe L& and formed its finite Fourier transform obtaining a function f E la.The question arises whether every element of lais representable as the finite Fourier transform of a functionfE LX,, that is to say, whether the finite Fourier transform defines a bounded linear transformation of L&, onto la. The answer is affirmative and given by the following theorem of RieszFischer.
Theorem 4.2.4. Let
a E la.Then there exists f E L,!
such that
a(k) =fW
(4.2.8)
and (4.2.9)
IlallIP = IIfllLB,
=
IIf"ll,a.
Proof. If we set
then we obtain for m > n
Thus the functions sn(x)form a Cauchy sequence in LX, and the completeness of LE, implies that there is f E Li, such that lim /Isn fII
(4.2.1 1)
n m
=
0.
Let k E I! be arbitrary and choose n > lkl. Then by Holder's inequality
If"(k)  4k)l
=
I&
MU) s n ( ~ > Ie'''
1
du 5
I l f  SnIILf,,
and (4.2.8) follows by (4.2.1 1). Moreover, this establishes the theorem by (4.2.6). Combining the results so far obtained for squaresummable functions we may state Corollary 4.2.5. The finite Fourier transform defines a bounded linear transformation of the Hilbert space L&, onto the Hilbert space la which preserves inner products, i.e. (4.2.12)
(6 g) = Uh,g")
for any f, g E L%n* Thus the map f +f" defines an isomorphism of the Hilbert space L& onto the Hilbert space 1'.
177
FINITE FOURIER TRANSFORMS
4.2.2 The Casep # 2 Until now we discussed the finite Fourier transform of functionsJE Lg, forp = 1,2. The transform is a function on Zbelonging to lp'. In particular, by (4.1.2) and (4.2.6), (4.2.13)
I f 71IS
(4.2.14)
=
I f II L
L
respectively. It is natural to inquire whether these results can be extended to exponents other than 1 or 2. This is partially possible by the M. RieszThorin convexity theorem.
Proposition4.2.6. Let 1 < p < 2 and f
E
LK,. Then f "
llfA1ll~~5
(4.2.15)
E
Ip' and
IIfllLL.
The assertion of this proposition is referred to as the Hausdorf Young inequality; in fact, this phrase will also be used to cover the cases p = 1 and p = 2 of (4.2.13) and (4.2.14).
Proof. To apply the convexity theorem (cf. Sec. 4.4), let R1 = ( T,T)with ordinary Lebesgue measure and R2 = Z where, in the usual way, Z is considered as a measure space in which each point has measure 1. For f~ LK, let T be defined as the finite Fourier transform: Tf = f". Then, since S,, C LK, it follows by (4.2.13) and (4.2.14) that 11 Th 1 I I 11 h /I,:L and 1 Th 11 ,a = 11 hll Lgx for every h E S,,. Thus T is of strong type (I ; a),(2; 2) on S,, with constants M I = M 2= 1, and (4.2.15) follows for every h E S,, by the M. RieszThorin convexity theorem. Let f E Lg, be arbitrary. Then (see Sec. 0.4) there exists a sequence {h,} of functions in S,, such that limj,m h,llL;. = 0. By Holder's inequality 1f"(k)  h;'(k)l 5  h,ll Lgn, k E Z,and therefore limj.+ h;'(k) =f"(k) for each k E Z. Hence by Fatou's lemma
]If
/If
where we have used the fact that (4.2.15) is already valid for functions in S2,. Thus (4.2.15) is completely established. The restriction to 1 5 p I 2 for (4.2.15) to be valid is essential. There isfe C,, such that 11f''lIlq = co for all q < 2 (see Problem 4.2.2). The HausdorffYoung inequality (4.2.15) may be regarded as an extension of the original Parseval equation (4.2.6) to exponents 1 < p < 2. But, in view of the RieszFischer theorem, the Parseval equation contains a further assertion, namely, for any a E l2 there is f E La,, such that (4.2.8) and thus (4.2.9) holds. Regarding this aspect we have the following
Proposition 4.2.7. Let a E Ip, 1 < p < 2. Then there exists f (4.2.16) and (4.2.17) 12F.A.
a(k)
=fW
E
Lgb such that
(kE Z)
178
FOURIER TRANSFORMS
Proof. If sn(x) is defined by (4.2.10), then we have by the Holder and HausdorffYoung inequalities for every h E Lg,
This implies by (0.8.5) that IIsn IIL
(4.2.18)
I ~I
I all 1'.
Since (4.2.18) is valid for each a E IP and each n E N,it follows that for m > n
Thus the functions sn(x)form a Cauchy sequence in Lgh, and hence the completeness of L$h assures the existence o f f € Lqi such that (4.2.19)
which together with (4.2.18) implies (4.2.17). Moreover, if k E Z is arbitrary and n E N such that n > lkl, then by Holder's inequality
and (4.2.16) follows by (4.2.19). Concerning the casep = 1, we recall the introduction to this Part. Indeed, to every a E I' one may assign the continuous function f(x) = xp=  a(k) elkx. Since this series converges uniformly, one easily deduces that a(k) = f"(k) and I(f((C2,I l[allIi. Again the restriction of Prop. 4.2.7 to 1 Ip I 2 is essential. For there is a sequence {a(k)}that belongs to l4for all q > 2 and yet is not the finite Fourier transform of any function in Liz (cf. Problem 4.2.3). In the introduction to this Part we defined the Fourier transform on 11, and the question arises whether it is possible to define a Fourier transform on Ip for other values ofp. Since lp c Iq for q > p, but not conversely, definition (11.6) does not apply. But Theorem 4.2.4 and Prop. 4.2.7 give us the feasibility of the following
Definition 4.2.8. The Fourier transform a" of
a E Ip,
In other words, the Fourier transform of quely determined function a" E Lgb given by
aE
n
(4.2.20)
lim n+ m
IIu"(0)

1 < p I 2, is defined by
Ip, 1 < p
I2,
2 n a(k) efkaIlp.= O
k=
is defined as the uni
179
FINITE FOURIER TRANSFORMS
(see (4.2.1 1 ) and (4.2.19)).It is an easy consequence (cf. Prop. 0.1.10) that the definitions (11.6) and (4.2.20) are consistent for a E I1n IP. The last few remarks are important in so far as they shall lead us (cf. Sec. 5.2) to the solution of the corresponding problems for the Fourier transform associated with the real line.
Problems 1. Let f, g E Ltn. Show that fg E Lgn and Lfg]"(k) = zT= m f"(j)g"(k  j ) . (Hint: [ 1, p. 231) HARDYROGOSINSKI 2. (i) Show that there are functionsf€ Czn such that Ilf"II,q = 03 for all q < 2. (Hint: ZYGMUND [7II, p. 1011) (ii) LetfE LZn, 1 < p < 2. Show that ~ ~ f " ~ ~ ,=p * IlfIILpn if and only iff(x) = c exp {imx} for some c E C and m E Z. (Hint: HARDYLITTLEWOOD [I], see also HEWITT[ I , P. 1101) 3. (i) Show that there are sequences {a(k)},", m that belong to Iq for all q > 2, yet are not the finite Fourier transform of any function in Lln. (Hint: ZYGMUND [S, p. 190, 711, p. 1021, HEWITT[l, p. 1101) (ii) Show that equality occurs in (4.2.17) if and only if a(k) = C8m.k for some c E @, m E Z, where 8 m . k is Kronecker's symbol. (Hint: HEWITTHIRSCHMAN [l], see also HEWITT [I, p. 1101) 4. Let f be a 2nhperiodic function belonging to L2(h, A), A > 0. Show that 22 If"(k)la = (1/2rA)J?,, If(u)la du, wheref" is defined by (11.4).
4.3 Finite FourierStieltjes Transforms
4.3.1 Fundamental Properties In Sec. 1.2.7 we introduced the kth complex FourierStieltjes coefficient of a (complexvalued) function p E BV,, by
(4.3.1)
!" , 1
p"(k) = 2n
ejkUdp(u)= b(o)]"(k)
the integral being a RiemannStieltjes integral. For every? p E BV,,, (4.3.1) defines a function p" on Z, called the$nite FourierStieltjes transform of p. We are going to derive some of its operational properties. Proposition 4.3.1. For p E BV,, we have (i)
b(o
+ h)]"(k)= eihkp"(k)
(h E R, k E Z), (k E p(n), f E Liz, then
(ii) [,G]"(k) = p"(k) (iii) if p is absolutely continuous, i.e. p(x) = s"_,f(u) du p"(k) = f^(k)for all k E Z, (iv) i f p is 2nperiodic, then p E L:, and (ik)p"(k) = p"(k) for all k E Z.
+
a
t Obviously, the integral defining p'(k) is also meaningful for every p E BV[  'II,n]. This will sometimes be used though some of the very elementary properties may then fail to hold (e.g Prop. 4.3.l(i)).
FOURIER TRANSFORMS
180
The proof is left to Problem 4.3.1. Further information about the finite FourierStieltjes transform is given by
Proposition 4.3.2. (i) (ii) (iii) (iv)
Ifp,
Y E
b + v]"(k) = p"(k)
BVan and c E 62, then
+
v"(k), (kE Z), [cpl"(k) = c p W (k E Z), IP"(k)l 5 ll~llev,, p"(k) = 0 on Z implies p(x) = const, there are sequences of class I" which are not thefinite FourierStieltjes transform of a function p E BVa,.
Concerning the proof, parts (i) and (ii) are immediate consequences of the definition (4.3.1), whereas (iii) follows by Prop. 1.2.12. Indeed, for the FejCr means of the corresponding FourierStieltjes series the hypothesis implies u,,(dp; x ) = 0 for every n E N. Therefore (1.2.59) yields h(x) dp(x) = 0 for every h E C,, from which (iii) follows by Prop. 0.6.1. For the proof of (iv) we refer to Problem 4.3.2. Part (iii) of the foregoing proposition only reveals a partial uniqueness property for the finite FourierStieltjes transform, since we did not normalize the function p E BV,, by p (  ~ )= 0. But if we regard p as an element of the Banach space BVg,, then p"(k) = 0 on Z implies that p is the zeroelement of BVg,, and thus we would have uniqueness. Hence the finite FourierStieltjes transform defines a onetoone bounded linear transformation of the Banach space BVg, into (but not onto) I". The RiernannLebesgue lemma does not hold for the finite FourierStieltjes transform. In fact, the FourierStieltjes transform of ti* E BV,, as defined on [P, T] by
cn
,;;{
(4.3.2)
0,   n < x < O
8*(x) =
x=o o<xs7r
is given by
[S*]"(k) = 1
(4.3.3)
(k E Z).
There are also continuous functions p E BV,, for which p"(k) does not tend to zero as Ikl f m.
Proposition 4.3.3. Let f by (0.6.8)) we have
E
L;,
pE
BV,,.
Then for the convolution (f* dp)(x) (defined
Lf * d P 1 W = f W P " ( k ) (kE Z). Proof. In view of Prop. 0.6.3,f * dp E,:L and thus both sides of (4.3.4) are meaning(4.3.4)
ful. Furthermore, by Fubini's theorem
establishing (4.3.4). For the proof of the general convolution theorem for the finite FourierStieltjes transform, which is to follow, we need
181
FINITE FOURIER TRANSFORMS
Lemma 4.3.4. Let
p E BV,,
(4.3.5)
h E R, and set q(x) = A x
Then q is 27rperiodic, belongs to ,:L n BV,
+ 4  Ax). and
(4.3.6) (4.3.7)
hp"(O),
k
=
0.
Proof. By the definition of the class BV, it easily follows that q is a 2nperiodic function, and thus q E L,: n BV., Suppose that p is monotonely increasing and h > 0. Then
In the general case, we use the Jordan decomposition of p, and (4.3.6) follows. Finally, by Prop. 4.3.1
ikq^(k) = q"(k) =
b(0 + h)  p(")]"(k) = (elhk l)p"(k),
and the proof is complete.
Theorem 4.3.5. For the convolution (p * dv)(x) of p, v E BV, (4.3.8)
we have
b * dv]"(k) = p"(k)v"(k)
(k E Z).
Proof. By Prop. 0.6.2, p * dv E BV, and thus both terms of (4.3.8) are welldefined. To prove the actual equality, let q be given by (4.3.5). Since q E L,; n BV, it follows that q * dv E ,L: n BV., Moreover,
+
by Prop. 4.3.3 and (4.3.7). On the other hand, (q * dv)(x) = (p * dv)(x h) (p * dv)(x).Since the righthand side is of type (4.3.5), p being replaced by p * dv,we may apply (4.3.7) to obtain
1
h[tL * dv]"(O),
k
=
0,
which, together with (4.3.9), proves the theorem. It follows from (4.3.8) and Prop. 4.3.2(iii) that (p * dv)(x)  (v * dp)(x) = const, a result which we already obtained directly from the definition of p * dv by partial integration (compare (0.6.7)). Thus BV:, becomes a Banach algebra under convolution as multiplication which is commutative. It even has a unit element which is given by
I82
FOURIER TRANSFORMS
(4.3.2). Furthermore, the finite FourierStieltjes transform is a non normincreasing isomorphism of the Banach algebra BVg, into I", the algebra of bounded functions on E with pointwise operations and supremumnorm. Proposition 4.3.6. I f p
E
BV,, is such that p" E I', then p is absohtely continuous and a,
(4.3.10)
r,
Proof. Since p" E I', the series on the right of (4.3.10) converges uniformly and thus defines a function f E C,,for which f"(k) = p"(k), k E Z. Therefore p(x) = f(u) du +p(lr) by Prop. 4.3.2(iii). Proposition 4.3.7. I f p
E
BV,,, and f E L,: is such that f
E
I', then for aN x
m
(4.3.11)
cf* dp)(x) = k =2 w f"(k)p"(k)
elkr.
z.k"=
Proof. By Prop. 4.1.5 we have f ( x ) =  f "(k) elkx a.e. Since the series converges uniformly, we obtain by termbyterm integration
which holds everywhere since both sides represent continuous functions. 4.3.2 Inversion Theory In view of the definition (4.3.1) of the finite FourierStieltjes transform of p E BV,,, we might expect by (4.3.10) that an inversion is given by m
p'(x) =
2 k=m
p"(k)
etkx.
But even if this series is understood as the limit for n + 00 of the symmetrical partial sums
(see Sec. 1.2.7), the limit does by no means exist for every x and p E BV,,; it must be interpreted in some generalized sense. Proposition 4.3.8. Let {B,(k)} satisfy (1.2.28) such that the corresponding kernel {O;(x)} of (4.1.18) is an approximate identity. Thenfor p E BV,,
for every h E CSn.
183
FINITE FOURIER TRANSFORMS
The proof follows by Prop. 1.2.12 by making use of (4.3.12)
U,(dp; x)
2 P)
=
O,(k)p"(k) elkx = 27
k=m
C(X  U)dPL(U),
,
which is valid in virtue of Prop. 4.3.7. The next proposition is an application of Prop. 1.4.8. Proposition 4.3.9. Let p E BV, and {O,(k)} be a elfactor such that the corresponding kernel { c(x)} of (4.1.18) is an absolutely continuous, even, positive approximate identity, monotonely decreasing on [0,n].Then m
(4.3.13)
lim P+PO
2
k=
e,(k)pv(k) elkx = p'(x) a.e. m
For the application of further results of Sec. 1.4 as well as for explicit formulae for some important examples of &factors we refer to Problem 4.3.3. If one wishes to obtain p instead of p', formula (4.3.13) suggests that an inversion of the type (4.3.14)
p(x
+ h)  p(x) = hp"(0) +
m
lim
PPo
2' k=m
eihk  1 OP(k)p"(k) ik
elkX
may be valid. But we already know (cf. Problem 4.1.5) that termbyterm integration converts the Fourier series of any f E L?j, into a (uniformly) convergent series. Thus we may expect that (4.3.14) holds without the convergencefactor {OP(k)}.In fact, we have Proposition 4.3.10. I f p (4.3.15)
E
BV,
then for all x, h
+ h)  p(x) = hp"(0) + lim 2
nI
p(x
nm
k=n
elhk p"(k) elkx. ik
Proof. Let h E R be fixed. If q is defined by (4.3.9, then q is a 2nperiodic function and thus by Prop. 4.1.5(ii) q ( x ) =  n q"(k) elkx belonging to L!j, n BV, for all x. (4.3.15) now follows by (4.3.7).
x;=
4.3.3 FourierStieltjes Transforms of Derivatives In this subsection r always denotes a natural number.
Proposition 4.3.11. I f f E AC',ia a n d f c r  l )E BV, (4.3.16)
(ik)lf"(k) =
then
v(r')]"(k)
(k E Z).
For the proof we observe that for r = 1 the assumptions are that f E BV, is 2nperiodic. In this case, (4.3.16) is Prop. 4.3.l(iv). If r 5: 2, then Prop. 4.1.8 shows that (ik)*'f''(k) = [f'rl)]"(k), k E Z, and (4.3.16) follows by an application of Prop. 4.3. I(iv) tof('l).
FOURIER TRANSFORMS
184
In order to establish the converse we introduce the following classes of functions:
{fE c,, (4.3.17)
{f E ,L: {f E Lg,
If€ Ac',;l,f(r)E La"n}
If If
4 a.e., 4 E AC',;', = 4 a.e., 4 E AC',;', =
In case of the reflexive spaces X,, = Lg, equal by definition. Proposition 4.3.12.
Iffor f E L,:
BV,,}
tfrll) E
$(,)
E
Lf;,}
(1 < p
Y > 01) 5 ( M [Ifll~l,/v)~ for all simple functions on R1 with M independent off and y. The least value of M may be called the weak ( r ; s) norm of T. Every linear operator Tof strong type is also of weak type, but not conversely. For the latter operators there holds the following convexity theorem of Marcinkiewicz : Let ( p l , q l ) and (pa, q2)be any two points of the triangle 0 Iq I p 5 1 sitch that q 1 # 4 2 . Suppose that a linear operator T is simultaneously of weak type ( l/pl ; l / q l ) and (~/Pz; with norms Ml and Mz,respectively. Then for any point ( p , 4) with (0 < t < 1) P = (1  t)Pl + tP2, 4 = (1  f)4l + tqa the operator T is of strong type ( l / p ; l/q), and llTf 5 K M ;  tMJIf JlliP, where K = Kt.p1,q1,p2.q2 is independent o f f and is bounded ifp,, q l , p 2 , qz are fixed and t stays away from 0 and 1. This theorem remains valid if T is only quasilinear. For this and further comments [7II, Chapter as well as for the proofs of the two convexity theorems we refer to ZYGMUND 121, EDWARDS [III, Chapter 131, BUTZERBERENS [I, p. 187 ff]. In the last book the above classical convexity theorems are discussed in the general setting of intermediate spaces and interpolation. Most of the results of this section generalize to general orthogonal series of functions; [I]. For further and more recent results on such series compare KACZMARZSTEINHAUS see the important work by the Hungarian school, in particular ALEXITS [31, FREUD[31 and the literature cited there. Contributions to harmonic analysis on the group E are found widely scattered in the
FINITE FOURIER TRANSFORMS
187
literature. As is the case in this text, they mainly serve as illustrations, at particular places, for the results to be expected on other groups. Sec. 4.3. The material of this section, though standard, is somewhat scattered in the litera
ture. As in the preceding sections the emphasis lies upon the fact that the finite FourierStieltjes transform defines a mapping from one function space into another. This has influenced the selection of the material given here, which is symmetrical with that of Sec. 4.1 and 5.3, so that the general references given there are applicable. In connection with (4.3.2), for an example of a continuous function p E BV,, such that p"(k) does not tend to zero; see ZYGMUND [71, p. 194 IT]. Another proof of the general convolution theorem is to be found in ZYGMUND [71, p. 391. For the results of Sec. 4.3.3 reference may also be made to BUTZERG~RLICH [l].
5 Fourier Transforms Associated with the Line Group
5.0 Introduction
In the preceding chapter we have regarded the finite Fourier transform as a transform of one function space into another. This emphasis is a useful one in order to give a unified approach to Fourier analysis on different groups. This chapter is devoted to the study of the line group. Parallel to Sec. 4.1, Sec. 5.1 is concerned with the operational rules of the Fourier transform in L'. The inversion theory will follow by the theory of singular integrals presented in Chapter 3. Included are results on generalized derivatives (Peano and Riemann) and connections with Fourier transforms and moments of positive functions. The relation between Fourier transforms and Fourier coefficients given by the Poisson summation formula is developed in Sec. 5.1.5. Sec. 5.2 is devoted to the definition of the Fourier transform for functions in Lp, 1 < p s 2, including the Titchmarsh inequality (Theorem 5.2.9), Parseval's formula (Prop. 5.2.13), and Plancherel's theorem (Theorem 5.2.23). The operational rules are developed, together with the central Theorem 5.2.21. Sec. 5.3 is concerned with a thorough investigation of the FourierStieltjes transform with its basic properties. We specifically mention the Lkvy inversion formula (Theorem 5.3.9) and the uniqueness theorem (Prop. 5.3.1 1).
5.1 L1Theory 5.1.1 Fundamental Properties
With every f E L1 we have associated (cf. (11.3)) its Fourier transform f" defined by 1 " (u E R). (5.1.1) fA(u) = f(u) e"'" du = [f(.)]^(u) = F;"fl(o) 427 In comparison with (4.1. l), there is a slight discrepancy regarding the constant factor
s
(0
FOURIER TRANSFORMS ASSOCIATED WITH THE LlN3 GROUP
189
in the definition off ". Again this factor is chosen such that it will lead to a symmetric inversion formula and Latheory. The first proposition which may be shown by direct substitution in (5.1.1) deals with some elementary operational properties of the transform. Proposition 5.1.1. For f E L1 we have (i) [f(o + h)]"(u) = eih"f"(v) (ii) [eihof(o)]"(u) = f " ( v + h)
(h, 0 E W), (h, 0 E R), (iii) [f!f(P o)l"(4 = f^(u/d (P > 0,?JE W Y (u E R). (iv) [f( o)l"(u) = f W It follows immediately by definition that the Fourier transformf^ o f f € L' exists for all u E R as a bounded function satisfying
IfWl 5 llflll
(5.1.2)
thus
Il f " IIaO
I
Ilf)ll.
(0
E
w,
Moreover,
Proposition 5.1.2. The Fourier transform deJines a bounded linear transformation of L' into C,. Indeed, we have for all h, u E W
1/% If"(u
+ h) f"(u)l
I/mm
lelhu
 11 If(u)l du.
Since the integrand is bounded by 2 If(u)l, tends to zero as h +0 for every u, and is independent of u, uniform continuity off * follows by Lebesgue's dominated convergence theorem, and thusf" E C. Furthermore f"(u) =
[ n u )  f ( u + i)] eiuudu, 2 . 6 Jm
and therefore (5.1.3)
By Prop. 0.1.9 this implies lim,,,, + mf"(v) = 0, a result which is known as the RiemannLebesgue lemma. Hence f" E C,. Theorem 5.1.3.
IfJ;g E L',
(5.1.4)
Proof. Since by Prop. have for every u E W
then
[f* gl"(u) = f W g "(0) 0.2.2 the convolution f * g exists a.e.
(u E
W).
as a function of L1, we
the inversion of the order of integration being justified by Fubini's theorem.
190
FOURIER TRANSFORMS
The latter theorem is known as the conuolution theorem for Fourier transforms in L'. It in particular shows that the Fourier transform converts convolutions to pointwise products. Proposition 5.1.4. Iff, g
(5.1.5)
E
L1, then
Jw
J00
Proof. It follows by Fubini's theorem that
1m
fA(u)g(u) du
=
In future we shall refer to a formula of type (5.1.5) as a Parseualformula.
Inversion Theory So far, given a function f E L', we defined its Fourier transform f " and considered some of its fundamental properties. We now take up the inversion, problem, i.e., the problem of reconstructing the original function f from the values,f^(u) off". In correspondence with the finite Fourier transform and with (11.5), i f f € L1 and
5.1.2
we might expect a formula like (5.1.6)
f(x) = Y I
1/2T
Im
f"(u) elxudu
m
to be valid. But just as for the finite Fburier transform we immediately encounter the problem of giving the Fourier inversion integral (5.1.6) a suitable interpretation since the Fourier transformf" off€ L' need not belong to L' (see Problem 5.1.4). Although we do not intend to give a general treatment of summability of integrals on the real line, we shall discuss the particular case concerned with (5.1.6) in some detail.
Definition 5.1.5. An even function 0 E L' is called a Ofactor (on the real line) if 6" E L1 such that (5. I .7)
["
P ( u ) do = I&.
Jm
If 6 is continuous, we call it a continuous 6yactor. Important examples of &factors are given by the Cesdro, Abel, and Gauss factor, i.e., by (5.1.8) (i) O,(x) = 1  1x1, 1x1 II (ii) &(x) = e'"', (iii) O,(x) = eXa, respectively (see Problem 5.1.2(i)).
FOURIER TRANSFORMS ASSOCIATED WITH THE LINE GROUP
Theorem 5.1.6. Let f for each p > 0 by
E
191
L'. Thenfor a Ofactor the 0means of the integral(5.1.6) defined
(5.1.9) exist for all x E R, belong to L', and satisfy
I I W ";dII1 5
(5.1.10)
(5.1.1 1 )
lim
0m
IIWf;
0;
p)

lI~Alllllflll
=
f(0)II'
0.
Thus the Fourier inversion integral is 8summable to f in L'norm.
Proof. Since 8 is even, we have by Prop. 5.1.1, 5.1.4, and Problem 5.1.1 for each P > O
(5.1.12) and hence the assertions follow immediately by Prop. 0.2.1, 0.2.2, Lemma 3.1.5, and Theorem 3.1.6 if we take x(x; p) = p8"(px). In contrast to the periodic case of Prop. 1.2.8 the definition of a &factor on the real line implies that the kernel {pP(px)}of the singular integral (5.1.12) is an approximate identity. Thus, via (5.1.9), a &factor on the real line always defines a summation process of the integral (5.1.6) with respect to convergence in L'space. As applications of the results of Sec. 3.2 we obtain
Theorem 5.1.7. Let f E L'. I f for a Bfactor P is moreover positive and monotonely decreasing on [0,a),then the Fourier inversion integral is 8summable a.e. to f ( x ) , i.e. (5.1.1 3)
lim
0m
Sm
1 4%
m
B(;)f"(v) elxe do
=f ( x )
a.e.
Theorem 5.1.8. Let f E L'. Iffor a dfactor 8" has a majorant ( Q M : IP(v)l I( B h ) M ( ~ ) such that (P), belongs to L' and is monotonely decreasing on [0,co), then the Fourier inversion integral is 8summable a.e. to f ( x ) . For the proof of the last two theorems we only mention that in view of (5.1.12) (5.1.14) U(f;X ; p)  f ( X ) =
jm + d2rr
[f(x
u)
+ f ( x  u )  2 f ( x ) ] P ( p u )du.
0
Therefore (5.1.13) follows by Prop. 3.2.2 and 3.2.4, respectively. Concerning the examples (5.1.8) of &factors we have at the same time shown
FOURIER TRANSFORMS
192
Corollary 5.1.9. For f E L1 the Fourier inuersion integral is Cesdro, Abel, and Gauss summable to f ( x )for almost all x E R, i.e. (5.1.15)
lim
o
(5.1.16)
&JIo y) (1 
lim

f*(u) elxudo = f ( x ) a.e.,
e~"~'Pf^(u) elxudu = f ( x ) a.e., e  ( u ~ O ) ~ "elxu ( u )dv = f ( x ) a.e.,
(5.1.17)
respectively, In particular, all these relations are valid at each point of continuity o f f .
In fact, in view of Problem 5.1.2(i) we have B;(x) = F(x), Q(x) = p(x), and G(x)= w(x), where the functions F, p and w are defined by (3.1.15), (3.1.39), and (3.1.33), respectively. Thus, if we denote the e,means of (5.1.6) by U,Cf;x ; p), k = 1,2, 3, we obtain via (5.1.12) the following correspondence with the singular integrals of FejCr, CauchyPoisson and GaussWeierstrass as introduced in (3.1.14), (3.1.38) and (3.1.32), respectively : (5.1.1 8) (5.1.19) (5.1.20) Therefore Cor. 5.1.9 is nothing but a collection of the results of Cor. 3.2.3 and 3.2.5. One observes that formulae (5.1.18)(5.1.20) indicate the important r61e that the singular integrals considered in Chapter 3 play in the theory of summation of the Fourier inversion integral. The preceding corollary on the pointwise summability of (5.1.6) gives three possible interpretations for the inversion formula (5.1 A). A further interpretation is given by Proposition 5.1.10. I f f and f
*
belong to L1, then
(5.1.21)
for almost all x E R. Therefore f is equal a.e. to a function in L' n C,. I f f is continuous on R, then the inversion formula (5.1.21) holds everywhere. Proof. By Cor. 5.1.9 the lefthand side of (5.1.21) is Abel summable to f a.e. On the other hand, as lelul/Pf^(u)etxul IIf"(v)l E L1 for all p > 0 and
FOURIER TRANSFORMS ASSOCIATED WITH THE LINE GROUP
193
it follows by Lebesgue's dominated convergence theorem that
for all x E R. This proves the result. Another interpretation of (5. I .6) follows from Jordan's criterion (see Problem 5.13iv)). A further consequence of Cor. 5.1.9 is the uniqueness theorem of the Fourier transform in L'. It states that two distinct functionsf l andfa in L' have distinct Fourier transformsf;' and f;. Indeed, Proposition 5.1.11. I f f € L' and f"(u)
= 0 on R, then f ( x ) = 0 a.e.
Together with Problem 5.1.3 we may therefore strengthen Prop. 5.1.2 to: The Fourier transform defines a onetoone bounded linear transformation of L1 into (but not onto) C,. However, the set of functions [L']" = { g E Co I g = f", f E L'} is dense in C,. Moreover, according to Theorem 5.1.3 the Fourier transform defines an isomorphism of the commutative Banach algebra L' (with convolution as multiplication) into the Banach algebra C, (with pointwise multiplication). Concerning the Banach algebra L1 itself, the properties of the Fourier transform so far obtained enable us to show the following proposition the proof of which may be regarded as a first example of the integral transform method. Proposition 5.1.12. The Banach algebra L' has no unit element. Howeuer, there exist approximate identities, i.e., there are sets of functions { ~ ( xp)} ; c L', p > 0, such that Ilx(0; p) * f  f 11' = 0 for euery f E L'. Proof. Indeed, if e E L' existed such that e * f = f for every f E L', then, in particular, e e = e. By Theorem 5.1.3 it would follow that (e"(u))a = e"(u), and thus, for each 27 E R, e"(u) would equal zero or one. But e"(u) is continuous, and therefore e"(u) would be identically zero or one, and in fact e"(u) = 0 since e" must vanish at 2 00. Then the uniqueness theorem implies e(x) = 0 a.e. which contradicts e * f = f for every f E L1. The existence of approximate identities is given by Lemma 3.1.5 and Theorem 3.1.6. As a consequence of Prop. 5.1.10 we now prove the following analogs of (4.1.8)(4.1.10). Proposition 5.1.13. I f f , g E L' and g is such that g"
E L1,
then
(5.1.22) (5.1.23)
( x E R>,
Cm f ( u ) z ) du = [" f " ( u ) c )du.
Ja,
Jm
Proof. According to Prop. 5.1.10, g is equal a.e. to a function of class C,. Therefore by Prop. 0.2.1 the convolutionf * g exists for every x E R as a continuous function. By the convolution theorem If*g]"(u) = f"(u)g"(u), and since the hypotheses assure I~F.A.
FOURIER TRANSFORMS
194
u*g]^ E L1,
a further application of Prop. 5.1.10 gives (5.1.22). Replacing g by g (  x ) and using Prop. 5.1.1(iv), we obtain (5.1.23) by setting x = 0. The particular case f = g is of some interest; formula (5.1.23) then states that the Parseual equation (5.1.24)
IlfIILP
is valid for allfe L' withf"
=
Ilf"llL~
E L'.
5.1.3 Fourier Transforms of Derivatives We have seen in Sec. 5.1.1 that certain operations on functions correspond to more suitable operations on their Fourier transforms. We continue these investigations on the operational properties of the Fourier transform by fitting in derivatives. Proposition 5.1.14. Let f E L1 n ACf&' andf") E L'. Then
(5.1.25)
(uI
(v E R).
If(')]^(u) = (iu)'f^(u)
Thus, if the hypotheses of the above proposition are satisfied, thenf"(u) = o(lu1 r) as t c o (compare also Problems 5.1.9(ii) and 5.3.5).
Proof. Since f~ ACT&1, integration by parts (rtimes) yields (5.1.26) for all x
k=O
E R.
( lY.(;)f(x
+ k) = f . . . lO1fr)(x+ u1 +   .+ u,) du,. . .du,
If we introduce the function m(x)
=~
thus
K [  ~ , ~ ~ ( X ) ,
(5.1.27) we may rewrite the righthand side of (5.1.26) as a convolution product. Indeed (cf. Problem 5.1.6(i)), (5.1.28)
k=O
( IY*($(x
+ k) = ([m*I' *f('))(x),
where the right side, being the convolution of the product (rtimes) m *. withf") E L', exists for all x E R by Prop. 0.2.1. Since (5.1.29)
m"(v) =
.* rn E L1
y=J, 1
iu
V # O
1,
u = 0,
the Fourier transform of equation (5.1.28) is given in view of Prop. 5.1.1 and Theorem 5.1.3 by
for all u # 0. Therefore (5.1.25) holds at all points u # 2kr, k E E. Since this set is
195
FOURIER TRANSFORMS ASSOCUTED WITH THE LINE GROUP
denumerable and since, in virtue of Prop. 5.1.2, both sides of equation (5.1.25) represent continuous functions, this equation is actually established for all u E R. Proposition 5.1.15.
Iffor f E L'
there exists g E L1 such that
(5.1.30) (iu)'f"(u) = gA(u) for some natural number r, then f E W[I Cfor the definition see (3.1.48)).
(0
E
w
Proof. Let r = 1. Then we may invert the single steps of the proof of the latter proposition to obtain the result. In fact, the assumption (5.1.30) together with (5.1.29) and the convolution theorem implies for each h > 0 and u # 0 1  elhv (5.1.31) [f(X)  f ( x  h)]"(~)= g"(u) = g(x u)duIn(u).
[Io
iu
+
h
Therefore it follows by the uniqueness theorem of Fourier transforms that for each fixed h > 0 f ( x )  f(x  h) = J   g(u) du
(5.1.32)
xh
for almost all x
Since f
E L',
lim
hm
E
R. We then have for every y > 0, h > 0
on the one hand we obtain for each fixed y > 0
IOv
Lf(x)  f ( x  h)] dx =
f ( x ) dx  lim
hrm
1
Yh
h
f ( x ) dx =
loY f ( x ) dx.
On the other hand, by Lebesgue's dominated convergence theorem
Therefore it follows that f ( x ) = J   g(u) du a.e.,
(5.1.33)
m
sf
g(u) du, Prop. 5.1.15 is established for t = 1. and if we set $(x) = Regarding the general case r 2 2, we first show that the assumption (5.1.30) implies that iuf"(u) is the Fourier transform of some L1function. Let (5.1.34) Then f ( u )
=
(1
+ i u )  I , and therefore by the convolution theorem (iu)' f^(U) E [L']". (I + iu)I' E[L']", (1 + iuy1
196
FOURIER TRANSFORMS
But for certain constant coefficients ak, 1 Ik 5 r (iu)'
(1 + iuy1f A ( u ) = iuf^(u)
+
 1, r1
+
 ( r  I)f"(u)
Now the lefthand side of the latter equation belongs to [L1]^ as well as the terms in curly brackets. This implies iuf*(u) E [L']^, i.e., there exists g , E L' such that iuf "(u) = g;'(u) for all u E R. Thus the case r = 1 applies, and (5.1.33) in particular shows that f0= J?g d u ) du a.e. The proof for r 2 2 now proceeds iteratively as follows. If (5.1.30) holds, then there exists gl E L' such that (iuylg;(u) = g"(u) and f ( x ) = JEm gl(ul) du,. But this in turn implies iug;(u) E [L']", i.e. iug;(o) = g$(u), say, with g2 E L1 and g l ( x ) = JtOD ga(U) du a.e. Hence OD
Applying this method successively we obtain a sequence of functions gk 1 5 k Ir  1 such that for almost all x E R
f (XI =
gl(u) du, 8k(x) =
and therefore (5.1.35)
f(x)
=
sx
m
j:m gk + l(u) du,
~ U 1 Y
5k 5
 2,
ju'du,. . /"'g(u,) ,
OD
du,
gr  1(x) =
/:
E
L' for
dU) du,
a.e.
OD
If we abbreviate the right side as &), then 4 E AC;', #k) = gk E L1, 1 I k Ir  1, = g a.e., and the proof is complete. As in the periodic case it is now convenient to introduce the class
+(r)
(5.1.36)
W';!&)I= {fE L' I YWf 7 0 ) = g
w , g E L'},
$(u) being an arbitrary complexvalued (continuous) function on 08. Then
Theorem 5.1.16. For f E L1 the following assertions are equiualent: (i) f E Wily (ii) f~ WIL1; (iu)'], (iii) there exists g E L1 such that the representation (5.1.35) holds, each of the iterated integrals existing as a function in L'. These results in particular show that iff E L' is rtimes continuously differentiable withf") E L', then all derivatives f c k ) ,1 I k I r  1, belong to L'.
Derivatives of Fourier Transforms, Moments of Positive Functions, Peano and Riemann Derivatives Whereas the last subsection dealt with the Fourier transform of derivatives of the original function f, we are now interested in results concerning smoothness properties of the transformf". Prop. 5.1.2 may be regarded as a first contribution in this direction.
5.1.4
197
FOURIER TRANSFORMS ASSOCIATED WITH THE LINE GROUP
Proposition 5.1.17. I f f E L' is such that x'f(x) E L1 for some r E P, then the Fourier transformf o f f has an rth derivative of class Co, and
1
(i)k (f")(k)(v)= 
(5. I .37)
4Z
uy(u) elUudu
(k = 0, 1,. .., r),
( f ")(yo)= (  i)"k
(5.1.38)
(k = 0, 1,. . ., r),
m
where Y k is the kth moment off, namely (5. I .39)
Yk
Proof. Suppose r f"(v
=
=
Im
4%
(k E P).
u"f(u)du
OD
1. Then
+ h)  f"(u) 
i
 Ihu/2
h
sin (hu/2)uf(u) etuudu. hu/2
Since the integrand converges to uf(u) exp{ ivu} a.e. as h f 0, and since it has an integrable majorant independent of h, given by IuI If(u)I, we obtain by Lebesgue's dominated convergence theorem that (f")' exists at each v E R and i (f")'(v) = uf(u) etuudu.
/ 6
W
m
The proof for general r follows by mathematical induction. We observe that the moment yr exists (as a Lebesgue integral) if and only if x'f(x) L', i.e., if and only if the absolute moment (cf. (3.3.6))
E
(5.1.40)
exists as a finite number. Furthermore, if yr exists, so do all the moments Y k of order k = 0, 1, . . ., r. It is important to note that for positive functions a certain converse of Prop. 5.1.17 is valid, and it is then possible to weaken the assumptions upon the differentiability of the transform f" considerably. For this purpose we need the following notations. For an arbitrary function f on the real line we define its first central difference at x with respect to the increment h E R by
6ff ( x ) = f (x
+ );  f("  );
and the higher differences by
(r E N).
&+tf(x) = n ; q f ( x )
It can be shown (cf. Problem 1.1.6) that (5.1.41)
2 (l).(;)f(x+
Gf(x)=k  0
(fk)h)
(h~R,reN).
As a generalization of the concept of ordinary derivatives we extend Def. 1.5.14 to
FOURIER TRANSFORMS
198
Definition 5.1.18. Letf ( x ) be defined in a neighbourhood of thepoint xo.The rth Riemann derivative o f f at xo (with respect to the central difference (5.1.41)) is defined by
(5.1.42)
f[rl(xo)
=
lim
h+O
h'
af(xo),
in case the limit exists and is finite. In other words, if the rth central difference quotient h'Tif(xo) off at xo has a limit as h f 0, then we call this limit the rth Riemann derivative o f f at xo. The notation f i r ]is used to distinguish the rth Riemann derivative from the rth ordinary derivative
f(I).
The concept of a Riemann derivative is more general than that of the ordinary derivative. Indeed, it follows by Problem 5.1.11 that if the rth ordinary derivative f (')(x0)exists, so does the rth Riemann derivativef [rl(xo), and we havef ['l(xo) = f (')(xo), but not conversely. We emphasize that, in order to definef ['I, we need not suppose the existence of Riemann derivatives of lower order whereas the ordinary derivative fCr) is defined successively by means of the derivatives.f(k),1 I k 5 r  1. After these preliminaries we may now state the following
Proposition 5.1.19. Let f the origin satisfies
E
L' be positive. If the 2rth central difference quotient o f f at
then the 2rth moment yzt o f f exists as do all moments Y k of order k < 2r. Moreover, the derivatives (f")(k), 1 5 k 5 2r, exist as functions in Co and are given by (5.1.37). (5.1.38) holds as well.
Proof. In view of the definition we have by Prop. 5.1.1(ii)
Hence we obtain for the 2rth central difference quotient off " at the origin
By Fatou's lemma it follows that
which is bounded by hypothesis. Thus Yar exists which, in view of Prop. 5.1.17, completes the proof. Upon combining the assertions of the last two propositions we obtain
FOURIER TRANSFORMS ASSOCIATED WITH THE LINE GROUP
199
Corollary 5.1.20. The 2rth moment yzrof apositivefunctionf E L1 exists ifandonly ifthe Fourier transform f has a 2rth ordinary derivative at the origin.
Moreover, if we pose the problem of finding conditions upon the Riemann derivative of a function which assure the existence of its ordinary derivative, we have as a first answer Corollary 5.1.21. I f a function is representable as the Fourier transform of a positive function of class L1, then its 2rth ordinary derivative exists i f and only i f it has a 2rth Riemann derivative.
Let us assume that the rth moment off E L' exists. Then by Prop. 5.1.17 and Taylor's formula the Fourier transformf^ admits the following expansion at the origin: f^(v) =
2
k0
vk
(f^)'"(O)
+ ;[(f^)'"(r]V)  (fn)(r)(o)]vr
(0 Ir] < 1).
Therefore Proposition 5.1.22. I f the rth moment y, of a function f E L1 exists absolutely as a finite number, then (5.1.43)
(v f 0).
Again a converse is valid for positive functionsf. In order to derive this result we introduce Peano derivatives. For an arbitrary function f on the real line an expression of the form (5.1.44)
(h E R, r E N)
with arbitrary (constant) coefficients 1, E @, 1 Ik I r  1, is called an rth Peano difference off at x with respect to the increment h (and the constants lk).
Definition 5.1.23. Let f ( x ) be defined in a neighbourhood of the point xo. I f there are constants &, 1 I k Ir  1, such that the corresponding rth Peano diference quotient h'O; f ( x o ) off at xo tends to a limit as h +0, then this limit is called the rth Peano derivafive o f f at xo and denoted by f(')(x0),i.e.
Again, in order to define a Peano derivative of order r we do not need any explicit assumptions on Peano derivatives of lower order. Only the existence of certain constants I,, 1 5 k I r  1, is assumed for which the limit (5.1.45) exists. But we still have to show that the definition of a Peano derivative of a function at a point is meaningful, that is to say, that f 0
Then by Prop. 5.l.l(iii) and (5.1.52) (5.1.58)
[x:ln(k) = X^(k/P)
(k E z, p > 0).
FOURIER TRANSFORMS ASSOCIATED WITH THE LINE GROUP
203
It is actually this property which makes periodic kernels {xp*(x)}generated by some x E NL' so useful in applications. Though the dependence of the kernel itself upon the parameter p may be badly arranged, the Fourier transforms of xz, p > 0, are obtained by one and the same function x'' by a simple scale change. If furthermore x E BV, we may apply Prop. 5.1.30 to deduce
which holds for everyfE X,, and all x E R, p > 0. Thus the singular integralJ(f; x ; p) forfe X,, may be obtained by introducing the factor x"(k/p) into the Fourier series of f i n which case we may consider J(f; x ; p) as a method of summation of the Fourier series o f f with summation function x"(k/p). A comparison with our definition (1.2.28) of a 0factor shows 'that this would correspond to the particular situation that one has the representation 0,(k) = 0(k/p) with A = (O,m), po = 03. But note that (5.1.59) does not assume x"(k/p) E 1'. Let x satisfy the assumptions of Prop. 5.1.29(i). Then the Poisson summation formula gives
2
m
fik =2 px(p(x + 2 k ~ )= ) x"(k/p) e l k x m k= m
(5.1.60)
( x E R, p > 0).
To illustrate the importance of this formula, we give proofs of (3.1.20), (3.1.36), and (3.1.43) which connect the periodic and nonperiodic kernels of FejCr, Weierstrass, and Poisson, respectively. (i) FejCr: Let x(x) = F(x) as given by (3.1.15) with p = n 1, ~ E PThen . by Problem 5.1.2(i) and Prop. 5.1.10
+
and since all the assumptions of Prop. 5.1.29 are satisfied, (5.1.60) is valid. For the righthand side we have by (1.2.24)
2Y
(k ~ e t k )x =
k=  m
k= n
for all x # 2jr, j
E Z,
m
2
V'Z k =  a ) (n
(1 
*)
=

and for the lefthand side
1 + l)F((n + l)(x + 2 k ~ )=) n+1
k=m
sina ((n ((42)
+ 1)x/2)
+ k#
Therefore for all x sina (n
+ 1)x  sina (n + l ) x sins x
'
which proves (3.1.20) (without using the theory of meromorphic functions).
'
204
FOURIER TRANSFORMS
(ii) Weierstrass: Let ~ ( x = ) ~ ( x as ) given by (3.1.33) with p w"(u) = exp {  ua}, and (5.1.60) turns out to be
J' 2
(5.1.61)
T7
t
,(x+Zkn)a/4t
k=m
=
f
= tl'a,
e  k a t ,fkx
k=m
t > 0. Then
( x E R, t > O),
which is (3.1.36). (iii) Poisson: Here we set x(x) = p(x), where p is given by (3.1.39), and p = y  ' , y > 0. Thenp"(t7) = exp{ lol} by Problem 5.1.2 and Prop, 5.1.10, and (5.1.60) gives
which proves (3.1.43). Apart from these connections between periodic and nonperiodic kernels, the results of this subsection (and of Sec. 3.1.2), in particular formula (5.1.58), may also be used to link the theory of Fourier integrals to that of Fourier series, thus to obtain many facts about the Fourier transform from the corresponding facts about the finite Fourier transform. For examples see Problem 5.1.16. Problems 1.
(i) Let f~ L'. Show that f"(v) is an even (odd) function on R if and only if f ( x ) is an
even (odd) function on R. Show that Lf(  o)]"(u)
= f^(  v).
(ii) Let f E L1 and h E 08, h # 0. Show that Lf(h o ) ] " ( u ) = Ihl 'f"(u/h). (iii) Let f, fn E L1 be such that limn+,,.  fnlll = 0. Show that limn+ f,^(v) = f ^ ( o )
[If
uniformly for u E W. In particular, show that if { ~ ( x p)} ; is an approximate identity, then limD+a f ( v ; p) = 1 for all D E R. (iv) Letf, g E L1 and suppose that g"(u) = 0 for 101 2 1. Show that [f*gJA(v)= 0 for
IvI 2 1. 2. (i) In the notation of (5.1.8) show that
[7, p. 581, HEWITTSTROMBERG [I, p. 4071) (Hint: See e.g. BOCHNER (ii) Given real numbers a c b and e 0, show that there exists g E L' such that g"(v) = 1 for a 5 u I: b, = 0 for u Ia  e, u 2 b e, and gA is linear on [a  e, a], [b, b + el. Evaluate the Fourier transform of (1  ix)a, and show that there exist functions g E L' such that gA(u) > 0 for u > 0, = 0 for D I 0. (Hint: For the first question use el, see also R. R. GOLDBERG [l, pp. 2326]) (iii) Show that the Hermite functions Hn(x) of (3.1.13) satisfy H,^(u) = (i)"Hn(u). Thus the Hermite functions Hn(x) are eigenfunctions of the Fourier transform with the corresponding eigenvalues ( iy. (Hint: Use Problem 3.1.3(iv), Prop. 5.1.14, 5.1.17, and show that inH,"(v) satisfies the same recursion formula as Hn(v);see also HEWITT[l, p. 1601, SZNAQY[5, p. 3501, TITCHMARSH [6, p. 81 I) (iv) Show that 1, a > O sin au 0, a = O 1, a < 0. (Hint: See e.g. BOCHNER [7, p. 161, HEWITT[I, p. 1631)
+
FOURIER TRANSFORMS ASSOCIATED WITH THE LINE GROUP
205
{;,
(v) If N(x) denotes the kernel of the singular integral (3.1.58) of Jacksonde La Vallke Poussin, show that (3/2W + (3/4) IuI3, IuI 5 1 Nn(u) = (2  1~1)'/4, 1 < 101 I2 I4 2 (see ACHIESER [2, p. 1381). Thus, in view of (5.1.58), the coefficients of the polynomial N:(x), constituting the kernel of the periodic singular integral (3.1 59) of Jacksonde La VallCe Poussin, are given through [N,*]"(k) = N"(k/n). 3. Show that there exist functions g E Co which are not the Fourier transform of some function in L', yet the set [L'] { g G Co I g(u) = f"(u), f E L'} is dense in Co.(Hint: As to the first part, R. R. GOLDBERG [ I , p. 81 examines the exampleg(x) = l/log x for x > e, =x/e for 0 5 x 5 e, = g(x) for x < 0; see also RUDIN[4, p. 1951. For the second part, one may use Problem 5.1.2(iii) and Cor. 3.1.9; see also LEVITAN [I]) 4. (i) Give examples of functions f E L1 for whichf" 4 L1 (cf. (5.1.27), (5.1.34)). (ii) Suppose that f E L' n L" and that f A is realvalued and nonnegative. Show that f" belongs to L'. (Hint: Use Lebesgue's monotone convergence theorem and Cor. 5.1.9; compare BOCHNERCHANDRASEKHARAN [ 1, p. 201, HEWITTSTROMBERG 11, P. 4091) (iii) Suppose that f E L' n L". Show that llfll2 = 11 f"Il2. (Hint: Use (ii); compare BOCHNERCHANDRASEKHARAN [1, p. 231) (iv) Let p > 0 and suppose that f is bounded and integrable in [  p , p], and that
'
A=
for all u E R. Show that 4 belongs to L'. (Hint: Compare also LINNIK[ I , p. 131) 5 . Let f E L'. (i) Show that for every p > 0 S ( f ; x ; p)
sin pu
=
du.
PU
If there is a point xo and 6 > 0 such that f(X0 + u ) + Ax0  u)  2f(XO) du < (5.1.62)
I
U
03,
show that Iim,,+" S ( f ; XO;p) = f(x0). (5.1.62) is Dini's condition for the convergence of the Fourier inversion integral. This result states that i f f E L', then the convergence or divergence of S ( f ; x ; p) at a particular point is governed entirely by the behaviour off in a neighbourhood of that point. This is Riemann's localization theorem. (Hint: See also BOCHNERCHANDRASEKHARAN [I, p. lo]) I f f is of bounded variation in an interval including the point xo, show that Iirn,,+" S(f; xo; p) = 2'[f(x,+) + f(xo)]. This is Jordan's rheorem for the convergence of the Fourier inversion integral. (Hint: By Problem 5.1.2(iv) sin pu du. S(f; xo; P)  f ( X 0 ) = Jo" [f(xo u) + f(xo  U )  2f(xo)] U
;
+
s,"
Splitting into the two parts J:, for some 6 > 0, the first tends to 2'lf(xO+) + f(xo)] as p + c1) (compare Problem 1.2.10), whereas the second tends to zero by the RiemannLebesgue lemma; see e.g. R. R. GOLDBERG [I, p. 10 ff]) Iff E L' n BV, then for every x f(x)
=
1"
lim 
o+m
4%
 p f
u) efx"do, (
which is referred to as Jordan's criterion (for Fourier integrals). Examine the particular case where f is the characteristic function of [ h, h], /z > 0, thus f(.r) = K [  h , h ] ( X ) . (Hint: Compare with Problem 5.1.2(iv))
206
FOURIER TRANSFORMS
6. (i) LetfE X(R) and h > 0. Show that (cf. (5.1.27))
+
thus j : f ( o u) du E X(R). (ii) I f f € L' and h E R, show that for all v E R
fW, hf^(O),
(iii) Let f~ L'. Show that j : f ( x and /ohf(x
u f 0 u = 0.
+ u) du (as a function of x ) is of bounded variation
+ u) du =
f "(u)
elxudv
for all x, h E R. In other words, one has the inversion formula
[2, p. 1441) (Hint: Use Jordan's criterion; see also ACHIESER 7. (i) Find two functions f, g E L', neither of which vanishes anywhere, such that (f*g)(x) = 0. (Hint: Consider O1 and use Prop. S,l.l(ii), the convolution and uniqueness theorem; see also HEWITTSTROMBERG [l, p. 4141) (ii) Suppose that f E L1 and (f*f)(x) = f ( x ) a.e. Prove thatf(x) = 0 a.e. (iii) Suppose that f E L' and ( f * f ) ( x )= 0 a.e. Prove that f(x) = 0 a.e. 8. Prove the Bernstein inequality for the derivative of a low frequency function, i.e. : Let x E W:l and suppose that f ( v ) = 0 for IvI 1 n. F o r f e X(W) show that Il ( f* x)'Ilx(~)5 An [ I f * xIIx(~),where A is a certain constant. (Hint: Let 1E Wtl be such that I^(u) = 1 for Iul I1. Show that x = x * (nl(n 0)) and apply Problem 3.1.5 to prove the assertion with A = lll'lll; see H. S. SHAPIRO [I, p. 951) 9. (i) Let f ( x ) , f(x)/x E L'. Show that [f(o)/o]^(u) = i J," fA(u) du. (ii) Let S be the space of all rapidly decreasingfunctions, i.e., the set of all those C"functionsffor which limlxl+m Ixlkf(')(x) = 0 for all k, r E P. Show that the Fourier transform maps S onto S. (Hint: See e.g. ZEMANIAN [l, p. 1821) 10. For f E L'(0, m) the Fouriercosine [sine] transform is defined for v E R by f?(u) =
2
[
Jomf(u)cos uu du f;(v)
=
,/lom :f(u) sin du] uu
Show that i f f € L1 is even [odd], thenf^(u) = f,;(v) Lfn(v) = zXA(v)]. 11. (i) Show that if the rth ordinary derivative f(')(xo) exists, so does the rth Riemann derivativef['](xo), and we have ~["(xo)= f(')(xo). (ii) For the functionf(x) = x sin ( x  ~ for ) x # 0, = 0 for x = 0 show that the second Riemann derivative exists at x = 0 but notf"(0). (Hint: CHAUNDY [l, p. 1261) 12. If the rth onesided difference ALf(x) off at x with increment h E R is defined successively by AXx) = f ( x h)  f ( x ) , A;f(x) = A;AL'f(x), it may be shown by induction that
+
ALf(x) =
2 (l).k(L)f(x
+ kh).
k=O
The rth Riemann derivative off at xo with respect to the onesided difference is then defined by 1 f['l(xo) = lim  A; f(xo), h0
h'
FOURIER TRANSFORMS ASSOCIATED WITH THE LINE GROUP
207
in case the limit exists. We use the same notation since no confusion may occur. The latter definition is particularly useful in Chapter 10. ( 9 Show that if the rth ordinary derivative f(')(xo)exists, so does the rth Riemann derivative with respect to the onesided difference, and both are equal. (ii) Show that x sin [(27r/log 2) log XI, x # 0 0, x = o has a second Riemann derivative (with respect to the onesided difference) at x = 0, but there exists no ordinary derivative at the origin. (Hint: CHAUNDY [I, P. 1261) (iii) Show that for the function f of Problem 5.1.11(ii) the second Riemann derivative with respect to the onesided difference does not exist at x = 0. 13. (i) Show that if the rth ordinary derivative f(')(xo) exists, so does the rth Peano derivative f(')(xO), and we have f(')(xo) = f(')(xo). (ii) For the function f(x) = exp {  x  ~ }sin (exp { x  ~ } )for x # 0, = O for x = 0 show that Peano derivatives of every order exist at x = 0 (and are zero), whereas the second ordinary derivative does not exist at x = 0. (Hint: CHAUNDY [I, p. 1191) 14. (i) Prove (5.1.49). (Hint: Use the formula
evaluate thejth derivative and set x = 0) (ii) Forf(x) = 1x1 x'l show thatfhas only Peano derivatives up to the order r  1 at x = 0 (and these are zero), but Riemann derivatives (with respect to the central difference) of higher order exist (and are zero). (Hint: See also BUTZER[lo]) 15. I f f is defined in some neighbourhood of the point xo and if the (r  1)th ordinary derivative f(' l)(x0)exists, then we call
f @ (xo) = lim
+ h) 
r1 hk
,f(k)(xo)] hr k=o k . the rth Taylor deriuatiue offat xo if the limit exists. The following shows that the Taylor derivative is more general than the ordinary but less general than the Peano derivative. h0
[f(xo
(i) Show that if the rth ordinary derivative f(')(xo) exists, so does the rth Taylor derivativef@(xo),and we havef@(xo) = f(')(xo).On the other hand, the existence of the rth Taylor derivative at xo implies the existence of the Peano derivative at xo, and both are equal. (ii) For the function (r, n E N, r, n 2 2)
show that there exist only the first (r  1) ordinary derivatives at x = 0 (which are zero), but the rth Taylor derivative f @ ( O ) exists (and is zero). On the other hand,fhas Peano derivatives up to the order [(r  l)(n 1)  13 at x = 0, whereas the (r + 1)th Taylor derivative does not exist at x = 0. (Hint: DENJOY [l], CHAUNDY 11, p. 1371) 16. Use (5.1.58) to deduce the uniqueness theorem [and RiemannLebesgue lemma] for L1Fourier transforms from that for the finite Fourier transform. (Hint: Apply Problem 3.l.lO(iii) and the uniform continuity of L'Fourier transforms; see KATZNELSON [l, p. 1291)
+
208
FOURIER TRANSFORMS
5.2 LpTheory, 1 c p I 2 The definition of the Fourier transform for a function f E LP, 1 < p I2, is more complicated than for f~ L'. In fact, recalling the definition (5.1.1) of the Fourier transform forfE L', (5.2.1)
fA(u) =
I J m f(u) e{""du 4%  m
we observe that for arbitrary functions f~ L p , p > 1, the integral defining the transformf" need not exist as an ordinary Lebesgue integral (see Problem 5.2.1), and thus the problem is to give a suitable interpretation of (5.2.1). The question of defining the Fourier transform on L P , 1 < p I 2, is divided into the cases p = 2 and 1 < p < 2. We first study p = 2 and then that for 1 < p < 2 as an application of the case p = 2 and the M. RieszThorin convexity theorem. The treatment is quite analogous to Def. 4.2.8 of the Fourier transform on Ip, 1 < p s 2. 5.2.1 The Casep = 2 We begin by discussing the case that f belongs to both L' and L2 and reduce the general case hereto. The following proposition will turn out to be an extension of (5.1.24).
Proposition 5.2.1.
Iff€ L1 n La, thenf" E La and
(5.2.2)

Proof. If we set f * ( x ) = f(x) and h(x) = ( f * f * ) ( x ) , then it follows by Prop. 0.2.1, 0.2.2 that h E Co n L', and an application of (5.1.16) at x = 0 gives

But according to Prop. 5.1.1(iv) and Theorem 5.1.3 we have h"(u) =f"(u)f"(u) = If"(u)12, and thus the integrand is a positive function which increases monotonely to h" as p + 00. Hence we may apply Lebesgue's monotone convergence theorem to deduce hn E L1 and h^(u)du = d/2nh(O). Since h(0) = 1 f li; and h"(u) = If"(u)la, the result follows.
Jrm
Proposition 5.2.2. Let f E La. DeJiningfp by (5.2.3) then f, E L' n La and f; E La for each p > 0. Moreouer, f; converges in Lanorm to a function in La as p + 00.
Proof. It follows immediately by Holder's inequality that f,E L' n La for all
p
> 0,
FOURIER TRANSFORMS ASSOCIATED WITH THE LINE GROUP
and thus fpA E L2 by Prop. 5.2.1. Moreover, since ji; off,,  f,,, (5.2.2) gives for p1 c p a
 f&
209
is the Fourier transform
Sincef E La, the latter sum tends to zero as pl, p 2 + co, giving limD1.Da+mlJ f ;  fGllz = 0. Thus by completeness of L2 there exists a unique function in La such that the functionsf: tend to it in the Lametric as p + co. We may now define the Fourier transform of any L2function. Definition 5.2.3. For f
E
La we dejne the Fourier transform F a M off as
Thus the Fourier transform off given by (5.2.4)
E
L2 is the uniquely determined function F2W E L2
lim IIFa[[fl(o)  
U)
P m
e'"" du 112 = 0.
The latter definition is meaningful by Prop. 5.2.2. Note that the Fourier transform F 2 M as an element of La needs to be defined only a.e. Using the same arguments as above, one might give the definition of the L2Fourier transform in the seemingly more general form
where pl, p 2 tend to co independently. But it is easy to see that both definitions would be equivalent. For the Fourier transform on La we obtain as an extension of Prop. 5.2.1
Theorem 5.2.4 The Fourier transform (5.2.4) defines a bounded linear transformation of La into La which preserves norms, i.e., for f E La (5.2.5)
IIF21flIIa =
llflla
IIF21f]  fpl12 = 0, it follows that Proof. Since 11f,^i12 = IIF2M112. On the other hand, 11fPl12= 11f 112, and since f , E L' n L2, Prop. 5.2.1 implies l\f:\\ 2 = )I f,\l 2 . Upon collecting results the proof is complete. We remark that we shall later on show that the Fourier transform (5.2.4) actually defines an isomorphism of the Hilbert space La onto itself. Equation (5.2.5) is again referred to as Parseval's equation (cf. (5.1.24)).
The Case 1 < p < 2 We have defined the Fourier transform in Lp for the valuesp = 1,2. It is natural to inquire whether this is also possible for other values of p . The answer given in the 5.2.2
14~.~.
210
FOURIER TRANSFORMS
following will be affirmative for 1 < p < 2. As a significant application of the M. RieszThorin convexity theorem we obtain
Proposition 5.2.5. Let 1 Ip I2. I f h E Soo, then h"
E
LP'
and
(5.2.6) Ilh^llP* IlhllP. Proof. In order to apply the convexity theorem (cf. Sec. 4.4), let R1 = Ra = R, p1 = pa = Lebesgue measure on R, and let T be the transformation which assigns to each simple function h its Fourier transform h" of (5.2.1):
Obviously, since So, c L', Tis welldefined as a linear transformation on So, which by (5.1.2) satisfies IIThIIm 5 Ilhlll,
(5.2.8)
and by (5.2.2), since So,
c
La,
IIThIla = Ilhlla. Thus T is of strong type (1 ;a)and (2; 2) on So, with constants Ml = Ma = 1. The M. RieszThorin convexity theorem then implies that T is also of strong type (p; p') satisfying (5.2.6). As an immediate consequence of Prop. 0.3.4, 0.7.1 we have (5.2.9)
Proposition 5.2.6. The operator Tas defined in (5.2.7) on Soo can be uniquely extended to a bounded linear transformation of LP, 1 I p I2, into LP' such that (5.2.10) IITfllP, 5 I l f l l P holds for all f E Lp, Thus we are able to assign to eachf E Lp,1 < p < 2, a uniquely determined function Tf E Lp' for which (5.2.10) holds and which is of the form (5.2.7) on Soo. We may furthermore prove
Proposition 5.2.7. Let f E Lp, 1 < p < 2, and f, be defined as in (5.2.3). Then (5.2.1 1)
lim IITf f,^IIp,= 0.
D m
Proof. By HiSlder's inequality it follows that f , defined by (52.1). Moreover,
E
L1 n LP, and therefore f,^ is well
(5.2.12)
Let p > 0 be fixed. Then by Prop. 0.3.4 we choose a sequence { h k } of functions in Soo which vanish outside of [  p , p ] and which approximatef, in the LPmetric, thus (5.2.13) lim 1 f,  hkllp= 0. k
(0
\If,
By Holder's inequality this implies limk4m  hk\Il = 0. Therefore we obtain limk+m1 f;  h;;'llc = 0 by (5,1.2), i.e. 1imk+,,,h;(u) = f,^(u) for all u E R.
FOURIER TRANSFORMS ASSOCIATED WITH THE LINE GROUP
21 1
Since "Tf,  hkhJIPI I /If, hkllp by equations (5.2.7) and (5.2.10), we have Iimk+" /IT&  h c ( I , = 0 by (5.2.13). Hence by Prop. 0.1.10 there exists a subsequence {hk,} such that limjrmh;;(u) = (Tf,)(v) for almost all u E R. Therefore it follows for f E Lp, 1 < p < 2, and for each p > 0 that
(5.2.14) (TfD)(o) = f,^(u) extending definition (5.2.7) on Soo to all functions f , . But by (5.2.10) this implies 1 Tf .,l^f 5 11f  fDllp from which the assertion (5.2.11) follows by (5.2.12). We are now justified in defining Tf as the Fourier transform off E LP, 1 < p < 2. Definition 5.2.8. For f
(5.2.15)
E
LP, 1 < p < 2, we define the Fourier transform F P [ f ]o f f by
(P') f,^(u) = 1.i.m. "') FP[fl(u) = 1.i.m. Pr"
P'"
1' d2T
f(u) ,IUU du,
D
In other words, the Fourier transform o f f € Lp, 1 < p < 2, is defined as the uniquely determined function F p [ f lE LP' given by (5.2.15). We have Theorem 5.2.9. The Fourier transform (5.2.15) defines a bounded linear transformation of Lpy1 < p < 2, into Lp' which contracts norms, i.e., for f E Lp
(5.2.16) IIFP"f1llP,5 I I f IIP' The inequality (5.2.16)is often called the Titchmarsh inequality. In the following we shall sometimes call (5.1.2) and (5.2.5) by the same name so as to cover inequalities of this type for all 1 5 p I2. Proposition 5.2.10. I f f E L1 n Lp, 1 < p I2, then F'lfl(u) = Fp"fl(u)a.e.
Proof. According to (5.2.4), (5.2.15), and Prop. 0.1.10 there exists a subsequence such that PP[fl(u) = lim k+*
Sincef
E
1 '* 
f ( u ) ecu du a.e.
%&/pk
L1, the limit exists for all u E R and FP[f](u= )
1
1 " f(u) esu du = F1[fl(u), d 2 n "
giving the result. Let us mention that the latter proposition extends (5.2.7) and (5.2.14) to all f E L1 n Lp, 1 < p < 2. Prop. 5.2.10is important as it shows that the definition (5.2.1) for p = 1 on the one hand and Def. 5.2.3,5.2.8for 1 < p I2 on the other hand are consistent. We are therefore justified in employing the more simple notation f to denote the Fourier transform for all spaces Lp, 1 Ip I2. It will be clear from the context in which sense f " will be taken. If any confusion should occur we shall return to FP[fl. As in case of the definition of Fourier transforms in Ipspaces (cf. Problem 4.2.3) it is not generally possible to define a Fourier transform for functionsf€ Lp, p > 2, by the methods so far employed (see Problem 5.2.2). See also Sec. 5.4.

212
FOURIER TRANSFORMS
5.2.3 Fundamental Properties Up to this stage we have defined the Fourier transform forfe Lp, 1 < p I 2, and showed that it obeys the Parseval equation (5.2.5) in case p = 2 and the Titchmarsh inequality (5.2.16) in case 1 < p < 2. We shall now study further properties of these transforms. Proposition 5.2.11. I f f € Lp, 1 < p I2, then for eachfixed h E R, p > 0:
+ h)]"(u) = e"vf"(u) a.e., [eth"f(o)]"(u) = f " ( u + h) a.e.,
[f(o
[f(./P)l"(0)= Pf"(P0) a.e.9 [f( o)]^(u) = f^(u) a.e.
The proof is left to Problem 5.2.3. The following result establishes the conuolution theorem for Fourier transforms in 1 < p I 2.
LP,
Theorem 5.2.12. I f f €
LP,
1 < p 5 2, and g E L1, then
FPlf*g](u) = Fplfl(u)F1[g](u) a.e.
(5.2.17)
Proof. In Prop, 0.2.2 we have already shown that the convolution f * g of YE Lp, 1 < p I2, and g E L1 exists a.e. and again belongs to Lp. Therefore both sides of (5.2.17) are welldefined. To prove equality we first supposefE L1 n Lp. Thenf* g E L1 n Lp by Prop. 0.2.2, and F1[f*g](u) = F'[fl(u)F'[g](u) for all u E R by Theorem 5.1.3. (5.2.17) now follows in virtue of Prop. 5.2.10. In general, iff€ Lp, 1 < p 5 2, then the functionsf, of (5.2.3) belong to L1 n L p for each p > 0 and thus FPv,* g](u) = Fp[f,](u)F1[g](u) for almost all u E R. This implies by the Minkowski and Titchmarsh inequality that for each p > 0
IIFPLf*81  FP[flF1[glllPf
 mf,* 8 1 1 1 P , + IIFP[f,lF"gl  Fp[flF"glllP, 1IUh) *gllp + IIF1[glll~lIFP[f~  f l l l p * 5 2 Ilgllillff~llp~
5 llF"[f* 81
5
which by (5.2.12) tends to zero as p +.co, proving (5.2.17). We next prove the Parseual formula for Lpfunctions. Proposition 5.2.13. Iff, g E Lp, 1 < p 5 2, then
Proof. For positive p1 and pa let& and gDzbe defined by (5.2.3). Since&,, g,, may apply Prop. 5.2.10, 5.1.4 to deduce
E
L1 we
FOURIER TRANSFORMS ASSOCIATED WITH THE LINE GROUP
213
Moreover, limpl,m 1 f "  fGllp, = 0, lim,,,, 1) f  f,,IIp = 0 by Titchmarsh's inequality and (5.2.12). Therefore, taking the limit as p1 + 03 for each side separately and using Holder's inequality, we obtain for each p2 > 0 Jm
Jm
from which the result follows by letting pa+ 03. We remark that in case p = 2 formula (5.2.18) reads
(f",a= (f,p).
Summation of the Fourier Inversion Integral Next, we turn to the inversion problem of the Fourier transform in Lp, 1 < p I 2. Whereas we have so far assigned to each f E Lp, 1 < p I 2, its Fourier transform f " and studied some of its fundamental properties, we are now interested in determiningf explicitly, given f " . As in the case p = 1 (see (5.1.6)), the formal inversion will be given by 1 " (5.2.19) f "(v) efxudu. f(x)= 5.2.4
j
r
m
But sincef " E Lp', the Fourier inversion integral (5.2.19) does not exist in general as an ordinary Lebesgue integral, and the problem again is to interpret (5.2.19) suitably. Theorem 5.2.14. Let f E Lp, 1 < p I 2. Then for a efactor the 8means of the integral (5.2.19), defined for each p > 0 by (5.2.20)
V ( f ;x
exist for all x E R, belong to
)Y 1/2r
;p LP
Sm
B(;)f"(u) erxudv,
m
and satisfy
II wf; P)IIP 5 IIB^IIlII f IIP limm I1 wf; P) = 0.
(5.2.21)
0 ;
(5.2.22)
0 ;
(P > 01,
f(0)IIP
0
Thus the inversion integral (5.2.19) is 8summable to f in LPnorm. Proof. Since I1811m I ~ ~ 8 by A ~(5.1.21), ~ l and as 8 E L1 n L" implies 8 E Lq, 1 < q < 03, it follows by Prop. 5.2.10, 5.2.11, 5.2.13 that
(5.2.23)
u(f; X ; P)
=
&
f ( x  u)8^(pu) du.
Since 0" E L1 n L", too, the proof is now an immediate consequence of Prop. 0.2.1, 0.2.2, Lemma 3.1.5, and Theorem 3.1.6. Theorem 5.2.15. Let f E Lp, 1 < p I2. If for a &factor 8" is moreover positive and monotonely decreasing on [0,m), then the Fourier inversion integral is 8summable a.e. to f ( x ) , i.e. (5.2.24)
lim PW1/%
jmB(:)f^(u) m
elxudu = f ( x ) a.e.
214
FOURIER TRANSFORMS
For the proof as well as for the explicit formulae of some important examples of 0factors we refer to Problem 5.2.4. Proposition 5.2.16. Let f
Lp, 1 < p I 2. V f " E L', then
E
1
zz
(5.2.25)
"
f"(v) elxudo = f ( x ) a.e.
1m
The proof is that of Prop. 5.1.10. One consequence of (5.2.25) is the uniqueness theorem for Fourier transforms in Lp, 1 < p I2. Proposition 5.2.17. Letf, g f(x) = g(x) a.e.
E
Lp, 1 < p 5 2. Zff"(v) = g"(v) for almost all v E R, then
This shows that the Fourier transform defines a onetoone bounded linear transformation of the Banach space Lp,1 < p I2, into the Banach space Lp'. If 1 < p < 2, the corresponding mapping is into, but not onto (see Problem 5.2.6), yet the set [Lp]" = {g E Lp' I g = f " a.e., f E Lp} is dense in Lp'. In case p = 2 the Fourier transform is a transformation of L2 onto L2 as we shall see in Sec. 5.2.6. Proposition 5.2.18. Let f
E
Lp, 1 < p I 2. I f g E L' is such that g"
E
L', then
(5.2.26) (5.2.27) In case f " E L1 we have furthermore (5.2.28) I l f I12 = The proof is left to Problem 5.2.7.
Ilf"II2.
Fourier Transforms of Derivatives In this subsection r always denotes a natural number.
5.2.5
Proposition 5.2.19. For 1 < p I2 let f (5.2.29)
If(')]"(v)
E
Lp n ACC' and f ( r )E Lp. Then
= (iv)'f"(v)
a.e.
Proof. As in the case p = 1,f E AC&l implies (5.1.26) for all x duce the function m by (5.1.27), we have (cf. (5.1.28)) (5.2.30)
k0
(l)'@(x
E
R, and if we intro
+ k ) = ([m*I'*fCr))(x).
Since m E L1 n I$ and therefore m E LQ, 1 Iq 5 00, the righthand side exists by Prop. 0.2.1 for all x E R as a function in C,,and by Prop. 0.2.2 as a function in Lp.
FOURIER TRANSFORMS ASSOCIATED WITH THE LINE GROUP
215
Taking the Fourier transform of equation (5.2.30), we obtain by (5.1.29), Prop. 5.2.11, and Theorem 5.2.12
for almost all u E R, which implies (5.2.29).
Proposition 5.2.20. If for f
E
Lp, 1 < p I 2, there exists g E Lp such that
(5.2.31)
a.e.,
(iu)'f"(u) = g"(u)
thenfe WLp(for the dejnition see (3.1.48)).
Proof. Let r = 1. In conjunction with the proof forp = 1 (cf. (5.1.31)) the assumption (5.2.31) gives for each h > 0
(5.2.32)
[f ( x )  f ( x  h)]"(u)
=
g(x [/:h
I^
+ u ) du 1 (u)
a.e.,
the integral being the convolution of m(  o / h ) E L' and g E Lp. As an application of the uniqueness theorem for Fourier transforms in Lp we obtain for each fixed h > 0
f ( 4 f ( x  h) = /xxh
(5.2.33)
g(u) du
Since5 g E Lp, 1 < p 5 2, impliesf g" E Lp', we have (1 + Iul)f"(u) E Lp' by (5.2.31), and since (1 + I u l )  l E L*, q > 1, it follows by Holder's inequality that f"(u) = ( I 1
+ I4
+
lUl)f*(U)E
Hence we obtain by Prop. 5.2.16 for almost all x
E
L'.
R
the latter integral defining a continuous function 4 which belongs to Co by Prop. 5.1.2. Now (5.2.33) delivers +(XI
 +(x
 h) = /x:hg(u)du
which holds for all x E R, h > 0, since both sides are continuous functions. But E C, implies limb+ +(x  h) = 0, i.e., the limit limb+ J :  h g(u) du exists and
+
the integral being in general only conditionally convergent (note that g E Lp, p > 1, only). Since f = a.e., Prop. 5.2.20 is established for r = 1. The proof for arbitrary integers r 2 2 follows along the same lines as for the case p = 1. Again, (1 + i u ) l  , E [L']" and (I + iu)lr(iu).f^(u)E [Lp]" by Theorem 5.2.12 and (5.2.31). This implies iuf"(u) E [LP]", i.e., there exists g, E Lp such that
+
216
FOURIER TRANSFORMS
iuf"(u) = gT(u). If we now apply the result for r = 1 and repeat this method successively, we obtain a sequence of functions gk E Lp, 1 < k Ir  1, such that f ( x ) = l? gl(u) du, gk(X) = j? gk+1(u) du, 1 I k Ir  2, gr i(x) = j? g(U) du, and hence (5.2.35)
f ( x ) =m:J
dul
1;:.
du,.
.
.I:'
g(ur)du, a.e.,
the integrals being in general only conditionally convergent. If we define the righthand side as the function $, then f = # a.e. and $ E ACf;', $(lC) = g k E Lp, 1I k Ir  1, and I$(') = g a.e. This proves the proposition. Note that # l C ) E Co,
Osksr1. The preceding results actually give equivalent characterizations of the function class W[. for 1 c p I 2. Indeed, extending the definition (5.1.36), let (5.2.36)
W[LP; $(u)] = {f
E
LP I $(o) f ^(u) = gA(u),g
E
LP}
(1 I p I 2)
for an arbitrary complexvalued (continuous) function $(u) on R. Then together with Theorem 5.1.16,
Theorem 5.2.21. Let f
E
LP, 1 Ip
5 2. The.following assertions are equivalent:
(i) f E Wlp, (ii)f E W[Lp; (iu)'], (iu) there exists g E Lp such that the representation (5.2.35) holds, each of the iterated integrals existing (for 1 < p I2 only conditionally) as a function in Lp. We again have the following consequence: I f f E Lp, 1 I p I 2, is rtimes continuously differentiable with f ( r )E Lp, then all derivatives f(lC),1 I k Ir  1, belong to LP.
5.2.6 Theorem of Plancherel We return to the inversion problem for Fourier transforms. We obtained the original function f E Lp, 1 < p I 2, from the Fourier transform f " by summation of the Fourier inversion integral. This led to the inversion formulae (5.2.22) and (5.2.24) which are also valid in L1. It is a very important feature of the theory of Fourier transforms in Lp, 1 < p I 2, that we may interpret the integral (5.2.19) in the following, more direct fashion which is completely symmetrical to Def. 5.2.3, 5.2.8: (5.2.37)
In this section we shall prove (5.2.37) only in case p = 2, whereas the proof for the cases 1 < p < 2 is essentially dependent upon the theory of Hilbert transforms and thus left to Sec. 8.3.4.
Theorem 5.2.22. I f f E La, then (5.2.3 8)
FOURIER TRANSFORMS ASSOCIATED WITH THE LINE GROUP
217
Proof. In view of Theorem 5.2.4 we have f E La and thus exists as an element of La. According to Parseval's formula (5.2.18) and Parseval's equation (5.2.5) the vanishes. Therefore f(x) = Lf^l^(x)a.e., and it (Hilbert space) product 117
w]Alli
follows by the definition of the Fourier transform in La that
which implies (5.2.38) by taking complex conjugates. In the following we summarize the results of Theorem 5.2.4, 5.2.22 to obtain the theorem of Plancherel.
Theorem 5.2.23. For eachf E La there exists a function f * E La, called the Fourier transform off, such that 1) f ,, f a + 0 as p + oc), wheref , is defined by (5.2.3). Similarly, the relation (5.2.38) holds. The functions f and f , satisfy the Parseval equation 1 f (12 = 11f"lla. Every f E La is the Fourier transform of a unique element of La.
In other words, the Fourier transform on La is a onetoone, normpreserving linear transformation of La onto La. Moreover, it preserves inner products, thus is unitary, as is shown by Proposition 5.2.24. Iff, g E La, then (f,g )
=
( f ", gn), i.e.
(5.2.39) The proof follows immediately by Parseval's formula (5.2.18). Indeed, since [F]"(x) a.e., we have
g(x) =
Therefore the Fourier transform defines an isomorphism of the Hilbert space La onto itself. Problems 1 . Show that the function [ d m ( l
+ llog lxIl)]l belongs to La but not to Lp for any other value of p. 2. Construct examples of functions in order to show that the definition of a Fourier transform as given in this section cannot be extended to Lpspace, p > 2. (Hint: TITCHMARSH [6, p. 1 1 11, ZYGMUND [7II, p. 2581) 3. Prove Prop. 5.2.1 1. 4. (i) Prove Theorem 5.2.15 and show that
Im
lim VO+
4%
euluyA(u) e ( x u c i u = f(x)
a.e.
 m
(ii) Formulate and prove the counterpart of Theorem 5.2.15 in case 6" satisfies the assumptions of Prop. 3.2.4. As a consequence, show that the Fourier inversion integral o f f E Lp, 1 c p I2, is Cedro summable to f ( x ) almost everywhere,
i.e. f"(u) elxu do = f(x)
a.e.
218
FOURIER TRANSFORMS
I2, such that f $ L' but f" E L'. (Hint: Consider the functionf(x) = (ix)'(eihx  l), h > 0. Show that f E Lp, 1 c p I2, and d%j"(u)= 1 for 0 I u I h, = 0 for u c 0, v > h) 6. Let 1 < p < 2. Show that there exist functions g E LP' which are not the Fourier transform of some function in LP, yet the set [LP]" = { g E LP' I g(u) = f"(u) a.e., f E Lp} is dense in Lp'. (Hint: cf. Problem 5.1.3, compare also with HEWITT[l, p. 1771) 7. (i) Prove Prop. 5.2.18. Show in particular that if x E NL' is such that x" E L', then for the singular integral J ( f ; x ; p )
5. Show that there are functions f E LP, 1 < p
for every f E Lp, 1 I p I2. Apply this to the standard examples of kernels x. (ii) Let f,g E La. Show that for all x E R 1
1
(f*g ) ( x ) = 7
"
f"(u)g"(o) elxudu. d2n m (Hint: See e.g. ACHIESER [2, p. 1161, ZYGMUND [7II, p. 2531) 8. Use the Parseval equation to evaluate
(Hint: Apply (5.2.5) to f ( x ) = q  l , l , ( x ) , f ( x ) = (1  ~ x ~ ) q  l , l l ( xrespectively. ), Compare with HEWITTSTROMBERG [l, p. 413 ff], ZYGMUND [7II, p. 2511) 9. LetfE LP, 1 c p I 2. (9 Show that for every x , h E R
f " ( ~ du ) =1/2=
!"
 m ,lhu i u
f ( u ) elXu du.
In particular,
which may be used to provide a further interpretation of (5.2.1) for functions
f~ Lp, 1 c p I2. (Hint: Use Problem 5.1.6(i) and apply (5.2.18) with g(u) = m((x  u)/h), m being given by (5.1.27)) (ii) Show that for every x , h
E
R
f "(u)
elxudu.
In particular,
which may be regarded as a further interpretation of the Fourier inversion integral for functions f e Lp, 1 < p I2. (Hint: Again apply (5.2.18) but with g(u) = (iu)l(efhu  l), cf. Problem 5.2.5.The formulae of this Problem are also derived [2, p. 1201, ZYGMUND [7II, p. 2541, HEWITT[l, p. 173 a, WEINin e.g. ACHIESER BERGER [l, p. 3141) 10. Let f~ Lp, 1 < p S 2, be such that (ix)f(x) E LP. Show that [f"]'(v) exists almost everywhere as a function in Lp' and [f"]'(u) =  [(i ~)f(o)]"(u) a.e. (Hint: Show that j~ L1 and thus f A E CO. Then use the preceding Problem to derive f"(u) =  SY [(i ~)f(o)l"(u) du, the integral being only conditionally convergent; see also [ 1, p. 1261) BOCHNERCHANDRASEKHARAN
FOURIER TRANSFORMS ASSOCIATED WITH THE LINE GROUP
21 9
E Coohas a piecewise continuous derivative, show that f " E L1 and thus f(x) = [fn]"(  x ) . (Hint: Use the proof for r = 1 of Prop. 5.2.19, 5.2.20) 12. Iff E W[Lp; (iu)'] for some r E N, show that f E W[Lp;(iv)j] for j = 1, 2, . . ., r  1 .
11. I f f
5.3 FourierStieltjes Transforms
5.3.1 Fundamental Properties For p E BV (cf. Sec. 0.5.) the FourierStieltjes transform p"(u) is defined by (5.3.1)
p"(u) =
d2rr
m
etuudp(u) = [p(o)]"(u)
(u E
W).
03
This integral is absolutely and uniformly convergent for all u E R and thus defines a function p" which is defined at each point of R. We shall first give some of its operational properties.
Proposition 5.3.1. For p E BV (i)
[ (+) ]"(u) p
=
( p > 0, h, u E
e'hv pv(pu)
(ii) [F]"(u) = p"(v) (iii) if p is absolutely continuous, i.e. p(x) for all u E R.
= Jf
f ( u ) du, f
E
w,
(0 E R), L', then p"(u) = f "(u)
The proof is left to Problem 5.3.1,
Proposition 5.3.2. The FourierStieltjes transform defines a bounded linear transformation of BV into C . Proof. The fact that p"(v) is a bounded function on R follows by (v E R).
Furthermore, we have: for all u, h E R
say. Given B > 0, we choose p so large that I2 < ~ / and 2 then choose h so small that Zl < 42. Thus p" is uniformly continuous, and the proof is complete.
220
FOURIER TRANSFORMS
We observe that the RiemannLebesgue lemma does not hold for FourierStieltjes transforms. Indeed, the FourierStieltjes transform of
I
0,  o o c x < o S(x) = 2/;;72, x=o
(5.3.3)
f i x ,
is given by
(5.3.4)
O<X 0 by (5.3.14)
exist for all x
U(dp;X ; p) E R,
=
jm O(i )

4%oo
belong to L' and satzkfy IIWdCl; ";P)II1 5
(5.3.15)
Il8^II1IItLllBY
lim
J
~ + m a
(P
> 01,
r m
r m
(5.3.16)
elxup"(v) do,
h(x)U(dp;x ; p) dx
=
J  m h(x) dp(x)
for every h E C. According to (5.3.11) we have for each p
= 0
and hence the assertions follow immediately by Prop. 0.5.5, Lemma 3.1.5, and Problem 3.1.9 upon taking ~ ( xp); = p8^(px). Thus the FourierStieltjes inversion integral is 0summable to p in the weak* topology of (C,)*. Concerning generalized pointwise convergence of (5.3.13) we have in view of (5.3.17) and Prop. 3.2.6 Proposition 5.3.8. Let p E BV. Iffor a Ofactor 8^ is moreover positive and monotonely decreasing on [0,a),then the FourierStieltjes inversion integral is 0summable a.e. to p'(x), i.e. (5.3.18)
Thus, for example, for the Abelmeans of (5.3.13) we have (cf. (5.1.8)) (5.3.1 9)
eY1"'eix"p"(v) dv = p'(x) a.e.
For further results see Problem 5.3.3.
223
FOURIER TRANSFORMS ASSOCIATED WITH THE LINE GROUP
We continue our investigations on the convergence of the FourierStieltjes inversion integral by establishing the theorem of Ltuy. In connection with (5.3.18) it is natural to look for inversion formulae delivering p instead of p'. For this purpose we may integrate (5.3.18), but Problem 5.1.6 then suggests that we may avoid the use of convergencefactors. These heuristic considerations are confirmed by
Theorem 5.3.9. I f p
E
(5.3.20)
+ h)  p(x) = lim 
p(x
BV, then for any h E R ( x E R).
etxup"(u)du
Furthermore, (5.3.21)
IOh
[p(x
1  cos hu
+ u)  p(x  u)] du = 
e
p
(4do,
the integral being absolutely convergent.
Proof. If we define q(x) by (5.3.6), then q E BV n L' and it follows by Jordan's criterion (cf. Problem 5.1.5) that (5.3.22)
j"
q(x) = lim p1/%
(x E R).
, , u) elxv du
0"
This, in virtue of (5.3.8), implies (5.3.20). Intuitively, (5.3.21) follows from (5.3.20) by a further integration. For a rigorous proof we introduce for a > 0, say, the function (5.3.23) which again belongs to BV n L1. By the convolution theorem for L1functions we obtain by (5.1.29) and (5.3.8) (5.3.24) If we apply (5.3.22) to q1 and set a = h, then
IOh
[p(x
1 + h + u )  p(x + u ) ] du = lim =
s"
Em Dm
6
0
 coshv el(" h)vp"(v)dv. ua
+
Since (1  cos hu)/uaE L1, the integral on the right converges absolutely. Passing to the limit and replacing x h by x, we obtain
+
which, after an obvious change of variables, establishes (5.3.21).
FOURIER TRANSFORMS
224
Proposition 5.3.10. Let p E BV be such that p" derivative p' E Cogiven by (5.3.13).
E
L'. Then p is uniformly contiriuous with
Proof. Since p" E L' and (exp {ihv}  l)/ihv is bounded, it follows from (5.3.20) that
h
(5.3.25)
efXup"(v)do.
Hence Ip(x + h)  &)I IIhl llp"Il1 and p is uniformly continuous. Moreover, since the integrand is dominated by Ip"I and converges to exp {ixv}p"(v) as h +0, we may apply Lebesgue's dominated convergence theorem to deduce that p'(x) exists for all x E R and is given by (5.3.13). By Prop. 5.1.2 this implies p' E Co. Now to the uniqueness theorem for the FourierStieltjes transform. Proposition 5.3.11. I f p E BV and p"("(u) = 0 on W, then p(x)
= 0 on W.
In other words, if p1 and pa are two functions of class BV such that p';(u) = &(v), then pl(x) = pa(x). For the proof we may use Prop. 5.3.10. Indeed, (5.3.13) implies p'(x) = 0, and thus p(x) is identically equal to a constant which must be zero since p, as an element of BV, is normalized by p( 00) = 0. Together with Prop. 5.3.2, Theorem 5.3.5, and Problem 5.3.2 we may therefore state Corollary 5.3.12. The FourierStieltjes transform defines a onetoone bounded linear transformation of BV into (but not onto) C. Moreover, it is an isomorphism of the Banach algebra BV (with convolution as multiplication) into the Banach algebra C (withpointwise multiplication).
Note that the Banach algebra BV is not only commutative, but also has a unit element given by (5.3.3), in distinction to the Banach algebra L' which has no unit (cf. Prop. 5.1.12). 5.3.3 FourierStleltjes Transforms of Derivatives Again, in this subsection r always denotes a natural number.
Proposition 5.3.13. Let f
E L'
nA C ~ G and ~ f ( r  l )E BV. Then (v E W).
(ivyf"(v) = Lf(' ')]"(u)
(5.3.26)
Proof. As in the proof of Prop. 5.1.14, integration by parts ((r  1)times)yields
This implies by (5.1.27) (5.3.28)
A',f(x) = Ioldu,... ~ o l d u ,  l ~ o ' ~ ~ , f ( r+ lu1 )(x
+ . . a +
= ( [ m *]'
* df(''))(x).
u,)
FOURIER TRANSFORMS ASSOCIATED WITH THE LINE GROUP
225
The last term is the convolution of the product (rtimes) in *.  .* m E L' withf('l) BV. Now it follows by Prop. 5.l.l(i), ( 5 . 3 3 , and (5.1.29) that
E
for all u # 0. This proves (5.3.26) in view of the continuity of all expressions involved. In order to prove the converse, let us introduce the following classes of functions:
{ f C~ I ~ AC;&' E n C'',f(l)~ L"} { f L'~ If= 9 a.e., 9 E AC;G~,#k) E L', 1 I kI r  1, #  1 1 ) E BV} { f Lp ~ If= 9 a.e., 46 E ACi&', + ( l r ) E LP, 1 I k I r} (1 < p < a).
i
(5.3.29) V;(,,=
Note that the classes Wk, X(R) = LP, 1 < p < co.
and Vk,,,
are equal by definition for the reflexive spaces
Proposition 5.3.14. If for f E L' there exists p E BV such that (5.3.30)
(iu)'f^(u)
= p"(v)
then f E Vll.
1. With q(x) as defined by (5.3.6) we have for u # 0 ethv  1 q^(u) = p"(u) = (efhu l)f^(u) = Lf(0 + h) f(o)]^(u).
Proof. Let r
=
iu
Therefore we obtain by the uniqueness theorem of the Fourier transform in L' for each 11 E R p(x h)  p(x) = f ( x h)  f ( x ) a.e.
+
+
On replacing h by h this implies rx
rx
rxh
Jt
for every x E R. Taking the limit for h + co, the righthand side tends to f(u) du since Y E L'. On the other hand, by Lebesgue's dominated convergence theorem
and thusf(x) = JZ dp(u) a.e. orf(x) = p(x) a.e. Let r 2 2. As in the proof of Prop. 5.1.15, (1 + E [L']", and hence (1 + iu)lr(iuyfA(u) E [L']" by Prop. 5.3.3 and (5.3.30). This implies iuf"(u) E [L']", i.e., there exists g, E L1 such that iuf"(u) = g;'(u). Thus the case r = 1 of Prop. 5.1.15 applies, givingf(x) = JZm gl(u) du a.e. Furthermore, (iu)'lgP(u) = ~ " ( 0 ) .If r > 2, this in turn implies iugP(u) E [L']", i.e., iug;'(u) = g$(v) for some g, E L', Hence Jm
15~.~.
Jm
226
FOURIER TRANSFORMS
Applying this method successively, we obtain a sequence of functions k I r  1, such that (iv)'kg:(u) = p"(v) and
Therefore p E L1 and
(5.3.31)
du,
f(x) =
fr: ../:" du,.
du,,
j:il
gk E
L1, 1 I
dp(u,) a.e.
If we abbreviate the righthand side by &), then C$E ACIG2, # k ) = gk E L1, 1 I k I r  1, # ,   l ) = p E BV, and the proof is complete. If we introduce the classes of functions
for an arbitrary complexvalued (continuous) function +(v) on Iw, the preceding results together with Theorem 5.2.21 may be summarized to Theorem 5.3.15. Let f E Lp, 1 I p I2. The following assertions are equivalent: (i) f E V ~ P , (ii) f E V[Lp;(iv)'], (iii) there exists p E BV i f p = 1 and g E L p i f 1 < p I 2 such that the representations (5.3.31) and (5.2.35) hold, respectively, each of the iterated integrals existing (for 1 < p I 2 only conditionally) as a function in L P . Problems (i) Prove Prop. 5.3.1. (ii) The dipole measure is the function 6, E BV defined by &(x) = 0 for x c  1, x > 0, = .\/%for  1 < x c 0, = 4 7 2 for x =  1, x = 0. Show that 61;(v) = (exp {iu}  1). (iii) The binomial measure is the function 6, E BV, r E N, defined by 6,(x) = 0 for x < rr x > 0, = d s (  l y  k ( L ) for  k < x <  k + 1, k = 1,..., r, = for x = k, k = 0, 1,. . ., r. Show that 6 r ( v ) = (exp{iu} l),. Show that there exist functions g E C which are not the FourierStieltjes transform of some function in BV. (Hint: Use the example of Problem 5.1.3, see also HEWITT [l, p. 1571 for a discussion of uniform approximation on R of elements g E C by FourierStieltjes transforms) (9 Let p E BV. Show that 1 " lim e'"p"(u) elxu dv = p'(x) ax.
.\/a(
t0+
1
4%
m
(ii) State and prove the counterpart of Prop. 5.3.8 for &factors for which 0' satisfies the assumptions of Problem 3.2.4.
(iii) Let p E BV. Show that
FOURIER TRANSFORMS ASSOCIATED WITH THE LINE GROUP
227
4. Let p E BV and f E L' be such that f " E L1. Show that 1 " ( f * dp)(x) = f ^ ( ~ ) P " ( velx" ) dv
xv1
for all x E R and that (see also KATZNELSON [l, p. 1321)
s_mmm fo
1
m 
dp(4 =
5. Let p E BV and suppose that p"(u) that
=

f ' @ ) p W do.
O(lvl') as
u
0 for some integer r 1 2. Show
(Hint: COOPER[2]) 6. Show that p~ BV is continuous if and only if limQ+,m (1/2p)k, Ip"(v)12du = 0. (Hint : LUKACS[I, p. 471, KATZNELSON [ 1, p. 1381) 7. Iff E V[Lp; (iv)'] for some r E N, show that f E W[Lp; (iu)'] for every j = 1, 2, . . ., r  1. 8. Let p E BV and p* E BVzn be given by (3.1.56). Show that the finite FourierStieltjes transform [p*]" is the restriction to the integers of the FourierStieltjes transform p", i.e. [p*]"(k) = p"(k), k E Z. (Hint: Use (3.1.57); see also KATZNELSON [l, p. 1341) 9. Let f E X(R) and 4 = p * dv, p, v E BV. Show that
'4Irn%
x mf(

U)
in jmm u 
d&u) =  /mm
f(x
Y ) C(U) dv(Y)
(ad.
In particular, (0.8.8) holds. (Hint: The result follows for X(R) = L1 by the convolution and uniqueness theorem. For X(R) = Lp, 1 < p < co, approximate f in Lpnorm by a sequence {fn} c L1 n LP (cf. (5.2.3)). To cover the case X(R) = C, let f E Coand take a sequence { fn} c Coowhich approximatesf (cf. Prop. 0.3.4); then it follows in particular for the Weierstrass integral that for every x E R, t > 0
this together with (3.1.53) gives the assertion for every f E C)
5.4 Notes and Remarks The main references to this chapter are the books by BOCHNER [7], WIENER[2], TITCH[6], CARLEMAN [l], BOCHNERCHANDRASEKHARAN [ 1 1 (see also BURKHARDT [l] and the literature cited there), ZYGMUND [7II, Chapter 161, HEWITT[l], R. R. GOLDBERG [l], WEISS[l], KATZNELSON [I]. Shorter accounts may be found in most of the texts on analysis, see for example the relevant chapters in ACHIESER [2], HEWITTSTROMBERG [I], RUDIN[4], SZ.NAGY[4], or the survey articles of WEISS[2] and DOETSCH [S]. MARSH
Sec. 5.1. For the RiemannLebesgue lemma see also ACHIESER [2, p. 1141 and BOCHNERCHANDRASEKHARAN [ 1, p. 31. In connection with the inversion theory of Sec. 5.1.2, summability of divergent integrals can of course be considered under a more general setting. If the integral Jyw g(u) dv does not exist in the ordinary (Lebesgue o r principal value) sense, various conventional definitions can be employed which assign to it a definite meaning, Such a definition should satisfy the condition of permanency, that is, if the integral exists in the ordinary sense with value I , then the value of the integral in the generalized sense
228
FOURIER TRANSFORMS
should also equal 1. Then the present integral will be said to be 8summable to the value I if limo+m j?m B(u/p)g(u)du = I, the integral'existing for each finite p > 0. Particular examples of convergence factors are 8' (Ceshro) and Oa (Abel) of (5.1.8).It follows readily that the CesAro summability of JZm g(v) dv to I implies its Abel summability to I. In this respect see HOBSON [2II, p. 384 ff],HARDY[2,p. 110 ff]. The present results on the Fourier inversion integral follow as easy consequences of the theory on singular integrals of Chapter 3. They are of course standard, see, for example, BOCHNERCHANDRASEKHARAN [11, HEWITTSTROMBERG [l,p. 400 R'J, WENS[l,p. 15 ff]. There is a theory of trigonometric integrals which corresponds to the theory of trigonometric series considered in Sec. 1.2.If g E L1(  p, p) for all p > 0, then j? g(u) elwudu is called a trigonometric integral, with value at x E R equal to limo+ g(u) etxudu wherever this limit exists. The integral {  i sgn u}g(u) efxudu is called the conjugate integral. (Note that g(u) is not necessarily a Fourier transform.) Correspondingly, one may consider the integrals
KO ~
som
1/v
g(u) cos xu du,
1/2/rr
so m
g(u) sin xu du.
Compare HOBSON1211, p. 720 ff], TITCHMARSH [6,p. 152 ff], ACHIESER[2,p. 1 1 1 ffl, ZYGMUND [7II, p. 244 ff],HEWITT [ 1, p. 152 ff].A class of functions for which a theory for the above integrals may be developed is that of locally integrable functions g(u) which tend monotonely to zero as IuI + 00, see BOCHNER [7,p. 1 ff]. There is an interesting connection between Fourier series and Fourier inversion integrals which often permits derivation of a result for one from the other. Indeed, let f~ L', a E R, and let f.(x) be the 2aperiodic function equal to f(x) in [a,a 274 Then
+
so
{ L dzr
2
[Dl
u) elxudo f,^(k) etkx} = 0 ( k = [PI uniformly in x for x E [a e, a 2a el, e > 0. The assertion remains valid if the hypothesisfE L' is replaced byf(x)/(l 1x1) E L', and a similar result also holds for con
lim
orm
of
+
+

+
jugate Fourier series and integrals. See ZYGMUND [7II, p. 2421, HEWITT[l,p. 1801. A further connection which links the theory of Fourier integrals to that of Fourier series is given by the results of Sec. 3.1.2and 5.1.5(cf. Problem 5.1.16).For a partial converse of the latter see Prop. 6.1.10. Prop. 5.1.11 on the uniqueness of Fourier transforms can also be deduced from the parallel result for the finite transform, see SZNAGY[5,p. 3161 (for further methods compare Problem 5.1.16 or BOCHNERCHANDRASEKHARAN [ I , p. 111). It can moreover be refined considerably. OFFORD[2, 31 showed that if f E L;oo and limo+m K Of(u) eruUdu = 0 for all v E R, then f(x) = 0 a.e. He was even able to weaken the hypothesis to
for all u E R; compare TITCHMARSH [6, p. 164 f]. In connection with Prop. 5.1.12, although the Banach algebra L1 has no unit element, every f E L' can be factorized into a convolution productfi *fa with fl, fa E L1. See RUDIN [l,21,and also [4,p. 192 ff]for the representation of the complex homomorphism on L'. An important topic not considered here is Wiener's theorem on the closure of translates in L1 (WIENER [2,p. 971). It has significant applications to Wiener's Tauberian theorem and enables a ready proof for the prime number theorem; it is treated in almost every book on Fourier analysis. The reader is referred to HARDY [2],WIDDER[l],BOCHNERCHANDRASEKHARAN [I]for the applications; we here follow R. R. GOLDBERG [l,p. 32 ff]for a brief formulation of the problems and results (see also ACHESER[2, p. 150 ff],REITER [l,p. 8 ff], EDWARDS [111, p. 6 ff 1) : Supposefs L'. Let TI denote the set of allg E L1 such that g is a finite linear combination
FOURIER TRANSFORMS ASSOCIATED WITH THE LINE GROUP
229
of translates off. That is, g belongs to T, if g ( x ) = 2 a k f ( x + ck) for some finite set of real and complex ah. The theorem of Wiener asserts: For f E L' the closure of T, in the L'topology, namely F,is equal to all of L' if and only iff^(v) # 0 for v E R. This is connected with the theory of translation invariant subspaces, initiated by BEURLING [2]. A (closed linear) subspace M of L' is said to be translation invariant i f f € M implies that every translate of f i s also in M. Then M is a translation invariant subspace if and only if M is a closed ideal of L' (ideal means: M is a n algebra with respect to the operations of L' and g * h E M whenever g E M, h E L'). Setting out to determine what are the closed maximal ideals in L', let MA,h E R, be the set of all f E L' such that f^(h) = 0. If M is any closed maximal ideal of L', then M = MAfor some A. A further interesting problem is posed if we assume that TI is a proper ideal for some f E L'. Then the question is: which f E L1 have the property that is precisely the intersection of the maximal ideals containing i t ? This problem, in a reformulation involving bounded functions, is often referred to as the problem of spectral synthesis. For the bounded function formulation see H. POLLARD [l] and the literature cited there (see also KATZNELSON [ 1 , p. 159 ff]). For results in L2 compare the notes and remarks to Sec. 5.2. A further classical result of Wiener not treated here is concerned with analytic functions of Fourier transforms (cf. R. R. GOLDBERG [ l , p. 26 ff]). As a particular result of this theory we cite: Let IE be a compact set of R. I f f € L' and f"(o) # 0 for v E IE, then there exists g E L' such that l/f"(v) = g^(u) for v E E. For Fourier transforms in the complex domain we refer to PALAYWIENER [l], TITCHMARSH [6], CARLEMAN [l]. For Sec. 5.1.3 see BOCHNERCHANDRASEKHARAN [1, p. 7 ff]. Prop. 5.1.14 is commonly treated (cf. RUDIN[4, p. 1821). The proof of case r = 1 of Prop. 5.1.15 makes use of ideas of COOPER[2] as modified by BERENS (unpublished, see also NE,SSEL [2, p. 102 ff]). The proof for r 1 2 is taken from BOCHNERCHANDRASEKHARAN [l, p. 281. For the material of Sec. 5.1.4 we refer to books on probability theory, for example, CRAMBR[2, p. 89 ff], LINNIK[ l , p. 491, and LUKACS[l, p. 27 ff]; it need not be emphasized that such coverage is often given from a somewhat different point of view. For Prop. 5.1.175.1.19 compare also BOCHNER [6, p. 70 ff]. For the connection between the existence of symmetric moments limp.+ukf(u) du and Riemann derivatives of f A we refer to ZYGMUND [4], see also PITMAN [l]. Concerning the literature to generalized derivatives see BUTZERBERENS [l, Sec. 2.21. The (second) Riemann derivative plays a fundamental rBle in RIEMANN'S Habilitationsschrift [ 13 on the theory of trigonometric series (see the account in ZYGMUND [71, Chapter 91 or BARI[lI, p. 192 ff]). The notion of a Peano derivative was introduced by PEANO [ l , pp. 2042091. For Lemma 5.1.25 see also OLIVER[ll. For Sec. 5.1.5 on the Poisson summation formula see BOCHNER [7, p. 39 ff], [6, p. 19, 30 ff]. ZYGMUND [71, p. 68, 1601, ACHIESER [2, p. 126 ff], FELLER [2II, p. 592f1, KATZNELSON [ I , p. 1281 (also GOOD[l]). For applications of Prop. 5.1.30, due to W. H. YOUNGand G. H. HARDY,to singular integrals see ACHIESER [2, p. 1371, BUTZER[6]. Formula (5.1.61) is the wellknown transformation formula for the theta funtion &(x, t ) ; see BELLMAN [ll, DOETSCH[4]. ck
t,
Sec. 5.2. The L2theory was first given by M. PLANCHEREL in 191015; the approach of Sec.
5.2.1 is due to F. RIESZ[2]. There are other approaches due to TITCHMARSH, BOCHNER, and WIENER;see the literature cited in TITCHMARSH [6, p. 69 ff]. The method of Wiener proceeds via Hermite functions, see WIENER[2, Chapter 11 or SZ.NAGY [5, p. 349 ff ]. For a generalization of the Plancherel theory due to WATSONcompare ACHIESER[2, p. 117 ff], BOCHNERCHANDRASEKHARAN [ 1, Chapter 51. Concerning the Lptheory for 1 < p c 2 which depends upon the M. RieszThorin interpolation theorem we mention especially WEIS [2, p. 168 ff], ZYGMUND [7II, p. 254 ffl, and [ I , p. 141 ff]. The inequality (5.2.16), due to TITCHMARSH [l], is also often KATZNELSON referred to as the HausdorffYoung inequality.
230
FOURIER TRANSFORMS
The present method cannot be used to define a Fourier transform on Lp, p > 2 (see TITCHMARSH [a, p. 1 1 11). However, there are several different ways to define the Fourier transform on Lp f o r p > 2, namely by a method of OFFORD[I], BochnerWiener's generalized harmonic analysis, and via distribution theory. The latter is the most reasonable way; one uses Parseval's formulae of type ( 5 . 1 3 , that is, duality, to define the Fourier transform of tempered distributions. Details are to be found in any text on distribution theory, for example, SCHWARTZ [1, 31, ZEMANIAN [11, BREMERMAN [11, DONOGHUE [ 13. See also KATZNELSON [ l , p. 146 ff] or Vol. I1 of this treatise. Whereas the Offord approach is classical, having the Ceshro means of the Fourier inversion integral as its starting point, that of BochnerWiener may be regarded as a precursor to the distributional approach (see the review by BOCHNER [5] of L. SCHWARTZ'S treatise on distributions). Let the set F, be defined by F, = ( j LLIf(x)(l ~ 1 ~ 1 E~ Ll},) ~r E~ P. Obviously, F, c F,,, and XSn,X(R) c Fa. for example. For f~ F, the Fourier transform is then defined by
+
This transform is uniquely determined up to an algebraic polynomial of degree (r  1) at most. After suitable modifications, many of the operational rules of the ordinary Fourier transform are transferable to this generalized one, Thus a uniqueness theorem holds, a convolution is introduced and transforms of derivatives are considered. However, for all details we refer to BOCHNER [7, Chapter 61, WIENER[2, Chapter 41; see also MASANI[l] (and the literature cited there), particularly for the connection with filter and prediction theory. For the operational properties of the Fourier transform in Lpspace as given in Sec. 5.2.3 we cite TITCHMARSH [6, Chapter 41. For Theorem 5.2.12 see also WEISS[ l , p. 321. The inversion theory of Sec. 5.2.4, despite of its independent interest, is not as necessary as in L1space. This is due to the standard theorems of Plancherel (p = 2) of Sec. 5.2.6 and of HilleTamarkin (1 < p < 2) of Sec. 8.3.4. For Sec. 5.2.5 see the references given to Sec. 5.1.3. For translation invariant subspaces in La see WIENER[2, p. 100 ff], BOCHNERCHANDRASEKHARAN [l, p. 148 ff], RUDIN[4, p. 190 ff], for example. One result states that, i f f € La and f"(u) # 0 almost everywhere, then every g E La belongs to the Laclosure of T,. For more on invariant subspaces see HELSON[l]. Sec. 5.3. Standard references to the material of this section are texts on probability theory such as LBVY[2], C R A M ~[2], R FELLER [2], L O ~ V[l], E BOCHNER [6], LUKACS[l], LINNIK[l], EISEN[ll; nevertheless, see also the accounts in BOCHNER [7, Chapter 41, ZYGMUND [XI, p. 258 ff], KATZNELSON [l, p. 131 ff]. There exist continuous functions p E BV such that lim,,, p"(u) # 0, see e.g. ZYGMUND 1711, p. 2591. The notation 'Dirac measure' for the function 6 of (5.3.3) will be cleared up in Vol. I1 while introducing the '&distribution'. For Theorem 5.3.5 see also WIDDER 1 1 , p. 2521. The summation of the integral (5.3.13) by a &factor is suggested by L1theory (compare with BOCHNER [6, p. 231). The results follow again as easy consequences of those of Chapter 3 on singular integrals. Concerning the theorem of LBvy (Theorem 5.3.9) we mention, among others, ZYGMUND [7II, p. 2601, LUKACS[l, p. 381. For Prop. 5.3.10 see LUKACS[I, p. 401. There are other proofs of the uniqueness theorem; for example, LUKACS [l, p. 351 uses the Weierstrass approximation theorem. For the results of Sec. 5.3.3 we refer to COOPER [2], BUTZERTREBELS [2, p. 36 ff]. As to Sec. 5.1.3, 5.2.5, the treatment in BOCHNERCHANDRASEKHARAN [11 supplied motivation. Finally, for global and local divisibility in the Wiener ring of FourierStieltjes transforms we refer to Chapter 13 of this volume.

Representation Theorems
6.0 Introduction
Suppose we are given a trigonometric series
2 f(k)elkX m
(6.0.1)
km
with arbitrary complex coefficientsf(k). How can one tell whether this series is the Fourier series of an X,,function, in other words, whether the numbersf&) are the Fourier coefficients g"(k) of some function g E X,,? The problem may be restated as follows: Given an arbitrary function f on Z, to determine conditions under which f admits a representation as the Fourier transform g" of some function g E X,, or as the finite FourierStieltjes transform of some p E BV,,. We have seen thatfon Z has to satisfy certain necessary conditions in order to be the finite Fourier or FourierStieltjes transform. Thusfon Z must be bounded in view of (4.1.2) and Prop. 4.3.2(ii). On the other hand, for L& we already know that a necessary and sufficient condition for a functionf on Z to be the finite Fourier transform of some g E L,2, is that f~ la. This is a consequence of the Parseval equation and the theorem of RieszFischer. But nothing as simple seems possible for other classes such as LE,, p # 2. However, if we consider the inversion formulae involving 8factors (see Sec. 4.1.2, 4.3.2), it is possible to give a reasonably satisfactory answer to the representation problem that includes necessary and sufficient conditions. Suitable sufficient conditions are also important. As a first contribution we recall Prop. 4.2.7. From the parallel point of view, it is frequently of interest to decide whether a given, complexvalued function f(u) on the line group R is, or is not, the FourierStieltjes transform of some p E BV or the Fourier transform of some g E Lp, 1 I p I 2. For p = 2 this problem is again completely solved. Indeed, the theorem of Plancherel states thatfis the Fourier transform of some g E L2 if and only i f f € La. In the other cases we have only established certain necessary conditions that f has to satisfy. Thus Prop. 5.3.2 asserts that every FourierStieltjes transform is bounded, while Prop.
232
FOURIER TRANSFORMS
5.1.2, Theorems 5.2.4,5.2.9 state that the Fourier transform of someg E Lp,1 I p I2, is necessarily of class Lp'. But it will be seen that the results of Sec. 5.1.2,5.2.4, and 5.3.2 on inversion theory again provide a method to solve the representation problem. Chapter 6 is composed of five sections. Each section treats a definite problem on the circle group and the corresponding one on the line group. Since the material on the circle group is rather standard, it will mainly serve as motivation for the counterparts on the line group. The latter are treated in detail, the results on the circle group being sometimes relegated to the Problems. Sec. 6.1.1 is concerned with classical necessary and sufficient conditions for representation of sequences as finite Fourier transforms, mainly due to W. H. and G. C. YOUNG,W. GROSS,and H. STEINHAUS. Sec. 6.1.2 deals with the analogous problem on the line group, including results of H. C R A M ~ofRfurther ; interest is a certain converse to the Poisson summation formula (Prop. 6.1.10). Sec. 6.2 is reserved to another type of characterization of FourierStieltjes transforms among all bounded and uniformly Theorem 6.2.3 is concerned with one continuous functions on R, due to S. BOCHNER; aspect of the socalled continuity theorem for FourierStieltjes transforms. As a matter of fact, the necessary and sufficient conditions for sequences or functions to be Fourier transforms are somewhat involved (except for the case p = 2). Therefore, apart from the principal importance of these results in theoretical problems, we shall treat further conditions for representation in Sec. 6.3. Although these are only sufficient, they are more readily applicable to special sequences or functions. They depend upon results on convex and quasiconvex functions defined on Z and R which are established in Sec. 6.3.1.The P6lyatype characterization of Theorem 6.3.11 deserves particular mention. Sec. 6.3.4 discusses an important reduction theorem, resting upon a lemma due to E. M. STEIN.It states that if for some a > 0 a n d f e L1 there exists p E BV such that Iul.f"(u) = p"(u), then for every 0 < < a there exists g, E L' such that IuIBf"(u) = g;(u). The first applications of Theorem 6.3.1 1 are to be found in Sec. 6.4. There it is shown that a general class of means of Fourier inversion integrals and Fourier series are representable as singular integrals. Sec. 6.5 gives a brief account on the representation of multipliers, including a theorem of K. DE LEEUWon multipliers of type (Lp, Lp), 1 < p < 2.
6.1
Necessary and Sufficient Conditions
6.1.1 Representation of Sequences as Finite Fourier or FourierStieltjes Transforms Given the finite Fourier transform g" of some g E Xan, the inversion formulae (4.1.19) and (4.1.20) show how to recapture the original function g. Thus starting off with an arbitraryfE I", the question arises whether there exist necessary and sufficient conditions upon the means (6.1.1)
u,(x) =
2 e,(it)j(it)
k  m
eckn
REPRESENTATION THEOREMS
233
which guarantee the existence of a function g, belonging to some definite class, for which the finite Fourier transform gAis equal tof. Theorem 6.1.1. Let f E I", and let {d,(k)} be a dfactor such that mate identity. Then the condition
{c(x)} is an approxi
is necessary and suflcient such thatf is thefinite FourierStieltjes transform of afunction E BV,, i f p = 1 ,andf is thefinite Fourier transform of afunction g E L$, if1 < p 5 00.
p
Proof. Necessity. Let 1 < p I 00. Iff admits a representation as the finite Fourier transform of some g E Lg,, thus 1 " f ( k ) = g^(k) 3 27r  * g(u) eSikudu (kE Z), then, according to (4.1.16) and (4.1.17),
2 W
(6.1.3) u,(x)
=
k=w
where
d,(k)g^(k) elkx = U,(g; x ) = 
c is defined by (4.1.18). Hence by (1,1.4)
which proves (6.1.2) since {dr(x)} is an approximate identity (cf. (1.1.5)). If p = 1 and thus f ( k ) = p"(k), k E Z, for some p E BV,,, then by (4.3.12)
and hence by (1.1.13) and (1.1.5)
Suflciency. Sincef E I" and (cf. (1.2.28)) 0, E I', p E A, the series (6.1.1) is absolutely and uniformly convergent, and thus defines, for each p E A, a function u, E C,, for which ur(k) = d,(k)f(k), k E H. Furthermore, since {C(x)} is an approximate identity, lim, + D o d,(k) = 1, k E Z, by Problem 1.2.14(iv). Therefore in any case
(k E Z).
(6.1.5)
On the other hand, if p = 1 and if we set p,(x) = J?,,u,(v) do, this defines a family of absolutely continuous functions p,(x) the total variation (over [ 7r, 7rl) of which is uniformly bounded as p t po according to (6.1.2). Moreover, p,(x 27r)  p,(x) = 2?rf(O) for all x and p E A by (1.2.28). Therefore by Prop. 0.8.13 there exists {p,} c A with lim, p j = po and p E BV,, such that
+
+
234
FOURIER TRANSFORMS
for every h E C2z. In particular, for h(x) = (1/24 exp {  ikx}, k E Z, we obtain limj+m O,,(k)f(k) = ~ " ( k )k, E H, which, together with (6.1.5), proves the theorem for p = 1. If 1 < p I co, we apply Prop. 0.8.12 and obtain { p j } c A with 1imj+" pj = po and g E Lg, such that
for every h E Lg;. Again the specialization for h(x) = (1/27r) exp { ikx}k E h, leads to limj.m O,, (k)f(k) = g"(k), k E k , which, according to (6.1.5), proves the required representation for 1 c p I00. In the preceding theorem it has been a priori assumed that the function f to be represented should be bounded, i.e. f E I". But this assumption has only been used to ensure the absolute and uniform convergence of the sum (6.1.1) which defines u,(x). Thus, if we consider rowfinite &factors (cf. Sec. 1.2.5), no boundedness hypothesis is needed. In particular, for the Fejtr means (6.1.6)
Theorem 6.1.2. An arbitrary (complexvalued)function f on Z is theJinite Fourier transform of a function g belonging to
A function f on Z is the finite FourierStieltjes transform of a function p belonging to (iv) BVan * IlOn(0)II 1 = O(1) (n 3 a). We only sketch the proof. Iff is the finite Fourier transform of some g follows by (6.1.3) and (4.1.19) that
E
X2n,then it
On the other hand, if the FejCr means u,(x) as defined by (6.1.6) form a Cauchy sequence in X2n, they converge to a function g E Xanin view of the completeness of Xan.Together with Problem 4.1.1(ii) and (6.1.5) this shows f ( k ) = g"(k), k E Z. Of course, if the given function f on Z satisfies one of the conditions of Theorem 6.1.2, then it in particular follows that f is bounded. But sometimes it is of advantage to decide whether a given f admits a representation without knowing explicitly that f is bounded. Indeed, to decide whether a givenfs XPnbelongs to the class W[X2,; r(l(k)] (cf. (4.1.24) for the definition) we have to examine whether for a fixed but arbitrary complexvalued function r(l (on Z) r(l(k)fA(k)is the finite Fourier transform of some g E Xzn.An example of interest (cf. Theorem 4.1.10) is #(k) = (ik)', r E N, which is in fact unbounded on Z. Nevertheless, iff belongs to W[Xzn;(ik)'], then (ik)'f"(k) is bounded on Z. We conclude with a characterization of the class V[X,,;+(k)], introduced by (4.3.19).
235
REPRESENTATION THEOREMS
Corollary 6.1.3. Let f (i) f
(ii)
k=
n
XzR.The following assertions are equivalent:
#(k)l,
'['2R;
(1 2
E
(1  $)#(k)fA(k)
elk0
(1
(n
= 0(1)
XIR

03).
In particular, for #(k) = (iky, r E N,we have by Theorem 4.3.13 Corollary 6.1.4. Let f
E
XzR.The following assertions are equivalent: (n

a).
Concerning the corresponding results for the classes W[XzR;#(k)J we refer to Problem 6.1.3, 6.1.4.
6.1.2 Representation of Functions as Fourier or FourierStieltjes Transforms Concerning the representation of a given functionf(v) on R as the FourierStieltjes transform of a function p E BV or as the Fourier transform of some g E Lp, 1 Ip I 2, we may proceed as in the periodic case. Then the results of Sec. 5.1.2, 5.2.4, 5.3.2 on the inversion theory again provide a method in solving the representation problem ; necessary and sufficient conditions are stated in terms of the means
1"
u(x; p ) = 
(6.1.7)
O(E) f ( v ) elxudv.
m
P
Here 6 is an arbitrary continuous &factor (Def. 5.1.5).
Theorem 6.1.5. A necessary and suficient conditionfor f everywhere on R as a FourierStieltjes transform, i.e. f(u) =
(6.1.8)
jmetuudp(u)
4%
L" to be representable almost
E
a.e.
m
with p E BV, is that for any continuous 8factor (6.1.9)
f(v)ei""du
Ill
=
0(1)
I f f is continuous, (6.1.8) holds for all u E R.
Proof. Necessity. Iff admits the representation (6.1.8), then
1" m
8(E)pv(v) elwudv = p
P
B^(p(x
 u)) dp(u)
by (5.3.17), and therefore by (5.3.15)
Suficiency. Let (6.1.9) be valid for some continuous &factor. Since B E L1, we have O(o/p)f(o)E L' so that Prop. 5.1.2 and the hypothesis (6.1.9) imply that u(x; p) as
236
FOURIER TRANSFORMS
defined by (6.1.7) belongs to C, n L1 for each p > 0. Moreover, [u(o;p)]"(u) = O(u/p)f(u) ax. by Prop. 5.1.10. Now if we set F = {+ E L'I + E Co,
(6.1.10)
+A
E
L'},
+
we obtain by the Parseval formula (5.1.23) and by (6.1.9) for every E F
Furthermore, since O is continuous, we have limp+ooO(u/p) = O(0) = 1, and since I O(u/p)f(u)+"(u)l 5 11 llfll l+^(o)I E L', Lebesgue's dominated convergence theorem gives
,I"&
(6.1.11) Thus the integral on the left defines a bounded linear functional on F, considered as a subspace of Co. Since F is a dense subspace of Co (cf. Problem 6.1.5(i)), we may, according to Prop. 0.7.1, extend this functional such that it is bounded on C,. Now the Riesz representation theorem for bounded linear functionals on C, applies and gives the existence of p E BV such that for all E F
+
In particular, we obtain for +(u)
=
exp {  tua + ixu}, t > 0, x E R, that
Taking t f 0 + ,the lefthand side converges by Cor. 3.2.3 to f ( x ) a.e. and the righthand side by Lebesgue's dominated convergence theorem to p"(x) for all x E R, establishing the representation (6.1.8). Theorem 6.1.6. Let 1 c p 5 2 andf
E LP'.
A necessary and suficient condition that
(6.1.13) with g E Lp, is that for any continuous Ofactor
(6.1.14) Let us recall that for p = 2 condition (6.1.14) is indeed superlluous since f E L' impliesf€ [La]^ by Plancherel's theorem.
237
REPRESENTATION THEOREMS
Proof. For the necessity we observe that, iff is the Fourier transform of g E Lp, we have in view of Theorem 5.2.14
To prove the sufficiency we may proceed as in the proof of Theorem 6.1.5 using the Riesz representation theorem for bounded linear functionals on LP', 2 Ip' < 00. But we may equally well use the weak* compactness theorem for Lp, 1 < p I 2. Thus, let (6.1.14) be valid for some continuous &factor. Since 6 E L1 n Co, we have e ( o / p ) f ( o ) E L' by Holder's inequality, and therefore u(x; p) as defined by (6.1.7) belongs to C, n L p for each p > 0. In view of the stated theorems, assumption (6.1.14) implies that there exist a sequence {p,} with lim,.+mpr = 00 and g E Lp such that
J  m h(u)u(u;p,)
lim
~
1m
for every h E Lp'. Therefore, if we take h (5.2.27), (5.1.5), and Prop. 5.1.1(iv)
=
lim f.m
[ . I
m
m
du = J = $,
m
h(u)g(u)do
$ E F, we have by Parseval formulae
$"(u)e(E)f ( u ) Pr
the last equality being valid by Lebesgue's dominated convergence theorem. If we again take $(u) = exp {  tua + ixu}, then for every t > 0 and x E R
and since f  g" E Lp', Cor. 3.2.3 finally implies that f ( x ) = g"(x) a.e. In the preceding theorems it has been a priori assumed that f E L P ' to ensure the absolute convergence of the integral (6.1.7) defining u(x; p). But this condition, though a necessary one, may be explicitly avoided. Thus, if we consider 0factors with compact support, we need only assumef E Lioo.In particular, for the CesZiro factor we have (for a further extension see Problem 6.1.10) Theorem 6.1.7. Let f be measurable on R and integrable ouer every finite interual. Then for 1 Ip s 2 the condition (6.1.15)
1 &
(1 
F)
f(u) e f o w du
I.
= 0(1)
is necessary and suficient such that f is almost everywhere equal to the FourierStieltjes transform of a function p E 0V i f p = 1, and f is the Fourier transform of a function gELPif1 < p s 2 .
238
FOURIER TRANSFORMS
Proof. We only have to show that the assumptionsfe Let us set
Lioo and (6.1.15)
imply f e Lp'.
and (6.1.16) It follows t'romfe Lfoo that FPE L1 for every p > 0. Ifp = 1, (6.1.15) states that the Fourier transform u (  x ; p) of &(u) belongs to L'. By Prop. 5.1.10 we then have
and hence IF,,(u)[ I IIu(0; p)IIl for every p > 0 and almost all u E R. Since there is a constant M such that IIu(.; p)II1 I M uniformly with respect to p > 0 and since lirn,,+" F,,(u) = f(u) a.e., we concludefE L". If 1 < p I 2, then u(0; p) E Lp by (6.1.15). Therefore we can form the Fourier transform [ ~ ( o ;p)]^(u) and obtain by the Parseval formulae (5.2.27) and (5.1.5)
By Cor. 3.2.3 the lefthand side tends to [ u ( o ; p)]^(x) a.e. as t +0+, because [u(o; p)]^eLp', whereas the righthand side tends to FD(x)a.e. as t +0+, because FPE L'. Therefore FD(x)= [u(o; p)]^(x) a.e., and by Titchmarsh's inequality llFDllp.5 IIu(0; p)IIp. Hence IIFDIIp,is uniformly bounded with respect to p > 0. Since lirnD+" FP(x)= f ( x ) a.e., Fatou's lemma givesf€ LP'. Now we may complete the proof by an application of Theorems 6.1.5 and 6.1.6. We observe that for p = 2 Theorem 6.1.7 is a true improvement of the corresponding assertion of Plancherel's theorem since we do not assumefe La explicitly. The fact that we may avoid the explicit assumptionf E Lp' is of interest if we look at the classes W[LP; $(v)], V[LP; $(v)] of (5.2.36), (5.3.32). Indeed, iff belongs to W[Lp; #(v)], then the representation $(o)f"(v) = g"(v), g E Lp, of course implies $(v)f^(v) E LP'. But starting off with somefe Lp, to decide whetherfbelongs to W[Lp; #(v)], it issometimes not convenient to commence with $(v)f"(v) E Lp', since #(v) may be unbounded, e.g. if we take $(u) = (ivy.
Regarding the classes V[Lp; #(u)] we have Corollary 6.1.8. L e t f E LP, 1 5 p 5 2. The following assertions are equivalent: (i) f E WP;#(u)l,
REPRESENTATION THEOREMS
(ii)
1 4% 1" 1

ro
 F)$(o) f"(o)
(1
etoU do
1
P
= O(1)
(P

239 a).
For $(u) = ( h y , r E N, it follows by Theorem 5.3.15 Corollary 6.1.9. Let f E Lp, 1 5 p 5 2. The following assertions are equivalent: (i) f E V:P,
(iv)
(1 & j:p
(ii) f E V[Lp; (io)'], (1
 :)(ioyfA(o) I
(iii) llu(r)(f ; 0 ; p)llp = 0(1), eloudo
)I
= O(1)
(p
P

a).
For corresponding results concerning the classes W[Lp;$(u)] we refer to Problems 6.1.11, 6.1.12.
Here we conclude with a certain converse to the Poisson summation formula. In Sec. 3.1.2 we have associated a periodic approximate identity {x,*(x)} via (3.1.28) with every x E NL'. In particular, it follows by Prop. 3.1.12 that the functions x,* are bounded and continuous in Lhnorm, uniformly with respect to p > 0. Moreover, [x:]^(k) = xA(k/p)for every k E Z , p > 0 by (5.1.58). After obvious changes of notation, this proves one direction of the following
Proposition 6.1.10. A functionf, defined and continuous on R, is an L'Fourier transform if and only il;for each r E N, the functions f ( k / r ) on E are the finite Fourier transforms of L?j,jiunctions which are bounded and continuous in Lk,norm, uniformly with respect to rE N. To complete the proof, let g,, r E N, be functions in L?j, such that (i) g,^(k) = f ( k / r ) for all k E Z, r E N, (ii) there exists a constant M > 0 such that IIgrllL;, I M for all r E N, (iii) given e > 0, there exists 6 > 0 such that IIgr(o + h)  gr(o)IIL;n< e for all Ihl < 6 and r E N. Then it follows by Problem 1.1.3 that for the integral of FejCr limntm~~cr,,,(g,; 0)  gr(o)1IL;.= 0 uniformly with respect to r E N. Therefore {unr(gr;x)};= is a Cauchy sequence in LB, uniformly for r E N. This implies that, given e > 0, there is 1E N such that for all n, m > I and r E N (cf. Theorem 6.1.2(ii))
By a change of variables
Since each term of the difference is an approximate Riemann sum of an integral, letting r f 03, it follows by Fatou's lemma that m m !
I 1' n
(1
)!
f ( u ) elx" du 
(1 
)!
I
f ( v ) elx" do d x
1. By Problem 6.1.8 this implies the representation f(u) = g"(u) with some g E L', and Prop. 6.1.10 is completely established. There is an analog of Prop. 6.1.10 if Fourier transforms are replaced by FourierStieltjes transforms. For this we refer to Problem 6.1.13.
240
FOURIER TRANSFORMS
Problems 1. Let f E I", {B,(k)} be a &factor such that {O;(x)} is an approximate identity, and let the means u,(x) be defined by (6.1.1). (i) Show that f is the finite Fourier transform of a function in Xan if and only if the means u, form a Cauchy sequence in Xan, i.e., lim,,,,,,,, Ilu,,(o)  upa(0)llxan=0. (ii) For 1 < p < co show that IIup(o)llp = 0(1), p + po, if and only if I ~ U , ~ ( O )  upa(o)llp = o(l), p1, p? Po. (iii) Show that f is the finite FourierStieltjes transform of a monotonely increasing function if and only if u,(x) 1 0 for all x and p E A, provided the approximate identity {Qx)} is positive. 2. (i) Complete the proof of Theorem 6.1.2 and show that Ilo,(o)IIp = 0(1), n + m , for 1 < p < co if and only if Ila,(o)  um(o)llp .= o(l), n, m +00. (ii) Show that a function f on Z is the finite FourierStieltjes transform of a monotonely increasing function if and only if o,(x) > 0 for all x and n. 3. Let f E Xan. Show that f belongs to W[Xan; $(k)] if and only if +
4. Show that for f E Xznthe following assertions are equivalent: (i) f E Wiln, (ii) f E W[Xan;(ik)'], (iii) Iimn,m,m I I u W ; 0 )  ug)(fio)llxZn = 0. 5. (i) Show that theset Fasdefined by(6.1.10) is dense in Co and Lp, 1 5 p < Q). (Hint: Use Problem 5.1.2(iii) and Cor. 3.1.9) (ii) Let f~ LP'. Show that if (6.1.9) [or (6.1.14)] is satisfied for some continuous 8factor, then the condition is satisfied for every continuous &factor. (Hint: CRAMBR [I, P. 1921) 6. Let f E LP' and the means u(x; p) be defined by (6.1.7) with some continuous &factor. (i) Show that f i s the Fourier transform of a function in Lp, 1 0, 4 coincides on (I, I) with a 21periodic, piecewise continuously differentiable function $ which may be expanded in its Fourier series (cf. (11.4)) $(x) =
(6.2.7) where $(x) = d(x) for x
5
$"(k) efknxlL,
k=m
E (
I, I ) and
(6.2.8) The series (6.2.7) converges uniformly on R. In fact, since 4' is piecewise continuous and has compact support, it follows by Prop. 5.1.14 that iu+^(u) = [4"''(v> for all u E [w, and therefore irk$^(k) = I[$']^(k) for all k E Z. By the Parseval equation for 21periodic functions (cf. Problem 4.2.4) we have
and hence by Holder's inequality (6.2.9)
Now we consider the auxiliary function (6.2.10) which, roughly speaking, approximates the integral (6.2.6). By Prop. 0.7.1 we may extend the bounded linear functional L as defined on T by (6.2.4) to a bounded linear functional on the set of all 21periodic and continuous functions, with norms remaining unchanged. This is possible since T contains all 21periodic trigonometric polynomials. By the definition of L and (6.2.8), (6.2.9) we obtain (6.2.1 1)
243
REPRESENTATION THEOREMS
Hence by (6.2.7) and (6.2.8) (6.2.12) We next consider
IQ(l; #)I
I
1141 II#IIm. (u > 0).
As a function of u, Q(ul; #) being the sum of measurable functions is measurable. Furthermore, by (6.2.8) and (6.2.9)
Hence the following integral exists and we can interchange the order of summation and integration
0
If we now define for (k  1 ) ~ I s
101
0 such that
for every finite set of complex numbers {ck}. (ii) Show that f is the finite Fourier transform of a function g E LP, 1 < p 5 and only if there exists a constant M > 0 such that
00,
if
[l, p. 31 ff]) for every finite set of complex { c k } . (Hint: KACZMARZSTEINHAUS 2. (i) A functionf(k) on Z is said to be positiuedefinite if for every finite subset cl, . . .,c, of complex numbers and all n E N the inequality c,Gf(i  k ) 2 0 holds. Show that f on Z is positivedefinite if and only iff is the finite FourierStieltjes transform of a monotonely increasing function p E BV,,. (Hint: See e.g. ZYGMUND [71, p. 1381, HEWITT [ 1, p. 29 ff]) (ii) Show that a functionfe Z is the finite Fourier Stieltjes transform of some p E BV,, if and only if there are positivedefinite functionsf,, . . .,f4 on Z such that f ( k ) = fl(k) f,(k) i[f3(k) f4(k)]. (Hint: Use the Jordan decomposition) 3. A function f ( u ) on R is said to be positivedefinite if (i) f e C, (ii) f ( u ) = f(  v), (iii)  l)k) 2 0 for every finite collection v l , . . .,u, E R, cl,. . .,cn E @, Iy= I;= cj and all n E N. Show that f(u) on R is positivedefinite if and only if f i s the FourierStieltjes transform of a monotonely increasing function p E BV. (Hint: BOCHNER [7, p. 92 ff], R. R. GOLDBERG [l, p. 59 ff]) 4. (i) Let f E L". Show that there exists p E BV such that f( u ) = p"(v) a.e. if and only if there exists a constant M > 0 such that I ~ ~ m f ( u ) ~ n ( o ) d5u kJ f ~ for ~ every ~ ~ ~ m
+ zf(u,
246
FOURIER TRANSFORMS
+
E F. Modify the class F so as to obtain a necessary and sufficient condition for the representation in case f E L:oc only. (Hint: Use Problem 5.3.4 and argue as for (6.1.1 l), (6.1.12); compare KATZNELSON [1, p. 1341) (ii) Let f E La. Show that there exists p E BV such that f ( u ) = p"(u) a.e. if and only if there exists a constant M > 0 such that IS?mf(u)+(u) dul IMllQ,"Ilm for every E L1. (Hint: Use (5.3.11); compare BERRY [l], SCHOENBERG [2]) 5. (i) Let 1 < p I2 and f E Lp'. Show that there exists g E Lp such that f ( u ) = gA(u) a.e. if and only if there exists a constant M > 0 such that I J? f(u)+"(u) dul IMII+IIP, for every # E F. Extend to functionsf€ L:oc. (Hint: Use (5.2.27) and proceed as for the preceding Problem, compare also with Theorem 6.5.7) (ii) Let 1 < p I2 and f E LP'. Show that there exists g E LPsuch that f ( u ) = gA(u) a.e. if and only if there exists a constant M > 0 such that 1 J? f(u)Q,(u) dul 5 Mll+^llP.for every E Lp. 6. Let f~ L". Show that there exists g E L1 such that f ( ~=) g"(u) a.e. if and only if (i) there exists a constant M > 0 such that I f(u)+(u) dul IMll+"l\ for every Q, E L', (ii) to each E > 0 there exists S > 0 such that I JZa f(u)+(v) dul IEII+"II for every Q, E F with Il+^II1 I81l+"llm. (Hint: BERRY[l]) 7. Prove the following representation theorem: A necessary and sufficient condition that f ( u ) be the Fourier transform of a function in Lp, 1 < p I2, is that f e LP' and
+

+
Jr
ffi
where M is independent of the subdivision  00 < x 1 < x 2 < . . . < x n < 00. (Hint: Use a criterion of F. RIESZ(RIESZSZ.NAGY [I, p. 751): fis an indefinite integral of afunctioninL4,l c q < 00,ifandonlyif 2 $ : : l f ( x k t 1 ) f(xk)lq/(xktl  xk)ll IM , see BERRY [l]) 8. Let { p , , ( ~ ) } : =be~ a sequence of functions in BVznsuch that /Ipnllevan IMuniformly for all n. If the finite FourierStieltjes transforms p;(k) converge to a function f(k) on Z for every k E Z, show that f(k) is the finite FourierStieltjes transform of a function p E BV2n.
6.3 Sufficient Conditions
In the preceding sections we have been concerned with necessary and sufficient conditions for representation. But sometimes one is more interested in sufficient conditions which are readily applicable to decide whether a given functionfis the Fourier transform of an L1function. To this end we start off by gathering some facts on convex and quasiconvex functions defined on H or R. 6.3.1
QuasiConvexity
+
First we deal with (even) functionsf(k) on E. If, as usual, Af(k) = f(k 1)  f ( k ) and Aaf(k) = f(k 2)  2f(k 1) f ( k ) , thenf is said to be convex on P iff(k) is real and Aaf(k) 2 0 for all k E P.
+
+ +
Lemma 6.3.1. Let f be bounded and convex on P. Then f is monotonely decreasing on P, i.e. Af(k) I 0 for k E P, limk+a kAf(k) = 0, and (6.3.1) k=O
(k + l)Aaf(k) c a.
REPRESENTATION THEOREMS
247
Proof. Suppose that Af(ko) = a > 0 for some ko E P. Then A f ( k ) 1 a for all k 2 ko since Af(k 1 ) 1 A f ( k ) for all k E P in view of the convexity off on P. Hence f(k) 2 (k  ko)a f(ko) for k 2 ko, and thus limk+mf(k)= co, which is a contradiction to the boundedness off on P. Therefore A f ( k ) 5 0 for k E P. Since f is bounded and monotonely decreasing on P, limk,, f(k) = f(m) exists as a finite number. Now x E = o Af(k) = f ( n + 1)  f ( 0 ) so that zT=oAf(k) = f ( 0 3 )  f ( 0 ) . Thus the series is convergent, and since the terms of the series are negative and monotonely increasing, it follows by Cauchy’s criterion that Iirnk+ k A f ( k ) = 0. Using Abel’s transformation (cf. Problem 1.2.7) we have
+
+
and the convergence of (6.3.1) follows. In fact,
5 ( k + 1)A2f(k) = f ( 0 )  f(m).
k=O
For examples of functions which are convex on P we refer to Problem 6.3.1. Definition 6.3.2. A functionf, defined on P and satisfying m
(6.3.2) is called quasiconvex.
Thus every bounded and convex function on P is quasiconvex, but not conversely. Proposition 6.3.3. Let f Proof. We have
E
be quasiconvex. Then limk+wk IAf(k)I = 0.
I;(P)
Xg,,A 2 f ( k ) =  A f ( n ) , and therefore
which tends to zero as n z co.
If we now turn to functions defined on R,any realvalued function f ( x ) for which 2f((x, xJ2) I f ( x l ) f ( x , ) for all x l , x2 with a I x1 < x2 I b is said t o be convex on the interval [a, b]. Equivalently, f is convex on [a, b), if A: f ( x ) 2 0 for every x E [a, b), h > 0, provided h is so small that x 2h c b (see also Sec. 3.6.1). We shall only consider convex functions which are continuous. This condition is not very restrictive, since a convex function is either very regular or very irregular, and in particular, since a convex function which is not entirely irregular is necessarily continuous (cf. Problem 6.3.5). Any continuous convex function is absolutely continuous. In fact,
+
+
+
(6.3.3)
f (XI
=
JaX
YXu) du
+f ( 4 ,
where t,b is monotonely increasing (cf. Problem 6.3.5). Lemma 6.3.4. Let f ( x ) E Co[O, co) be convex for x > 0, and let 4 be given through (6.3.3).Then I&) 5 0 for x > 0 andlim,,, x$(x) = 0. Moreover, f ( x ) is rnonotonely decreasingfor x > 0 and (6.3.4)
Som
u d+(u)
0, and f ( x ) is monotonely decreasing for x > 0. Furthermore,
f(4  f(x)
[$(u)l du
=
2 (x
 a)[$(x)l,
and hencef(a) 1 lim supxm [ x$(x)]. In other words, lim supx+mx I$(x)l If ( u ) , and since this is true for every u > 0, we conclude hx+ x I#(x)l = 0 since lima+m f ( a ) = 0. In view of the continuity offat x = 0+, the integral Jit $(u) du exists and (D
f(0)  f ( e ) =
IOc
[$(u)l du
Therefore 0 2 E$(E) 1 f(&)  f ( O ) , and thus lim,+ot by parts it follows that
1”
u d#(U) = p $ W
 6
2
444.
E$(&)
 e $ G )  f@)
=
0. By a repeated integration
+ f(&).
Hence lim,,+mn.c+ot J: u d$(u) exists, and since the integrand is positive, (6.3.4) is established. Under the assumptions of Lemma 6.3.4 we have by (6.3.3) that f ( x h)  f ( x ) = J! #(x u ) nu. Therefore f ’ ( x ) exists almost everywhere. In fact, f ’ ( x ) = #(x) for all x > 0 except perhaps for the denumerable set of discontinuities of the first kind (finite jumps) of $. But, after a suitable convention concerning the value off’ at these points, we may assume without loss of generality thatf’(x) is monotonely increasing for x > 0 and u df’(u) < co. This motivates the following
+
+
jr
Definition 6.3.5. A functionf, dejned on (0, co), is called quasiconvex i f f with derivativef’ E BVl,,(O, co) such that
jom Idf‘(u)l
(6.3.5)
E ACl,,(O,
co)
< 00.
In many cases of interestf’ is furthermore continuous on (0, m), except perhaps for a finite set of discontinuities of the first kind, and absolutely continuous in every bounded subinterval of (0, co) which does not contain any of these points. In this case, (6.3.5) is satisfied if J: u If”(u)l du c 00. Proposition 6.3.6. Let f E C,[O, 00) limE+ot ealf’(e)I = 0. Proof. If 0
0. We shall see that Cor. 6.3.12 is very convenient and useful for applications, more so than Cor. 6.3.9.
A Reduction Theorem In the following chapters it turns out that the classes W[Lp; #(u)], V[Lp; #(u)] of (5.2.36), (5.3.32) for #(u) = IuIa, a > 0, play a fundamental r61e in our investigations on approximation theory. Preparatorily, we are here concerned with the following question which we elucidate for the class WIL1; IuIa]: If for f E L1 there is some g E L1 such that Iul"f^(v) = g"(u), does there exist g , E L1 for each 0 c /3 c a such that lvlsf^(u) = g,^(u)? In other words, does f E WIL1; Iul"] imply f E WIL1; Iuls] for every 0 c /3 c a? In Sec. 11.2 we shall see that g may be interpreted as an ath derivative off in some generalized (fractional) sense. In the light of this, an affirmative answer to the above question would therefore assure that the existence of an ath derivative implies the existence of the /3th derivative for every 0 c /3 c a. This corresponds quite naturally to the procedure by which we proved Prop. 5.1.15. There it was shown that i f f o r f ~L' there exists g E L1 such that (iu)*f"(u)= g'+) for some r E N , then there exist g k E L1 such that (iv)kf*(u) = g;(u) for every
6.3.4
REPRESENTATION THEOREMS
253
k = 1,2,...,r  l;thatistosay,WIL1;(iu)'] c W[Ll;(i~)~]fork = 1,2, ..., r  1. There, the key was the function r] of (5.1.34), here for the reduction of the power 101" we use the functions G, of Problem 6.3.8 in connection with the following lemma of E. M. Stein.
Lemma 6.3.13. Let a > 0. There exist t q a E BV, j (6.3.18)
(up= (1
=
1,2, 3, such that f o r all u E R
+ u2)"zpy,a(u),
( 1 + u y 2 = &,&) + l~lUP;,m. Proof. In view of Problem 6.3.8 there exists G2 E L1 such that G;(u) = ( 1 v2)l for every u E R. In fact, G,(x) = q 2 exp {  1x1)by Problem 5.1.2(i), and thus IIG2111 = 1. We now consider the binomial expansion
(6.3.19)
+
m
(6.3.20) k= 0
the series being absolutely and uniformly convergent for It1 I1 since (cf. Problem 6.3.1 2) (6.3.21) Setting t = ( 1
(a > 0).
+ u ~ )  (6.3.20) ~, gives
(6.3.22) the series being absolutely and uniformly convergent for all u E R. Considering the partial sums
these are, by the convolution theorem, Fourier(Stieltjes) transforms of (cf. (5.3.3))
It follows that limntm[8"(u)
+ sc,,(u)]
=
lul"(1
+ u2)cr/2for all u E R and
uniformly for all n E N since IIG2111 = 1. Therefore Theorem 6.2.3 applies and guarantees the existence of some pl,"E BV such that limn+ [8"(u) sc,,(u)] = &,(u). This proves (6.3.18). To establish (6.3.19), let a > 0 be a given fixed number, and choose n E N and 0 < y < 2 such that a = ny. We have
+
254
FOURIER TRANSFORMS
Now u2/(l + u2) E [BV]" by (6.3.18) and (1 + u ~ )  ( "  ~  ( ~E/ [L1]^ ~ ) ) by Problem 6.3.8 since n  k  (y/2) > 0 for k = 0, 1,. . .,n  1. Therefore a repeated application of the convolution theorems (Theorems 5.1.3, 5.3.5, Prop. 5.3.3) gives the existence of some pa," E BV (in fact, t ~ may ~ be , assumed ~ to be absolutely continuous) such that
Again, [u2/(1 + U ~ ) ] ~ ( ~  ( Y ' ~ ) )[BV]" E by (6.3.18) since n(k  (y/2)) > 0 for k = 1 , 2 , . . .,n. Therefore the latter sum is the FourierStieltjes transform of some p3," E BV, which proves (6.3.19). Now to the problem posed at the beginning of this subsection.
Theorem 6.3.14. Let 1 I p I 2 anda > 0. I f f € W [ L p ;IuI"], thenfg W[Lp;Iu18]for eueryO < p < a. Moreouer, i f f E V [ L p ; Iula], thenfE W [ L p ; lu1B]for euery 0 < /3 < a. Proof. LetfE VIL1; IuIa],i.e., for f E L1 thereexists p E BV such that Ivl"f"(u) By (6.3.19) there are p2,", p3,#E BV such that (1
+ V2).9"(~)
=f W P ; , a M
= p"(u).
+ tL"(4PXa(U).
+
Therefore (1 u2)a/afA(u)E [BV]" by the convolution theorems. Since by Problem 6.3.8 (1 u ' )  ( "  ~ ) E~ ~[Ll]" for every 0 < < a, it follows that (1 + u2)olaf"(u) E [L']" by Prop. 5.3.3. Now (6.3.18) implies the existence of pl,s E BV such that
+
IuIflf"(u) = (1
+ u"fl'"fA(v)p';.fl(u).
Thus Iulsf"(u) E [Ll]" by Prop. 5.3.3, i.e., there exists some goE L' such that Iulflf"(u) = g;(u). This proves the second assertion for p = 1. The proof of the other cases follows along the same lines. Concerning the classes W[X2,;lkl"], V [ X 2 , ; lkl"] of periodic functions, the corresponding problem may be treated in the same manner using the periodic version of Lemma 6.3.13 (see Problem 6.3.10). But the reduction will now be more elementary since for every p > 0 there exists vB E L:, such that &(k) = 1klB for k # 0, 0 for k = 0 (see Problem 6.3.1). Compare Problem 6.3.11 for the formulation of the result. Problems 1.
(i) Let f ( u ) be a realvalued function defined on [0, 00) and suppose that f ' , f " exist and f"(u) 2 0 for v E [0, 00). Show that the sequence {f(k)};=o is convex on P. (ii) Let a > 0. Show that the sequence {ka},",l is convex on N. (iii) Let a > 0. Show that there exists an even function pa E L&, such that v;(k) = Ikl  a for k # 0, 0 for k = 0. In fact,
255
REPRESENTATION THEOREMS
2. (i) Let {ak}?=O be a convex sequence on P such that limk+mak = 0. Show that the series (ao/?) 2psla k cos kx converges on [  T , n],save possibly at x = 0, to a positive and integrable sum f ( x ) , and is the Fourier series off. (Hint: Theorem 6.3.7, compare with Problem 1.2.7; see also ZYGMUND [71, p. 1831, EDWARDS [11, D. 112ff1) (ii) Show that (logk)' cos kx is the Fourier series of an L:,function. (iii) Show that there are series (ao/2) ak cos kx with coefficients monotonely decreasing to zero such that the sumf(x), which exists save for x = 0 by Problem 1.2.7, does not belong to L?iz. (Hint: ZYGMUND [71, p. 1841) 3. Let {a&}?= be a sequence of real numbers monotonely decreasing to zero as k 03.
+
z&z
+
f
(i) The sum f ( x ) of X k l a k sin kx, which exists save for x = 0 by Problem 1.2.7, (Auk)log k < a.If this condition is satisfied, belongs to Lin if and only if then 2 b l a k sin kx is the Fourier series off. (Auk)log k < co if and only if k'ak < CO. (ii) Show that (iii) Let a > 0. Show that there exists an odd function E Lin such that $:(k) = {isgn k} lkl" f o r k # 0, 0 f o r k = 0. In fact,
z;p=l
2' (isgnk}JkJ"eik"
m
m
1,6.(x) =
k=m
=
2
2 k"sinkx.
k=l
(iv) Show that (log k)l sin kx is not the Fourier series of an Lizfunction. (Hint: HARDYROGOSINSKI [l, p. 331, BARI[lII, p. 2011) 4. Let f
2;= 
E :I
be even and quasiconvex. Show that f is of bounded variation on Z, i.e.,
IAf(k)l < co. (Hint: BARI[lII, p. 2021)
5. (i) Show that iff(x) is twice differentiable on (a, 6 ) such thatf"(x) 2 0 on (a, b), then f(x) is convex on (a, b). (ii) Suppose thatf(x) is convex on (a, b) and bounded from above in some subinterval of (a, 6). Show thatf(x) is continuous and has lefthand and righthand derivatives on (a, b). The righthand derivative is not less than the lefthand derivative and both derivatives increase with x. (Hint: HARDYLITTLEWOODP~LYA [l, p. 911, ZYGMUND [71, p. 221) (iii) Prove (6.3.3). (Hint: HARDYLITTLEWOODP~LYA [l, p. 941, ZYGMUND[71, P. 241) (iv) Let f(x) be a realvalued function on (a, b), 03 Ia < b I03, such that f((1  h)xl + Axz) I(1  A)f(xl) + hf(xz) for every xlr xz E (a, b), 0 Ih I1. Show that f is convex and continuous on (a, b). (Hint: RUDIN[4, p. 601) (v) Let f(x) be convex and continuous on (a, b). Show that /((I  h)xl + Axz) I (1  h)f(xl) + hf(x,) for all xl, xz E (a, b), 0 IA I 1. (Hint: KRASNOSEL'SKIF RUTICKI~ [l, p. 11) (vi) Let f, g E L1 and suppose that g is positive. Iffis convex on R, so is the convolution f * g. Thus, iff is convex, so is the integral of FejCr. 6. Let f ( u ) E Co be an even function such thatfe ACL,,(O, 03). Furthermore, suppose that f' is piecewise continuous on (0, CO) with n discontinuities of the first kind (finite jumps) at 0 < u1 < u2 < . . . < v,, < 03 and absolutely continuous on every compact subinterval of (0,co) which does not contain any of these points. If u If"(u)l do < CO, show that there exists an even function g E L' such that f(u) = g"(u). In fact, for x # 0
s,"
[I]) which is to be compared with (6.3.15). (Hint: BERENSGORLICH
256
FOURIER TRANSFORMS
7. (i) Prove the theorem ofPdlya: Let f(x) be an even function of Cowhich is convex for x > 0. Thenf is the Fourier transform of a positive function of L1. (Hint: Theorem 6.3.11; see also TITCHMARSH [6, p. 1701, L m c s [l, p. 701, LINNIK[I, p. 371) (ii) Show that there is an even function f(x) of Co monotonely decreasing on (0, a) which is not the Fourier transform of an L1function. (Hint: See TITCHMARSH [6, P. 1711) 8. (i) Show that, for every y > 0, (1 + tra)7 is the Fourier transform of an L1function. ~ (ii) Let a > 0. Show that there exists G, E NL' such that GZ(v) = (1 u ~ )  ' / for every tr E R.Furthermore, I[Gall1 5 M with constant M , independent of a. (Hint: Use (6.3.16)) (iii) Let a > 0. Show that there exists G,* E Liz such that [G,*]"(k) = (1 ka),la for every k E H. 9. Let a > 0. Show that (1 + Itr1")' is the Fourier transform of an L1function. 10. Let a > 0. Show that there exists p:, E BV2,,j = 1,2, 3, such that for all k E Z lkl" = (1 k2Y'2[p~,,l"(k), (1 kaYa = [p&X(k) + Ikl'[p&J"(k). (Hint: Either copy the proof of Lemma 6.3.13 by using Problems 6.2.8, 6.3.8(iii) or apply Problem 5.3.8) 11. (i) Show that iff E W[Xan; \k\"] for some a > 0, then f e W[X2,; lk18] for every
+
+
+
+
O 0 is an arbitrary but fixed number. These are of type (6.4.3)with 7,(u) = exp {  Iul"} and p = t'/,, t + O + . Obviously, 7, is an even function belonging to C, for each K > 0. Moreover, vK is quasiconvex. Indeed, 7, E AC,2,,(0, co) and u I$(u)l du < 00 for each K > 0. Therefore by Theorem 6.3.1 1, Prop. 5.1.10,and (5.1.22),(5.2.26) Proposition 6.4.1. For each exp { IulK}. Infact,
K
> 0 there exists an euen w,
E
NL' such that wf(u)
=
,.m
I
(6.4.5) For every f
(6.4.6)
E
Lp, 1 I p I2, and t > 0,
1
"
f
jm etl"l"
% ,
tl/K
(u) elxudu =
f

dg
mf(
x  U)W,(t %) du.
Since w, E NL', the righthand side of (6.4.6)is now defined for any f E X(R). It is an integral of type (6.4.1)with p = t'lK, denoted by x; t), thus
(6.4.7)
tl/K
W,(J x; t ) = d27r
f
02
f(x  u)w,(t 'h)du.
m
It is called the general singular integral of Weierstrass. Theorem 6.3.11 assures w, E NL' for each K > 0; but in view of (6.4.5)we may moreover assume w, E C,. It is furthermore a positive function for 0 < K I 2. For 0 < K I 1 this follows immediately by Theorem 6.3.11 and the convexity of w;(u) on (0, a).However, the extension to 1 < K I 2 is by no means trivial (cf. Problem 6.4.1(i)). On the other hand, w, fails to be positive for 2 < K < CO (see Problem 17F.
A.
FOURIER TRANSFORMS
258
6.4.1(ii)). The integral (6.4.7) subsumes those of CauchyPoisson and GaussWeierstrass as the particular cases K = 1 and K = 2. Indeed, wl(x) = 1/2/n(l va)l and wz(x) = 2lIa exp {  2 / 4 } by Problem 5.1.2(i). For the representation of w, in terms of an infinite series we refer to Problem 6.4.1(iii).
+
Since W , E NL' for each K > 0, it follows by Lemma 3.1.5 and Theorem 3.1.6 that (6.4.8) lim 1I W K ( A t )  f(.)Ilxm = 0 t+o+ 0 ;
for every f E X(R). It is now natural to ask for an order of approximation in (6.4.8) and to study its dependence upon structural properties off. Here the results of Sec. 3.4 and 3.5 will be of use. Indeed, we have the following inverse approximation theorem:
Proposition 6.4.2. Let f E X(R) and K > 0. Then 11 W,(f; 0 ; t )  f(o)Ilx(lw) = O(tn"9, t 3 O+ ,for some 0 < a < K implies that f E *Wz&, where r E P and 0 < /I < 2 are such that a = r + /I. Proof. Since p = r l l r , the assertion will be a consequence of Theorems 3.5.1, 3.5.2. Indeed, we show that w, belongs to W & :: To this end, it follows by Theorem 6.3.11 that vzswf(v)is the Fourier transform of an L1function for each s E P. Thus w, E W[L'; ( i ~ ) ~ ~ ] implying that W, E W$ for each s E P by Theorem 5.1.16. In view of Problem 5.2.12 this gives W, E W;Za for every r E P. Moreover, wa E Co by (6.4.5), and since the absolute moments of exp {  Ivl,} exist for any positive order, it follows by Prop. 5.1.17 that w, has derivatives of every order belonging to Co. Thus w K E W:n", for every r E P, and Prop. 6.4.2 is completely established. In applying the direct approximation theorems of Sec. 3.4 we immediately encounter the problem of the existence of the ath absolute moment of w, for 0 < a < K . To this end, the next three lemmata supply sufficient conditions upon quasiconvex functions f such that the ath absolute moment of g exists, g E L1 being such that f ( v ) = g^(v). We distinguish two cases: For 0 < a < 1 we obtain a result for arbitrary quasiconvex functions, whereas for a 1 1 we have to assume the absolute continuity off for a sufficiently high order. Lemma 6.4.3. Let f E CObe even and quasiconvex, and let g be such that f ( v ) = g*(v). Then for 0 < a < 1 condition$," u' Idf'(u)I < 00 implies the existence of theath absolute moment m(g; a) of g. (I
Proof. SincefE Cois even and quasiconvex, Theorem 6.3.1 1 assures the existence of some g E L' with f ( v ) = g^(v). In fact, the estimate (6.3.16) holds. Moreover, since g is even,
=
{yIorn(y)' so " ta
dt}
ulu[df'(u)l,
which is bounded by hypothesis. This proves the assertion. Since the ath moment of the FejCr kernel exists only for 0 < a < 1, one cannot immediately use the estimate (6.3.16) in case a z 1 . Indeed, the assumptions and proof of Theorem 6.3.1 1 will be modified; the proof is given for a < 3, the case a arbitrary being left to the reader. Lemma 6.4.4. Let f E Co be even and quasiconvex, and let g be such that f ( v ) = g"(v). Suppose thatf E AC;oo(O, 00) with lime+rnf("(p) = 0, lim,+o+ d  ' f t r ) ( e ) = 0 f o r j = 1 , 2, 3. Then for 1 < a < 3 condition I, u" 3  7 f(*)(u)I du < 00 implies the existence of the ath absolute moment of g .
259
REPRESENTATION THEOREMS
Proof. The proof follows by a straightforward modification of that of Theorem 6.3.11. Indeed, let x # 0 be arbitrary, fixed. Since f E ACfaC(O,co), a repeated integration by parts gives for 0 < e < p < 00


sp Y
f (4)(u)du.
Letting p + 00, E O f , the lefthand side tends to d/.r12g(x) as given by (6.3.15). On the other hand, the present hypotheses assure that the terms in brackets tend to zero. Therefore
Srn
4 sins (xu/2)f ( 4 ) ( u ) du, d2s 0 x4 the integral being absolutely convergent since for 1 < a < 3
(6.4.9)
g(x) =
The representation (6.4.9) implies (see (3.3.6))
5
{yso m
sina t d t }
Iom
u3"IfC4)(u)1du,
the constant in brackets being finite for 1 < a < 3. This proves the lemma.
Lemma 6.4.5. Let f E Co be even and quasiconoex, and let g be such that f ( v ) = gA(o). Suppose for some r E N that f E AC&+,'(O, 00) with limo+: f'"(p) = 0 for j = 1,2,. . ., 2r + 1 , Iim,,ot f(j)(e)= 0 for j = 1 , 2,. . ., 2r  1 , lim,+ot E ' ~ ( ~ ~  ~ + ' ) ) (=E ) 0 for j = 1,2. Then for 2r  1 < a < 2r + 1 condition uart1'l f(2r+2)(u)Idu < 03 implies the existence of the ath absolute moment of g. In this case, g admits the representation
s,"
(6.4.10) Obviously, if the assumptions of Lemma 6.4.4 or 6.4.5 are satisfied for some a, then the flth absolute moment of g exists for every 0 c fl < a. Let us now return to the direct approximation problem for the general singular integral of Weierstrass. By an elementary calculation it follows from Lemma 6.4.3 if K I1 , and Lemma 6.4.5 if K > 1 (choosing r E N such that 2r 1 < K ) :
Proposition 6.4.6. Zf K > 0, the ath absolute moment of wk exists for every 0 < a < K. This result and Prop. 5.1.17 imply that the kernel wKfor K > 1 belongs to Ms(cf. (3.4.5)) for every s E N with s c K. Now we may state as immediate consequences of Prop. 3.4.3 and 3.4.6
Proposition 6.4.7. Let f E X(R) and K > 0, a > 0 being such that a < K . (i) Zf 0 < a 5 2 and f E Lip* (X(R); a ) then (6.4.1 1) 11 w K ( f ; ;t,  f ('1 Ilx(w) = (t + O+). (ii) Zf a = r + fl with r E N, 0 < fl s 1, then f E Wg&) implies (6.4.11). (iii) Zf a = r + fl with r E N, 0 I fl I 2, a n d i f r + 2 < K , then f E *W&, implies (6.4.11). (iv) Zf a = 2s + fl with s E N, 0 I: fl I: 2, then f E *W;& implies (6.4.1 1). It is easy to combine the results of Prop. 6.4.2 and 6.4.7 so as to obtain equivalence theorems for the approximation off by the integral WK(f;x ; t). As an example we state
FOURIER TRANSFORMS
260
Corollary 6.4.8. L e t f e X(R) and 0 c a c
11 W,(A
0 ;
t)
 f(o)IIx(i~)
K
I2. Then
= O(r'lK)
f
E Lip*
(X(W;a).
Certainly it is possible to obtain equivalence theorems for K > 2. But we have to emphasize that Prop. 6.4.2, 6.4.7 do not cover all cases simultaneously. Thus for K = 3, a = 2 the methods so far employed do not enable us to establish an equivalence theorem. Indeed, Prop. 6.4.2 states that the approximation 11 W d f ; 0 ; t )  f(o)IIx(a)= O(t2'3) implies j~ *W&, but conversely, Prop. 6.4.7(ii) shows that only the stronger hypothesis f~ W&b, implies the latter approximation. This gap will be removed in Sec. 13.3.3 as a consequence of results on saturation (the case a = K ) for the integral W,(A x; t). We conclude with a brief introduction of the corresponding periodic integral. As above, K > 0 is a fixed parameter. Via (3.1.28) we may associate with w, E NL1 a periodic kernel which, with a slight abuse of notation (omitting the star), is given for each t > 0 by
(6.4.12)
wt,K(x)=
d~
2
k=m
tl/KwK(tl/K(x+ 2k?r)) a.e.
In view of (5.1.58), wLK(k)= exp{t lklK}.Thus the Fourier series of wt,k E L:, converges absolutely and uniformly in x for each t > 0, and (see Prop. 4 . 1 3 9 , 5.1.29) rn
(6.4.13) In the following we shall regard M J ~ as , ~ given for all x by (6.4.13). In view of Prop. 3.1.12 and Problem 3.1.10, { w ~ , ~ ( is x )then } an even continuous periodic approximate identity with parameter t tending to O+ . The corresponding singular integral
is called the general (periodic) singular integral of Weierstrass. The particular case = 2 has already beenintroduced in Problem 1.3.10.Since wCK E I1 for each t > 0, it follows by Prop. 4.1.6 that f o r f e Xzn K
m
(6.4.15)
wt.K(.f;
=
2
etlkl"f"(k) eIkxX.
k=w
Thus W,,,(f; x ) is a summation process of type (4.1.16) for Fourier series with 0factor (cf. (1.2.28)) given by {exp (  t lkl")}.In this form, Wt,,(f; x ) is also known as the AbelCartwright means of the Fourier series off. Sometimes one sets e* = r which turns (6.4.15) into
(6.4.16)
2 rlkiy*(k) efkx,
k=m
where r E (0, l), r + 1 . For K = 1 we recognize the Abel means as introduced in Sec. 1.2.4. Indeed, w l o g ~ l ~ r ~corresponds ,l(x) to pr(x) and Wlog(llr),lcf; x) to Pr(f; x ) , the integral of AbelPoisson; wlog(llr),l(x) admits the elementary representation (1.2.37).
26 1
REPRESENTATION THEOREMS
Concerning the approximation off by the integral Theorem 1.1.5 that for every f E x2, (6.4.17)
lim
i+o+
Wt,,(f; x ) for t '0+, it follows by
1I W d f ;  f ( o > t l x , , 0)
= 0.
Regarding direct approximation theorems, by Problem 3.1.10 and (6.4.12)
which is finite for 0 < a < K by Prop. 6.4.6. Therefore, if 0 < a < K and, in addition, 0 < a I2, f~ Lip* (Xan;a) implies 1) W&; 0)  f ( o ) l l x 2 , = O(taiK)by Prop. 1.6.3 (see also Problem 6.4.2(i)). Concerning higher order direct approximation theorems it is not hard to establish a periodic version of Prop. 3.4.6, applicable to the integral Wt.,(f; x) (compare Problems 6.4.2, 6.4.4). To prove inverse approximation theorems one may proceed as in Sec. 2.5.1 and 3.5 (see Problem 6.4.3 for the formulation of the result). Thus, keeping in mind that the periodic kernel {wf,x(x)}is generated via (6.4.12) by wx E NL', one may develop results for the periodic integral W,,,(f;x ) to the extent given for the nonperiodic integral W,(f; x ; t). These concern order of approximation with a subject to the restriction 0 < a < K . In Sec. 13.3 we shall return to this nonoptimal approximation case, the saturation case a = K being treated in Sec. 12.2.3.
6.4.2 Typical Means The typical means of the Fourier inversion integral are given by (6.4.18) K > 0 again being an arbitrary but fixed number. The integral (6.4.18) is of type (6.4.3) with ~ , ( v )= (1  IvlK)for IuI 5 1, 0 for IvI > 1, which is an even and quasiconvex function belonging to Co. Therefore by Theorem 6.3.11, Prop. 5.1.10, and (5.1.22), (5.2.26)
Proposition 6.4.9. For each (1  Iul")for I 1, Ofor
101
K
101
> 0 there exists an even rc E NL' such that r;(u) 1. Znfucr,
=
(6.4.19)
For every f (6.4.20)
E
Lp, 1 I p 5 2, and p > 0,
1' I
dz;;
P
(1 
r)f"(v)
efxudo =

x  u)r,(pu) du.
The righthand side is meaningful for any f E X(R) and defines a singular integral of FejCr's type :
We also call R,(J x ; p) typical means.
FOURIER TRANSFORMS
262
In view of (6.4.19) the kernel r, may be assumed to belong to Co. It is positive for 0 < K 5 1 since r,^ is convex on (0, co). The particular case K = 1 gives FejCr's kernel F(x) in which case rl(x) admits the explicit representation (3.1.15) (cf. also Problem 5.1.2(i)). Since r, E NL' for each K > 0, by Lemma 3.1.5 and Theorem 3.1.6 (6.4.22) lim I &(f; p)  f ( o ) I I x ( w = 0 0 ;
P+ m
for every f E X(R). As an inverse approximation theorem we have as for Prop. 6.4.2 Proposition 6.4.10. Let f E x(W) and K > 0. Then I I f t k ( f ; 0 ; p)  f(o)Ilx(w) = o(p"), p f co, for some 0 < a < K implies f E *W;& where r E P and 0 < fl < 2 are such that
a=r+fl. Concerning direct approximation theorems, Lemma 6.4.3 gives that the ath absolute moment of r, exists for 0 < a < 1, a < K. Therefore Prop. 3.4.3 is applicable. This leads to the following equivalence theorem: Corollary 6.4.11. Let f E X(R) and 0 < a < K 5 1. Then Lip* (1 '; IIRK(f; O; p)  .f(')IlX(R) = O(p") e f
However, Lemmata 6.4.4,6.4.5 are not applicablesince [r;]'doesnot belong to ACI,,(O, co) (cf. also Problem 6.3.6). This implies that the results of Sec. 3.4 do not apply at this stage in case a 2 1. Therefore for the typical means higher order (1 5 a < K ) direct approximation theorems are entirely postponed to Sec. 13.3.3. The saturation case a = K is discussed in Sec. 12.4.5. We turn to the periodic counterpart. As for the Weierstrass integral we may associate with r, via (3.1.28) an even periodic approximate identity. It being customary to replace the continuous parameter p > 0 by the discrete (n 1) with n E N, n +. m, with slight changes of notation we are led to
+
m
(6.4.23)
r,,,,(x) =
6k =2 m (n + l)r,((n
+ l)(x + 2kv))
a.e.
+
Sincer,?,(k) = (1  lk/(n 1)IK)for Ikl In,Ofor Ikl > n by (5.1.58), the Fourier series of r,&) reduces to a trigonometric polynomial of degree n ; we may regard (6.4.24) as the definition of r,,,(x) for all x. This implies that {r,,,&)} is a continuous kernel (for each fked K > 0). The corresponding singular integral o f f € X,, is denoted by (6.4.25)
Rll.KCf;
=
2v
s'
fi
.f(x
u)rlI,K(u)
du;
in view of (4.1.7) and (6.4.24) it may be rewritten as (6.4.26) known as the typicalmeans of the Fourier series off E XSn.These are of type (1.2.30) with rowfinite 0factor given by r;,. Obviously, K = 1 corresponds to the FejCr means of Sec. 1.2.2.
263
REPRESENTATION THEOREMS
By Theorem 1.1.5, for everyfE Xln, (6.4.27) lim llRn,w(f; n m
 f(o)Ilxl, = 0. Since the ath absolute moment of rk exists for 0 < a < 1, 277
1,,
Iul"lrn,,(u)l
du 5
1
0)
"
lul"l(n
+ l)r,((n + 1)u)l du = o(n")
by Problem 3.1.10 and (6.4.23). Therefore, if 0 < a < K and, in addition, 0 < a < 1, then 0 )  f(o)llxan = O(n"), n + co, by Prop. 1.6.3. As for the nonperiodic integral, higher order direct approximation theorems for R,,,(f; x ) are postponed to Sec. 13.3. Concerning inverse theorems, the situation for the typical means is quite different to that for the AbelCartwright means. Since for every f E X,, R,,,(f; x ) is a trigonometric polynomial of degree n at most, the theorems of Bernstein on best approximation by trigonometric polynomials are applicable. Indeed, if IIRn,,(f;  f(.) / I x an = O(n") for some 0 < a < K , then E,(X,,;f)= O(n"). Theorem 2.3.6 now asserts f E *W?,S,,where r E P and 0 < t!3 < 2 are such that a = r 8. Thus there only remains the saturation case a = K which is investigated in Sec. 12.2.2.
f~ Lip* (Xzn;a) implies l1Rn,,(f;
0)
+
Problems
1. Let K > 0 and w, E NL1 be given by (6.4.5). (i) Show that w,(x) is a positive function of x for each 1 < K 5 2. (Hint: LBVY[2, pp. 2722741, BOCHNER [3, 41) (ii) Show that w,(x) is not a positive function of x in case 2 < K < 00. (Hint: BOCHNER [7, p. 961, CRAMER[2, p. 911) (iii) Show that for x # 0
(Hint: HARDY[2, p. 3851, BERGSTROM [l], FELLER [l], LUKACS [l, p. 1051) 2. Let the periodic approximate identity {x:(x)} be generated via (3.1.28) by a n even x E NL1 and let Z,*(f; x), J ( f ; x ; p) be the singular integrals corresponding to the kernels { x X x ) } , {px(px)}, i.e.
(then Z,*(f; x ) = J(f; x ; p ) (a.e.) by (3.1.31) for each f E Xznand p > 0). Establish the following periodic counterparts of Prop. 3.4.3, Lemma 3.4.5, Prop. 3.4.6: (i) Let f E Xzn. If the ath absolute moment of x exists for some 0 < a 5 2, then f E Lip* (Xzn;a) implies llZ:(f;  f(0)llxan = O(p"), p +. 03. (ii) If x E M' (cf. (3.4.5)), then f~ WZznimplies 0)
Srn
d ( fx;)  f ( x ) = d2?r
m
N x , u)x(pu) du.
Here R ( x , u) is given by
+
[ p ( t )  # r ) ( ~ ) ] ( ~ u  t)l'l dr, R(x, u) = (r  l)! where $ E AC:; with $(I) E Xznis such that f ( x ) = $(x) (a.e.). In particular,
IlZ3f;
0)
 f(0)llxan
5
2
2 m(x; r)ll~(r)llx2nPr.
264
FOURIER TRANSFORMS
(iii) Let f E Xan, r E N, and 0 I a I1. If the (r exists, then f E W ,;; implies
+ 4 t h absolute moment of x E M'

(6.4.28) IICYf;  f(o)Ilxz, = o(P(r+")) (P a). (iv) Let f E X,,, r = 2s, s E N, and 0 < a I2. If the (2s + a)th absolute moment of x E M' exists, then f E *Wk& implies (6.4.28). (v) Let f E Xa,, r E N, and 0 < a I 2. If x E Mr+a,then for f E Wi2= 0)
In particular, f E *Wk& implies (6.4.28). 3. Let x, x t , J ( f ; x; p), I,*(f; x) be given as in Problem 6.4.2. Prove the following periodic counterpart of Theorems 3.5.1, 3.5.2: Let x satisfy the assumptions of these theorems. If, for some r E P, 0 < a < 2, f E X,, can be approximated by I,*(f; x) such that IlZp*(f; 0)  f(0)Ilx,, = O(P('+~)), p + Q), then f E *W;;,. 4. Apply the preceding Problems so as to obtain equivalence theorems for the approximation off E Xan by the general (periodic) integral Wt.,(L x) of Weierstrass. Prove, for example, that if a = r with r E N, 0 < j? < 2, and if r 2 < K , then 11 Wt,,(f;0 )  f ( o ) I l x 2 , = O(talK),t + 0 + , if and only iff E *Wk,",. 5. Let K > 0 and C, E NL' be such that c;(u) = (1 IulK)l (cf. Problem 6.3.9). The
+
+
+
integral
is called the generalized singular integrd of Picard (see BERENSGORLICH [l I). (i) Let K > 1. Show that c,(x) = (l/d%)J:m (1 f€LP,I I P 5 2 ,
+ IvIK)lefxuduand
that for
(ii) Show that c,(x) is a positive function of x for each 0 < K I 2. (Hint: For 0 < K I1 use the convexity of c;(u) on (0, a),for 1 < K 5 2 see LINNIK [l, p. 401) (iii) Show that ca(x) = d a exp {  Ixl}. Thus Ca(f; x; p ) is the special singular integral of Picard as introduced in Problem 3.2.5. (iv) Try to apply the results of Sec. 3.4, 3.5 so as to obtain equivalence theorems for the approximation off E X ( W ) by the integral C,(f; x; p) as p * 00. (v) Discuss the periodic version of the generalized singular integral of Picard as obtained via (3.1.28), (3.1.31) from the nonperiodic CJf; x; p). 6. Let K > 0 and G, E NL1 be such that G;(u) = (1 + ~ ~ ) (cf.  ~Problem l ~ 6.3.8(ii)). The integral
is called the Besselpotential off. (i) Let K > 1. Show that C,(x) = (l/d%)J:m (1 fELP, 1 I p 5 2 ,
+ u 2 )  K / a e r x u dand v
that for
(ii) Show that G, admits the representation G,(X) = [2""a~(K/2)]11X1(K ""K ( 1  w a ( 1x1)
(K
> 01,
265
REPRESENTATION THEOREMS
where K y is the modified Bessel function of the third kind of order y :
(Hint: ARONSZAJNSMITH [I, p. 414 ff], C A L D E R ~[ lN ] , CALDER~NZYGMUND [3]) (iii) Show that CK(x) is a positive function of x for each K > 0. (Hint: For this and further properties see the above literature as well as ARONSZAJNMULLASZEPTYCKI [I]) (iv) Formulate and prove counterparts of Problem 6.4.5(iv) [(v)] for [periodic] Bessel potentials. 7. Let K > 0, h > 0. Show that there exists rK.AENL1 such that rGd(u) = (1  1 ~ 1 for ~ ) ~ ) u J 5 1, 0 for JuI > 1. (Hint: For K > 0, h 2 1 use Theorem 6.3.11, for K > 0, 0 c h < 1 Problem 6.3.14) 8. For f E X(R) the Riesz means are defined by
where the kernel rK.ais given for K > 0, h > 0 by Problem 6.4.7.
s',
Show that rK,a(x)= (l/d%)
(I 
1 ~ 1 efxu ~ ) du ~ and that f o r f e
Lp, 1 I p I2,
Show that r K J x ) is a positive function of x for each 0 < K II , h 2 1 . Try to apply the results of Sec. 3.4, 3.5 so as to obtain equivalence theorems for the approximation off E x(R) by the Riesz means RK,l(f;x ;p ) as p + co. (Hint: Compare the discussion on the typical means in Sec. 6.4.2; these are indeed the particular case h = 1 of the Riesz means) Discuss the periodic version of the Riesz means. In this case, we employ the notation
is generated by rK,avia (3.1.28) by setting where the periodic kernel {r,,,K,L(x)J + I , n E N, n + m , thus
p =n
m
rn,K,a(x)=
(n + 1)rK,d(n+ l)(x + 2kn)). 4%k =2 m
Show that r,,,K,a(x)=
I:= ,, (1
 Ik/(n
+ l)lK)Aelkxand
which are known as the Riesz means of the Fourier series of f . Moreover, h,,./lRII.K.~(A  $(.)lx, = 0 for everyf e Xzn. 0)
9. The particular Riesz means for K = 2 are known as the singular integral of BochnerRiesz. In this case we use the notation
where the kernel ba is defined through bA(x) = r2,&). Thus 61 E NL' for each h > 0. Show that br(x) = 2"(1 + A)lx\ (w)+ A)J~l,z~+a(\x\), where J,(x) is the Bessel function of order y. (Hint: BOCHNER [2], V. L. SHAPIRO [I])
266
FOURIER TRANSFORMS
6.5 Multipliers
The representation problem of the preceding sections is connected with the problem of the representation of multipliers. Indeed, it will be shown that a function X on Z is the finite FourierStieltjes transform of some p E BV,, if and only if for each f E Czn h(k)f^(k) is the finite Fourier transform of some g E Can.A function X on Z satisfying this condition will be called a multiplier for Fourier coefficients of type (CZn,C2=). Correspondingly, it will be seen that a function h on R is a FourierStieltjes transform if and only if, for each f E L', h(u)f"(u) is the Fourier transform of some g E L', in other words, if and only if h(u) is a multiplier for Fourier transforms of type (L', L1). So one is led to investigate multipliers of various classes of functions and their representation.
6.5.1 Multipliers of Classes of Periodic Functions
Let Y, and Y, be one of the spaces Can,Lfi,, 1 I p I00. A function X on Z is said to be a multiplier of type (Y,, Y,) if for each f E Y, there exists a function g E Y, such that X(k)f "(k) = g"(k), k E Z. In other words, a sequence of complex numbers is a multiplier of type ( Y1, Y,) if for everyf E Y, the (formal) trigonometric  X(k)f"(k)etkxis the Fourier series of a function g E Y,. series
=:x
Proposition 6.5.1. Let A be a multiplier of type ( Y,, Y,), and let U be dejined as the transformation of Y, into Y2 which assigns to each f E Y, the function Uf = g where g E Y, is given by X(k)f"(k) = g"(k), k E h. Then U is linear, bounded, and translationinvariant. An operator Uis called translutioninvariant on Y, if U w ( o + h)](x) = ( U f ) ( x for every f E Y1 and h E R.
+ h)
Proof. It follows immediately by Prop. 4.1.1(i), 4.1.2(i), and the uniqueness theorem that U is welldefined as a transformation of Y1 into Y2 which is linear and commutes with the group of real translations. According to the closed graph theorem U will be bounded if it can be shown that U is closed. For this purpose suppose that
Then by Problem 4.1.1 lim f;(k) = f"(k),
n+ m
lim [Ufn]"(k)= g"(k)
n m
(kE Z).
But X(k)f c ( k ) = [Ufn]"(k),k E Z, by definition. Therefore h(k)f "(k) = g"(k), k E Z, i.e. Uf = g, and thus U is closed. Having associated with each multiplier X the corresponding multiplier operator U (= UA),the problem is to characterize the functions h of a definite multiplier class (Yl, Y,) and to develop a representation of the operators UA.There are complete solutions of numerous important particular cases, but here we shall confine ourselves to two representative examples. For further results we refer to the Problems.
267
REPRESENTATION THEOREMS
Proposition 6.5.2. A function A on h is a multiplier of type (L;,, X E I". In this case, we have for the multiplier operator (6.5.1)
II~II[L&,L,a,l =
L$,) if and onl'y if
II~Illrn.
Proof. Let h be a multiplier of type (L&, L&). Then by Prop. 6.5.1 there exists a constant M I M 1 f [IL;, for all f E L;,. Therefore by the Parseval equation such that 1 Uf
If we take f ( x ) = exp {imx}, m E Z, this implies that IA(m)l I M for all m E Z. Conversely, if A E Im, then Af" E l2 for every f E L:, and thus it follows by the theorem of RieszFischer that A is a multiplier of type (L;,, L&). Moreover, I1 Uf IIL;,I l l A l l ~ mIlf IlL& by the Parseval equation. On the other hand, for f ( x ) = exp {imx}, m E Z, we obtain 11 V f,;LI/ = IA(m)l, which proves (6.5.1).
Proposition 6.5.3. A necessary and suflcient condition for a function A on Z to be a multiplier of type (Can,C,,) is that A is the finite FourierStieltjes transform of a function p E BV,,. In this case, (6.5.2)
for every f E C,, and
'
(6.5.3)
I I [Can,Canl = 1 1 ~ 1 B1 V ~ ~ '
Proof. Necessity. Let o,(f; x ) be the Fejer means of the Fourier series of f (cf. (1.2.25)), and let us denote the Fejkr means of the series  A(k) exp {ikx) and 2;=  ,A(k) fA(k) exp {ikx} by TAX) and T,,(x), respectively, i.e.
z&
(9
r,(x) =
(ii)
T,(x) =
(6.5.4)
2 (1  Ikl 2  $)A(k)fA(k) (1
eikx
elkx.
k  n
Then by (4.1.7) (6.5.5)
Fl)~(/c)
k  n
rn(X)
=
& /:,f
(X
 u)Tn(u) du.
If A is a multiplier of type (C,,, C,,), then for every f E C,, there exists g E C,, such that = a,(g; x). Therefore I ~ T , , I ( I ~ ~ ~I 11g11c2n which in particular implies by (6.5.5) that for each f E C,, r,(x)
where the constant M depends on f but not on n. For each n E N the integral on the left defines a bounded linear functional on C,, the norm of which is given by l T,ll (see Prop. 0.8.8). Hence (6.5.6) and the uniform boundedness principle imply that the sequence { 1 1 ~ ~ 1 1is~ )bounded, that is to say that
(n

a).
Theorem 6.1.2 then shows that A is representable as the finite FourierStieltjes transform of a function p E BV,,.
268
FOURIER TRANSFORMS
Sufficiency.If there exists p E BVansuch that A(k) = p"(k), k E H, then for every f E C,, (6.5.7)
n,(x) =
1 1" an(f;x 271 I
 u) dp(u).
Since [o,(f; x )  o,(f; x ) ] tends uniformly to zero as m,n + co by Theorem 6.1.2, so does [n,(x)  .rm(x)]. Hence Theorem 6.1.2 applies again and shows that A(k)f^(k) are the Fourier coefficients of a C,,function. Now the representation (6.5.2) is valid since both sides have the same finite Fourier transform, whereas the relation (6.5.3) is a consequence of Problem 1.3.1. We finally note that the function p may be expressed in terms of the multiplier A by (6.5.8)
This is given by Prop. 4.3.10.
6.5.2 Multipliers on Lp
A function X(v) on R is said to be a multiplier of type (Lp, Lq), 1 I p , q I2, if for everyf E Lp there exists a function g E L4 such that h(v) f "(0) = g^(u). As in the periodic case one may prove Proposition 6.5.4. Let A be a multiplier of type (Lp,,)'L 1 5 p , q I 2, and let U be dejined as the operator of Lp into Lq which assigns to each f E Lp the function Uf = g where g E Lq is given by A(v)f"(o) = g^(v). Then the multiplier operator is linear, bounded, and translationinvariant. Parallel to the treatment in the preceding subsection we now discuss the important case of multiplier classes (Lp,Lp), 1 Ip I2.
Theorem 6.5.5. A function X is a multiplier of type (L2, L2) if and only if A E L". In this case, (6.5.9) Proof. Let X be a multiplier of type (La, La). Then by Prop. 6.5.4 there exists a constant M > 0 such that 11 Uf 112 IM 11 f [la for all f E La. Hence by the Parseval equation
Moreover, by Plancherel's theorem we may choose any convenient L2function for f ". Thus, if we put f" = K ~  , , , ~ , the characteristic function of the interval [  p , p], then it follows from (6.5.10) that lA(u)12 du I2Map, i.e. A is measurable and integrable over every finite interval. On the other hand, if we substitute
E,
K [  , , , , ( u ) ( ~ x ~lI4) exp {  ( x
for fA,we obtain for Ap(u)
3
~)~/8t}
( x E R, t > 0 )
A ( V ) ~  , , , ~ ( V )that
By Cor. 3.2.3 the left side tends to IA,(x)la ax. as t 0 + , since A, E L' for each p > 0. Therefore llApII IM , and since this is true for all p > 0, we have llAllm 5 M. f
269
REPRESENTATION THEOREMS
Conversely, if E L", then h f " ~La for every f E L2, and it follows by Plancherel's theorem that there exists g E L2 such that A(u)f"(u) = g"(u), i.e. A is a multiplier of type (L2, L2). Moreover, IIUfll, = llAfA112 I IIAIlmllfli2 by the Parseval equation, and therefore IIU/I I llAllm. Together with the opposite inequality llAllm I )IUII obtained above, this proves (6.5.9). Theorem 6.5.6. A necessary and suficient condition for a function h to be a multiplier
of type (L', L') is that A is the FourierStieltjes transform of a function p E BV. In this case,
for every f
E
L' and
II q L 1 , L q
(6.5.12)
= IlPllev.
Proof. Necessiry. If A is a multiplier of type (L1, Ll), then the corresponding multiplier operator U is a linear, bounded, translationinvariant operator of L' into L' by Prop. 6.5.4. Letf, g E L'. Thenf* g E L' by Prop. 0.2.2,and thus ( U [ f * g ] ) ( x )is welldefined as a function in L1. By the definition of Uand the convolution theorem we have [U[f*g]]"(u)= A(u)fA(u)g"(u) = [f*Ug]"(u), and hence by the uniqueness theorem (6.5.13) ( U [ f * g l ) ( x ) = (f* &)(x) a.e. Let { ~ ( xp)) ; be a positive approximate identity. Then limp+mIII(f; 0 ; p)  f(o)II1 = 0 for every f~ L' by Theorem 3.1.6, and since ( U [ I ( f ;0 ; p ) ] ) ( x ) = (f* U[x(o;p)])(x) a.e. by (6.5.13), we have for every f E L1 (6.5.14)
lim
I l f * U[x(o;p)1 
Uflll = 0.
P+m
If we set pp(x) = ST ( U [ x ( o ;p)])(u) du, this defines a set of absolutely continuous functions pp which are of uniformly bounded variation since Ilx(0; p)\\1 = 1 and (6.5.1 5)
( p > 0). IIU[X(";p)l(o)II1 I /lull Therefore by the weak* compactness theorem for Cothere exists {p,} with limj, pI = co and p E BV with IjpllSvI 11 U 11 such that for every f E CO IIpPllSV
=
This together with (6.5.14) and Prop. 0.1.10 shows that (6.5.11)holdsforevery f E CoA L1. However, both members in (6.5.11) define bounded linear operators of L' into L1 (cf. Prop. 0.5.6), and thus, since Con L1 is dense in L1, it follows that (6.5.11) holds for all f~ L1. In particular, we have 11 U 1 f llpl!Bv,which together with the opposite inequality obtained above shows that (6.5.12) is valid. Furthermore, by the definition of U and by Prop. 5.3.3, the representation (6.5.1 1) implies that Af" = [Uf]"= f"p" for everyfE L'. Therefore h is the FourierStieltjes transform of p E BV. Suficiency. Suppose there exists p E BV such that h = p". Then with each f E L1 we may associate the function g(x) = (f*dp)(x) which belongs to L1 by Prop. 0.5.6. In view of Prop. 5.3.3 we then have p"f" = gA. Thus A is a multiplier of type (Ll, L'), and the proof is complete. As a consequence of LCvy's theorem we mention that p is uniquely determined by the inultiplier h and given by A(u) du.
270
FOURIER TRANSFORMS
We now turn to the problem of deriving necessary and sufficient conditions for a bounded measurable function to be a multiplier of type (Lp, Lp), 1 < p < 2.
Theorem 6.5.7. Let A be a bounded measurable function on R. Then A is a multiplier of type (Lp, LP), 1 < p < 2, i f and only if there is a constant M such that (6.5.16)
for all dl, E F. In this case, satisfying (6.5.16).
)I UIIILp,Lp]
= M*,
where M * is the smallest constant
Proof. Let A be a multiplier of type (Lp, Lp), 1 < p < 2. Then, in particular, for each E F (cf. (6.1.10)) there exists gl E Lp such that A+,; = g;. Since E L" and $2 E L ', we have g; E ,'L and hence by Prop. 5.2.16
+1
gl(x) =
1" 2/2r
A(u)+;(u)
elx" do
a.e.,
m
i.e. gl(x) = [A+;]^(  x ) a.e. Therefore, if U is the multiplier operator corresponding to A, then U+l = g,, and we have by Parseval's formula (5.2.18) and Holder's inequality for
5
l l ~ ~ + ~ l A l l P l l + 2 1 1= P ~ l l ~ + l l l P l l + 2 l l P ~5
II UII
II+lIIPIldzllP~*
Hence (6.5.16) is valid with M * I 1 U 11. Conversely, if (6.5.16) holds, then for every fixed +1 E F the integral there defines a bounded linear functional over F considered as a subset of Lp'. Since F is dense in Lp', it may be extended to all of Lp'. Therefore by Prop. 0.8.11 there exists gl E Lpwith jigl IIp 5 M*ll+lllpsuch that for every d2 E F .I'm
W+;(Mmdo =
s_mmm
42(4g1(  4 do =
/mm
g ; ( w M du
upon using Parseval's formula (5.2.27). If we take b2(u) = exp { ru2 + ixu}, x E R, t > 0, as in the proof of Theorem 6.1.6, it hence follows that to every dl E F there exists g, E Lp such that A& = g;. Let T be the operator on F (considered as a subspace of L P ) into L p given by T$l = g,. Upon taking the supremum over all E F with l\+l\Ip I 1, we have by (0.7.5) JIT(IIF,L~l = sup IIT+lIIp = sup llglllp IM*.By Prop. 0.7.1, T may be 5 M*. extended to all of L P with the same norm, thus IITJlrL~,L~I Let f e Lp and Tf = g. Then there exists a sequence {C,,}c F such that limn+" \If+,,\Ip = 0. Furthermore, (ITf T+,,I(p= 0 by the continuity of T, and limn+m  +clip, = 0, limn+mIIg" = 0 by Titchmarsh's inequality. Since by definition A+; = [T+,,]^, we finally obtain IlAf"  gnllp,= 0. Hence A is a multiplier of type (Lp, Lp). Moreover, T is nothing but U,and )I U ) ) r L ~=. LM~*,.
\If^
Theorem 6.5.8. Let A be a bounded continuous function on 04. Then A is a multiplier of type (Lp, Lp), 1 < p < 2, if and only ifthere is a constant MI such that
27 1
REPRESENTATION THEOREMS
for aNJinite sets of real numbers { u k } and complex numbers {c,}, {dk}.In this case,
I LJ
(6.5.18)
II[LP,
L']
=
M:,
where M: is the smallest constant satisfying (6.5.17). Proof. Necessity. Let the bounded continuous function h be a multiplier of type (Lp, LP). Then there is a constant M such that (6.5.16) holds. If for any n E N,arbitrary sets of real numbers u l , . . ., u, and complex numbers cl,. . ., c,,, d,, . . ., dn,and any t > 0 we take &(x) = etp'x'
i ck
e"kX,
&(x) = etpxa
5 dk
etukx,
k1
k1
then by Problem 5.1.2(i)
Therefore by (6.5.16)
Let x1 # xz, xl, x2 E R. Then (6.5.20)
,  ( x ,  ~ ) ~ l 4 t ~ 'e  ( x a  u ) a f 4 t ~
du = 0,
and therefore for every A E L" (6.5.21)
h(u) e  ( X 1  U ) a f 4 t P '
eWau)a14tP
du = 0.
Since we may assume without loss of generality that v j # vk for j # k, 1 5 j, k In, we obtain by Prop. 3.2.1
and upon setting l/t = papp'
Thus taking lim supe ", an application of Lebesgue's dominated convergence theorem shows that
Since the expressions containing the dk satisfy a similar relation, inequality (6.5.17) is a consequence of (6.5.19) and (6.5.22) with M? 5 IIUII.
272
FOURIER TRANSFORMS
Sufficiency. Suppose (6.5.17) holds. In order to show that h is a multiplier of type (Lp, LP), 1 < p < 2, it suffices to establish (6.5.16) for any two functions $1, $, E F or even for any $1, $2 E B (Cf. (6.2.5)). To this end we proceed as in the proof of Theorem 6.2.2. Let C1,4, E B be arbitrary and 1 > 0 be such that the supports of $1 and 4, are both contained in (  I , I). If $1, $2 are the 21periodic extensions of dl,$, then (6.2.7)(6.2.9) hold for $,, $Ji,i = 1, 2. Furthermore, for
we have in view of (6.2.15)
On the other hand, if (6.5.17) holds for arbitrary finite sets UI, . . ., un E R, c1,. . ., Cn, dl, . . ., dnE @, it moreover holds for all countable sets in case the series occurring are absolutely convergent. Therefore, in virtue of (6.2.7)(6.2.9)
(lim sup 
I$2(u)lp' duy"'.
If p = In, n E N, then in view of the 21periodicity of $1

lirn pm2p
1" 0
1 n1 I $ ~ ( u ) l ~ d= u lim n  m 21n k n O
2
1
nI t 2(k t 1N
I$l(u)(pdu
nlt2kl
which now is easily seen to be a bound also for lirn supp+a,in case p > 0 is arbitrary. Since a similar relation holds for 4, and since $J,(U) = +i(u), i = 1, 2, for lit1 5 I, it therefore follows by (6.5.24) that
for every $1, 4, E B. Thus h is a multiplier of type (Lp, Lp). Moreover, 11 U 11 IM:, and since we have already seen that the opposite inequality is valid as well, (6.5.18) holds.
We observe that in case p = 1 Theorems 6.2.1 and 6.5.6 state that a bounded continuous function h is a multiplier of type (L1, L1) if and only if h satisfies (6.2.1). In this respect the result of Theorem 6.5.8 may be regarded as a certain extension of the representation theorem of S. Bochner to 1 < p < 2. Problems 1.
(i) Give the definition of a multiplier of type (BV,,, BV,,), (BV,,, Yl), (Yl, BV2,). (ii) Show that h E (BVzn, BV,,) if and only if h is the finite FourierStieltjes transform of somep E BVzn, i.e. (BV,,, BV,,) = [BV,,]". (Hint: For the necessity use (4.3.2), (4.3.3); see also ZYGMUND[71, p. 1761)
REPRESENTATION THEOREMS
2.
3.
4.
5.
273
(iii) Show that (L,",, L,") = (Lh, L!d = ( G n , BVzn) = (Can, L,",) = [BV,,]" and that for the corresponding multiplier operator U one has the representation (6.5.2) (a.e.). (Hint: See e.g. EDWARDS [III, p. 254 IT]) (i) Show that (L,",, La",) c (Lg,, Lg,) for every 1 < p < m. (ii) Show that 1' c (BVZn, Gn), l2 c (BVZ,, L&), and I p c (BVzn,,L[A), 1 < p < 2. (Hint: Use Prop. 4.3.2(ii), the theorem of RieszFischer, and Prop. 4.2.7) (iii) Let h E I," be even and quasiconvex. Show that h is a multiplier of type (Lg,, Lg,) and that the corresponding multiplier operator U is expressible as the convolution with an integrable function. (Hint: Theorem 6.3.7) Let h be a multiplier of type (Yl,Y,) and U the corresponding multiplier operator. (i) Show that U(f*fn) = Uf*t, = f * U t , for every f E Y1, t , E T,. (ii) If Y1 = Yz = L L show that U ( f l * f i ) = Ufi*fz = fl*Ufi for every fl, f a E Lg,. (i) Let 1 < p 5 m. Show that h E LB,) if and only if h E [Lp]^. In this case there exists g E Lg,, such that Uf = f *g for every f E L;,. (ii) Let 1 Ip 5 co. Show that h E (I$,, CZ,) if and only if h E [LE,]^. In this case there exists g E Lgj, such that Uf = f * g for every f E LE,. (iii) Show that h E (BV,,, C,,) if and only if h E [C,,]^. In this case there exists g E Czn such that Uf = g* df for every f E BV,,. (iv) Show that h E (BVzn, Li,) if and only if h E [L:,]^. In this case there exists g E L,: such that Uf = g* df for every f E BV,,. (Hint: ZYGMUND [71, pp. 1751771, EDWARDS [ I I I , pp. 2552571) (i) Give the definition of a multiplier of type (BV, BV). (ii) Show that h E (BV, BV) if and only if h E [BV]". In this case there exists p E BV such that Uf = f* dp for every f c BV. (Hint: For the necessity use (5.3.3), (5.3.4))
6.
7. 8. 9.
10.
(i) Show that (Ll, L') c (Lp, Lp) for every I < p 5 2. (ii) Let h E Co be even and quasiconvex. Show that h is a multiplier of type (LP, Lp), I s p I2, and that the corresponding multiplier operator U is expressible as the convolution with a functiong E L'. (Hint: Theorem 6.3.11) Let h be a multiplier of type (Lp, Lp), 1 Ip I2, and U the corresponding multiplier operator. For g E L1 show that U(f *g) = Uf*g for every f E Lp. Prove (6.5.20), (6.5.21). State and prove (if necessary at all) counterparts of Theorems 6.5.7, 6.5.8 for the limiting cases p = 1, p = 2. be the set of those f E Czn for which the Fourier series converges uniformly. Let if and only if If h is an even function on h, show that h E (CZn, /On
Ih(0) +
$ h(k) cos kuldu = O(1)
k=l
(n f co).
(Hint: Use the theorem of BanachSteinhaus and Prop. 1.3.1; see also KARAMATA [l], GOES[ 11) 1 I. Let 1 < p, q < 00. Show that (Lg,, L9,) = (L$, L%). In particular, (LL, LI,) = (Lg',, Lb,). (Hint: See ZYGMUND [71, p. 1781, EDWARDS [III, p. 2651)
6.6 Notes and Remarks Sec. 6.1. The material of Sec. 6.1.1 is standard and may be found in ZYGMUND [71, p. 136 ff], HEWITT[ l , p. 89 ff], HOFFMAN [I, p. 22 ff] (see also the references cited there).
In many of the books the conditions for representation are expressed in terms of the particular Cesaro or Abel means. In the latter case see also FICHTENHOLZ [I]. 18F.A.
214
FOURIER TRANSFORMS
The representation problem is connected with the moment problem for a finite interval. Given a sequence of numbers { a k } F = O , it is required to find necessary and sufficient conditions which determine f~ Lp(O, 1) such that u k = ji u k f ( u )du for all k E P. More generally, the powers { x k } b o are replaced by an arbitrary sequence of functions { t p k ( x ) } z = O c LP'(a,b). Correspondingly, the problem is to find conditions for the existence of p E BV[a, b] such that Cxk = J: 9)&) dp(u) for all k E P, { q k } b obeing a sequence in C[a, b]. In the particular case that 9)&) = (I/%) ,Ikx and [a, b] = [ P, P] one speaks of the trigonometric moment problem (see ACHIESER [l, p. 178 ff]). Our conditions are connected with those given by HAUSDORFF [l]. Thus CYk = ( 1 / 2 ~ p, ) efku dp(u), k E b, with p E W a n if and only if for x1 = xo ( h / ( n l)), xoE W arbitrary,
+
+
(n + a). For further details see KACZMARZSTEINHAUS [l, p. 31 ff], WIDDER[I, p. 100 ff], LORENTZ [I, p. 57 f, p. 77 f], ACHIESER [l], ACHIESERKREIN [l]. The term moment problem is first found in the work of T. STIELTJES of 18941895. He considers the case that (a, b) is the infinite interval (0, a),while H. HAMBURGER was concerned with the interval (a, a). The problem of Sec. 6.1.1 has also been considered in case the system {eikX}F= is replaced by a general orthogonal system of functions, thus determining necessary and sufficient conditions such that an orthogonal series is the Fourier series (with respect to the underlying orthogonal system) of a function belonging to a definite class. See KACZMARZSTEINHAUS [l, p. 214 ffl. Concerning Sec. 6.1.2, Theorem 6.1.5 is due to CRAMBR [l] (1939). His proof made use of Theorem 6.2.3; the present proof (see PFLUGER [l 1, BUTZERTREBELS [2, p. 15 f]) rests upon the Riesz representation theorem for bounded linear functionals on Co and certain Parseval formulae. It is indeed a straightforward modification of the method used originally by F. RIESZ[3] (1933) for the representation of positivedefinite functions. Doss [I] also treats the criterion of Cram& but he proves the sufficiency by reduction to a theorem of BERRY[I] of 1931. The Cramtr result is a standard topic in most books on probability theory, see e.g. LUKACS [l, p. 651 and LINNIK[l, p. 451; see also KATZNELSON [l, p. 1321. The proof of Theorem 6.1.6, the LPversion of the Cramer criterion, was suggested by work of OFFORD [l] devoted to the problem of defining a Fourier transform for Lp, p > 2. It is given in COOPER [3] and makes use of a selection principle, namely the weak* compactness theorem for Lp plus the Parseval formulae. Although it is related to the Riesz method, we regard it as a third method of proof of the representation theorems. It is, of course, also possible to prove Theorem 6.1.5 using the theorems of Helly and HellyBray. For functions of several variables the problem is considered in CRAMBR [l], BUTZERNESSEL [2], and Vol. 11. For related results concerning the representation as a Laplace transform see WIDDER [l, p. 276ff1, BERENSBUTZER [2], BERENSWESTPHAL [l]; for the representation as a Weierstrass transform see HIRSCHMANWIDDER [5, p. 170 ff 1 and NESSEL [ 11. Prop. 6.1.10 in the formulation of Problem 6.1.13 is given in KATZNELSON [ I , p. 1351 (who used Problem 6.2.4(i) in his proof). The present proof which rests upon the Cramer criterion is due to SUNOUCHI [8]. Sec. 6.2. The proof of the sufficiency part of Theorem 6.2.1 was originally given by BOCHNER [l] (1934) via Theorem 6.2.3. Our proof of the sufficiency is that of R.S. PHILLIPS [l ] and is valid iff is only bounded and measurable on R. It proceeds via the Riesz method; it has
also been generalized by PHILLIPS so as to give a representation theorem for Banach spacevalued functions. The results of S. BOCHNER in 19321934 actually initiated the many papers on representation theory and its widespread application in various branches of harmonic analysis. Thus SCHOENBERG [2] using the Riesz method gave an integral analog of Theorem 6.2.1 which may be compared with a paper of BERRY [l] on the representation as LPFouriertransforms
REPRESENTATION THEOREMS
275
(cf. Problems 6.2.4(ii), 6.2.6, 6.2.7). The results of BERRY,seldom cited, are some of the earliest on the subject. Theorem 6.2.1 plays an important r61e in abstract harmonic analysis; it may be restated in terms of the Bohr compactification of the real line: a bounded function f on R is the FourierStieltjes transform of a bounded measure on R if and only if it is continuous and the FourierStieltjes transform of a measure on the Bohr compactification [l] and RUDIN[3, p. 321. of R. See EBERLEIN The periodic version of Theorem 6.2.1, due to F. RIESZand S. BANACH,is given by Problem 6.2.1(i). Since exp ( ikx) E L& for all 1 I q < co, an analog for the representation as an Lg,Fourier transform is possible (see Problem 6.2.1(ii)). This result may also be regarded as a contribution to the trigonometric moment problem. A further fundamental representation theorem is the one connected with positivedefinite functions. For functions on the circle group this is to be found in Problem 6.2.2, on the line group in Problem 6.2.3. In the former case it is due to CARATHEODORY [I], HERGLOTZ [I], and TOEPLITZ [I] (all papers appearing in 1911), in the latter to BOCHNER[7, p. 92 ff, 325 ff]. It is commonly treated in books on the subject, e.g. KATZNELSON [I, p. 38, p. 1371, R. R. GOLDBERG [I, p. 59 ff], LUKACS[ l , p. 62 ff], LINNIK[l, p, 42 f]. BOCHNER was the first to recognize the key r d e of Problem 6.2.3 in harmonic analysis; he originally assumed f to be continuous. F. RIESZ[3] first showed that measurability was sufficient. For results concerning an integral analog of the positivedefinite condition see also COOPER[I]. Problem 6.2.3 has furthermore been carried over (WEIL [l]) to locally compact abelian groups, see RUDIN[3, p. 171, REITER[ l , p. 1061 and the literature cited there. Theorem 6.2.3 is connected with the fundamental continuity theorem of probability theory due to P. LBVY[ l ] (1922). A detailed discussion is given in BOCHNER[7, p. 85 ff, p. 321 ff; 6, p. 16 ff] and CRAMBR [2, p. 96 ff]. The following formulation is to be found in LUKACS [ 1, p. 541 : A sequence {pn(x)},"= of distribution functions converges weakly to a distribution function p ifand only if the sequence {pr(v)}of characteristic functions converges for every v to a function f ( v ) which is continuous at v = 0. The limiting function f is then the characteristic function (i.e. FourierStieltjes transform) of p. Here p is called a distribution function if p E BV increases monotonely with IIpllsv= 1. A sequence {pn(x)}of distribution functions is said to converge weakly to a distribution function p if limn+ pn(x) = p ( x ) for
all continuity points of p. Thus the continuity theorem indicates that the onetoone correspondence between distribution functions and characteristic functions is continuous. A further criterion for characteristic functions is the MATHIASKHINTCHINE theorem found in all books on probability. ROONEY [l, 21 is concerned with representation conditions of a somewhat different type. Sec. 6.3. The parallel remarks in Sec. 6.3.1 on convex and quasiconvex functions on Z and R and especially their applications in the subsections to follow once again exhibit the common structure of harmonic analysis on different groups. For functions defined on Z these notions are to be found in BARI[II, p. 3 f, 11, p. 2021, HEWITT[I, p. 61 ff], EDWARDS [lI, p. 110 ff]. Quasiconvexity (on P) together with f ( k ) 0 as k a, does not imply that f ( k ) is monotonely decreasing on P; but it follows that f is of bounded variation on P (see Problem 6.3.4). Consequently Problem 1.2.7 cannot be applied in the proof of Theorem 6.3.7 as would be the case iff were convex on P. On the other hand, iff E I," is even and monotonely decreasing on P, f is not necessarily the Fourier transform of an Li,function (see Problem 6.3.2). For convex functions on R see especially HARDYLITTLEWOODP~LYA [I, p. 70 ff], ZYGMUND [71, p. 21 f]. Frequently, f on [a, b] is said to be convex if for any two points pl, p2 of the curve off all the points of the arc plp2 lie either below the chordplp2 or on it. This (stronger) definition per se implies that f is continuous on (a, 6) (see Problem 6.3.5(iv)). For Lemma 6.3.4 and Prop. 6.3.6 see also BERENSGORLICH [l]. The material of Sec. 6.3.2 is standard and much of it due to A. N. KOLMOGOROV and W. H. YOUNG.See BARI[lII, p. 202 ff], EDWARDS [II, p. 113 ff], ZYGMUND 171, p. 182 ffl. For applications of Theorem 6.3.7 to factorization problems see EDWARDS 111, p. 117 ff I,
 
276
FOURIER TRANSFORMS
to saturation theory TABERSKI [2], SUNOUCHI [3]. Although the material will not be made use of in this text, it may motivate the results of Sec. 6.3.3. The latter are also wellknown but are treated less commonly. For Prop. 6.3.10 and Theorem 6.3.11 compare BEURLING [l]. The present proof of Theorem 6.3.11 is taken from BERENSGORLICH [I], a paper largely concerned with applications, in particular to saturation theory (see Sec. 12.2.2, Sec. 12.4.5). For a more general version (Problem 6.3.14) of the theorem see SZ.NAGY[3], TELJAKOVSKI~ [2]. The related result on convex (double monotonic) functions (see the final assertion of Theorem 6.3.11 and Problem 6.3.7) is to be found in ACHIESER [2, p. 154 ff], T~TCHMARSH [6, p. 1701. In probability theory it is known as Polya's condition; for extensions and applications to probability see LUKACS[I, p. 70 ff], LINNIK[I, p. 37 ff]. For further sufficient conditions concerning the representation as L1Fourier transform the reader is referred to Sec. 11.3.2. Lemma 6.3.13 of Sec. 6.3.4 is due to STEIN[II]. A proof was communicated to the authors in April 1967. The present proof of (6.3.19) is more elementary and due to H. JOHNEN (unpublished). For the periodic version in Problem 6.3.10 see also TAIBLESON [I]. Sec. 6.4. For background material on the general singular integral of Weierstrass as well
as the other examples of this section, the reader is referred to BERENSGORLICH [I] and the literature cited there. The wK of (6.4.5) are stable density functions in the sense of P. LBvY. Lemmata 6.4.36.4.5 are elementary versions of general results which were obtained by B. SZ.NAGYin his fundamental paper [3]. Concerning the periodic version of (6.4.14) of the Weierstrass integral, since the particular case K = I gives Abel's method of summation of the Fourier series off, the means (6.4.15) are often referred to as the AbelCartwright or (A, K ) means of the Fourier series(see CARTWRIGHT [I], HARDY[2, p. 71 ff, p. 380 ff]). The typical means (6.4.18) of the Fourier inversion integral are the particular case A = 1 of the Riesz means defined in Problem 6.4.8. For background material in the periodic case see HARDYRIESZ [I] and CHANDRASEKHARANMINAKSHISUNDARAM [I]. First detailed investigations on approximation by the typical means are to be found in ZYGMUND [3]. In the case K E N he showed, for example, that if f ( l ) E Lip (Czn; a) for r 5 K  1, 0 c a 5 1, then llRn,K(f;0)  .f(o)IICln = O(nlb)r except if K is odd and r = K  1, a = I , when the (multiplicative) factor log n must be added to the 0term. For fractional K see AUANCIC[2]. Further important contributions to the approximation by typical means may be found in SZ.NAGY[31; see also EFIMOV [3], TEUAKOVSKI~ [2], Guo ZHURUI[I], CHEN TIANPING[I]. Concerning the generalized integral of Picard defined in Problem 6.4.5 see BERENSGORLICH[I] and LINNIK11, p. 391. Bessel potentials (see Problem 6.4.6) play an important [ l , p. 297 ff]. For the rale in modern analysis. For a recent discussion see DONOGHUE original literature compare Problem 6.4.6 as well as GORLICH[2,3], WESTPHAL [2]. Periodic Bessel potentialsthese are generated via (3.1.28), (3.1.31) by G,do not seem to receive as much attention; nevertheless see TAIBLESON [I], WAINGER [ l ] as well as NESSELPAWELKE [l]. The results of this section on nonoptimal approximation (a < K) of the different processes may also be obtained, without any restrictions, as consequences of general theorems on interpolation spaces given in BUTZERBERENS [I], BUTZERSCHERER [ 11, and BERENS [3]. Sec. 6.5. For the classical results on multipliers (or factor sequences) for finite Fourier transform the reader is referred to ZYGMUND [71, p. 177 ff], for a detailed modern treatment to EDWARDS [111, pp. 2432991. For these reasons our account is brief. In Prop. 6.5.3 the functions h of the multiplier class (Can,Czn)are characterized, the norm of the corresponding multiplier operator U is evaluated and even Uf is represented as the convolution f * dp off E Czn with some p E BVz,, determining U. On the other hand, in Prop. 6.5.2 no representation of Uf as convolution is given. Yet it can be shown that there is a unique distribution (actually a pseudomeasure) A such that Uf = f* A for allfE L:,. For specific multiplier classes the real task is indeed to represent U as convolution with some distribution A from which A is easily recaptured as the Fourier transform of A, and to characterize
277
REPRESENTATION THEOREMS
A as closely as possible. However, in this section we have confined the discussion to some of those elementary situations where A belongs to one of our traditional function classes (compare also Problems 6.5.1, 6.5.4). For multipliers of type (Lgn, L&) there is as yet no complete and effective solution to the problem. Inclusion relations are known between classes (LE,, Ll,) for different values of p and q some of the proofs depending upon interpolation theorems such as the RieszThorin convexity theorem; see EDWARDS [III, p. 263 ff], HIRSCHMAN [I], and STEIN[2]. A related multiplier problem is the following: Which function h on Z has the property that, for each f E CZn,the series 2;=  A(k)f”(k) efkx has uniformly bounded partial sums? For an answer the reader is referred to EDWARDS [III, p. 257 ff]. A further problem is to characterize those functions h such that, for each f E Czn, h(k)f”(k)is the finite Fourier transform of some g €Lip (Czn;a). For this socalled [I] Lipschitz multiplier problem see EDWARDS [III, p. 2601 as well as ALJANCICTOMIC and the literature cited there. For a recent extended report on related problems see especially BOAS[I]. Concerning Sec. 6.5.2 the first treatment in bookform of multipliers (or factor functions) for Fourier transforms of functions in Lp, 1 Ip I2, seems to be that of HILLE[2, p. 361 ffl, see also HiLLEPHILLlPs [ I , p. 5661. Theorem 6.5.6 is to be found there as well as in RUDIN[3, p. 731. Theorems 6.5.7, 6.5.8 are due to DE LEEUW[3]. The extension of result of Theorem 6.5.8 to bounded measurable functions A, analogous to the PHILLIPS Theorem 6.2.2, is equally valid and given there. The paper by DE LEEUWis written in the setting of abstract harmonic analysis; he considers multipliers on subgroups and quotient groups. The assertions of Theorem 6.5.8 (in particular also (6.2.3)) are connected with the theory of almost periodic functions on the line. To each such function fthere corresponds a unique number M ( f ) , called the mean value off, such that Iimp. (1/2p) I) f(u) dn = M ( f ) . See WIENER [2, pp. 1851991, KATZNELSON [ I , pp. 1551691 (compare also Problem 5.3.6). Multipliers of type (Lp, Lg) have been the subject of many recent investigations. It may be mentioned here that most of these speak of translationinvariant operators. Thus, in his notes that t o each bounded linear fundamental and detailed paper [I], L. HORMANDER translationinvariant operator U from L P to L q there may be associated a unique tempered distribution A such that Uf = A *ffor all f E S (see also LOOMIS 13, p. 2531). The space of tempered distributions A such that IIA * f i l , 5 C 1 f for all f E S is denoted by L;. The set of Fourier transforms A^ of A E L; is denoted by Mi. The elements of M: are called studies begin on this basis. Obviously, this demultipliers of type (p, q). HORMANDER’S finition of multipliers of type (p, q) is not restricted to 1 Ip , q I 2 as was the case in our classical setting. For further literature on multipliers a very incomplete list is added. Those [I], SCHWARTZ [2], preceding the milestone paper by L. HORMANDER are MARCINKIEWICZ BEURLINCHELSON [I], MlKHLiN [I, 21, KRABBE[I]; for those following see FICATALAMANCA [l, 21, KRBE[l], CORDES [l], BRAINERDEDWARDS 111, C A L D E R ~[I], N GAUDRY 111 PEETRE[3], LITTMANMCCARTHYRIVI~RE [I, 21, RIEFFEL[I], EDWARDS121, FICATALAMANCAGAUDRY [l], LARSEN [I]. Compare also the literature cited in EDWARDS [I].
r
\Ip
7 Fourier Transform Methods and Second Order Partial Differential Equations
7.0 Introduction
Integral transform methods have proven of great utility in the solution of initial and boundary value problems in the theory of partial differential equations. The general method is to transform a given partial differential equation, involving an unknown function, into an equation involving the transform of this function. This transformed equation is then solved, and an appropriate inverse formula is applied to obtain the solution in terms of the original function space. More specifically, we have seen that the Fourier transform converts differentiation of order r into multiplication by (iu)’ (cf. Prop. 5.1.14), so transforming an ordinary differential equation into an algebraic equation (if the range of the independent variable is R). In the case of partial differential equations the method reduces by one the number of variables with respect to which differentiation occurs. Since such equations have several independent variables, which may also range over intervals of different type, the first question is concerned with respect to what variable one transforms the original equations and what transform can be used at all. Since the interval of integration of the transform must coincide with the range of a variable of the differential equation in question, one may generally apply: (i) the Fourier transform with respect to a variable x if it varies in the interval  00 < x < co, (ii) the finite Fourier transform (the Fourier transform in the form (11.4)) in the case that one or more of the independent variables, say x, ranges over a finite interval  L Ix IL, say, (iii) halfrange transforms (such as the cosine transform) for intervals of type 0 Ix IL, (iv) the Laplace transform with respect to t if it varies in t 2 0, this transform being particularly suitable for solving initial value problems. Of further importance are Mellin, Hankel, and Legendre transforms.
TRANSFORM METHODS AND PARTIAL DIFFERENTIAL EQUATIONS
279
In order to describe the method, let us consider a nonhomogeneous linear secondorder partial differential equation in two independent variables x , y
+E+ F u ( x , y ) = G(x, y ) aY with constant coefficients A , B, . . ., F and domain 00 < x < co,yo < y < y , , say (yo,y , may be 00, +a,respectively). In general, the unknown function u(x, y ) is not uniquely determined by the differential equation alone. Furthermore, 'onepoint' (or initial) conditions and 'twopoint' (or boundary) conditions must be prescribed. As the variable x ranges over R, the first step of the procedure is to apply the Fourier transform to both sides of equation (7.0.1) with respect to x. Setting [ ~ ( o , y)]^(u) = uh(u, y ) , [G(o, y)]"(u) = Gh(u,y ) , this yields
(7.0.2)
+ E
+
Fu"(u, y ) = G"(u, y ) , aY provided one assumes u(0, y ) , G(0,y ) E L1 for every yo < y < y , and imposes further assumptions such that for r = 1 , 2
Thus the secondorder partial differential equation converts into the equation (7.0.2), which is really a continuous system with one (ordinary differential) equation for each value of the parameter u E R. In order to apply (7.0.3), the coefficients A , B, . . .,F in equation (7.0.1) may be functions of y . They may even depend on x provided that rules corresponding to (7.0.3) can be established, converting differentiation with respect to x into more suitable operations. The second step is to solve this secondorder ordinary differential equation for u^(u, y ) , u being a parameter. According to classical theory, this equation has a unique solution provided that either two initial conditions or twopoint (boundary) conditions are prescribed. These must be deductible from corresponding conditions upon a solution of the original equation (7.0.1). Thus we further assume, for example, that (7.0.1) satisfies the onepoint conditions
which must be interpreted in such a way that they convert into the following onepoint conditions for (7.0.2) lim u"(u, y ) = &(u), uvo+
auc'(u, v) = (f(u). lim aY
YYOf
280
FOURIER TRANSFORMS
The third step is to reconstruct the solution u(x, y ) of the original equation (7.0.I) from the values of uA(u, y ) . For this purpose the inverse Fourier transform of u^(u, y ) is evaluated via a suitable inversion theorem (such as Prop. 5.1.10). The method as a whole may be indicated by the following diagram: Partial Differential Equation conditions
+ initial and boundary
   +
I
Solution U(x.3 Y )
f
Transform
Inverse Transform
.1
I
Ordinary Differential Equation boundary conditions
+ initial
and/or +
Solution
w,Y )
Instead of solving the equation directly in the original function space, we take the indirect way indicated by the arrows. The essential point of the method is that the original equation is converted into the transformed equation and that the latter should be more readily solvable. The advantage of the transform method is that it greatly facilitates the solution of such problems to a standard procedure. Finally, the method is not only concerned with the construction of the solutions in question but it delivers the uniqueness at the same time, The method to be presented not only provides a formal procedure but also a completely rigorous approach to the subject. By analogy with conic sections, in the theory of differential equations the general form of the equation (7.0.1) is said to be hyperbolic, parabolic, or elliptic according as the discriminant B2  4AC is positive, zero, or negative. One can show that the standard forms of the terms of secondorder of a hyperbolic, parabolic, or elliptic equation are given by a2u
a2u
a2u ,
aga  q’ a q
a2u + a2*Z u a p a,,
+
respectively (by affine transformation 6 = a x + Isy, 7 = y x 6y, a8  j3y # 0). For this purpose it suffices to study three standard forms of secondorder equations. This fits in with the preceding discussion since the method of solution depends not only upon the auxiliary conditions but also upon the nature of the coefficients and the domain of the independent variables. Sec. 7.1 is concerned with the finite Fourier transform method in the solution of three standard equations which are of parabolic, elliptic, and hyperbolic type. These are, in order, the equation of heat flow, Laplace’s equation, and the equation of motion. Further, the nonhomogeneous problem is considered in connection with the first equation, the Neumann problem with Laplace’s equation. Sec. 7.2 is devoted to the Fourier transform method as applied to the problem of heat flow on an infinite bar, to Dirichlet’s problem for the upper halfplane as well as to the equation of motion for an infinite string. Fourier transform methods are also applicable to certain difference and integral equations and to mixed problems that are of Fourier convolution type. Several such examples are given in the Problems.
TRANSFORM METHODS AND PARTIAL DIFFERENTIAL EQUATIONS
28 1
7.1 Finite Fourier Transform Method
7.1.1 Solution of Heat Conduction Problems
Given a homogeneous isotropic ring of unit radius whose diameter is small in comparison with its length, let f ( x ) ,  n Ix In, be any prescribed temperature distribution at time t = 0 with f( n) = f(n).Fourier'sproblem of the ring is to determine the temperature at any point of the ring at time t > 0 if there is no radiation at the surface. This temperature function u(x, t) satisfies the differential equation of heat conductiont (7.1.1)
(  n Ix In, t
> 0)
with the boundary conditions (7.1.2) and the initial condition (7.1.3)
u(x, 0) =
f(4
(  n Ix I n ) .
We interpret this problem in the following sense: 7.1.4) Given a function f E XZn,we call for a function u(x, t ) , defined and periodic in x for all x and t > 0, such that (i) au(x, t)/ax, aau(x, t)/ax2, au(x, t)/at existfor all x and t > 0, (ii) u(x, t) satisfies equation (7.1.1) for all x and t > 0, (iii) limt+,,+ I[u(o,t ) f(o)Ilx,, = 0, (iv) for each t > 0, u(x, t ) E Wga,,as a function of x, t)/at E Xznas a function o f x and (v) for each t > 0,
We remark that the boundary conditions (7.1.2) are contained in (7.1.4) in view of the periodicity of the functions involved. In particular, u(x, t) belongs to C,, as a function of x for each t > 0. To solve this problem we assume that there exists a function u(x, t) with these properties and apply the finite Fourier transform to each side of the equation (7.1.1). Setting [ ~ ( o ,t)]"(k) = ~ " ( kt),, we have for the righthand side by (7.1.4)(iv) and Prop. 4.1.8 (t > 0,k E Z),
whereas for the lefthand side (t > 0,k E Z), The physical constants of density, specific heat, and conductivity have been omitted since they do not affect the nature of the solution.
FOURIER TRANSFORMS
282
since by (7.1.4)(v) and Problem 4.1.1
I
u"(k, t
+
T) 7
 u"(k, t )
 [a,!;
I
')]^(k)
Hence au"(k, t)/at exists for each k E Z and t > 0 and is equal to the finite Fourier transform of &(x, t)/att. In particular, for every k E Z, u"(k, t ) is a continuous function of t on (0, a). Thus the finite Fourier transform of a solution u(x, t ) of (7.1.4) satisfies for each k E Z the ordinary differential equation (7.1.5)
having the classical solution uh(k, t ) = A(k) ekat with constant A(k) independent of t . The initial condition (7.1.4)(iii) yields IU^(k, t>  f"(k)l 5
Ib(0, t>  f(o)llxp,
=
o(1)
(t +O+);
therefore A(k) = f"(k) and (7.1.6)
uh(k, t ) = f"(k) ekat
(k E z, t > 0).
Hence, if our problem has a solution, then its finite Fourier transform is given by (7.1.6). In order to represent it in terms of the original functions we have to apply a suitable inversion theorem. According to Prop. 4,1S(i) we have for all x and t > 0
2 m
(7.1.7)
u(x, t ) =
eka'fn(k)etkX,
k=m
since the series converges absolutely and uniformly in x for each t > 0. Thus, if a solution of (7.1.4) exists, it is unique by Cor. 1.2.7. To prove that (7.1.7) actually is a solution of our problem, it remains to show that the function u(x, t ) given by (7.1.7) satisfies (7.1.4). This fact is left to Problem 7.1.1. Finally we may represent the solution of (7.1.4) in form of a singular integral. Indeed
t We note that instead of (7.1.4)(v) it would have been sufficient to suppose that the limit on the left side of the last inequality is zero, for we here only need the fact that the finite Fourier transform commutes with differentiation with respect to the timeparameter t . In other words, whereas (7.1.4)(v) assumes that the vectorvalued function u(x, r ) , defined on r > 0 with values in the Banach space Xan, is strongly differentiable for t > 0, it is sufficient to suppose that the difference quotient converges in the weak topology towards au(0, t)/at.
283
TRANSFORM METHODS AND PARTIAL DIFFERENTIAL EQUATIONS
the interchange of integration and summation being justified by the uniform convergence of the series. Thus in view of Problem 1.3.10 we have
Theorem 7.1.1. Fourier'sproblem (7.1.4)of the ring has a unique solution. It is given by the singular integral of Weierstrass, i.e.
03(x, t ) being the Jacobi thetafunction.
In Sec. 12.2.1 we shall study the dependence of the solution of (7.1.4) upon its initial value f and examine the rate at which it approximatesf for small values o f t . Now let us briefly discuss a problem with sources distributed along the ring. This situation is covered by the nonhomogeneous equation
(7.1.10) Suppose that, for each t > 0, F(x, t) belongs to X,, as a function of x and satisfies
(0 < t I l),
IIF(0, f)llx,, I M
(0
(7.1.11) (ii)
lim IIF(0, t Z0
+  F(0, t)llXz, = 0 T)
( t > 0).
Then we seek for a solution u(x, t) of (7.1.10)which satisfies (7.1.4)(i), (iii)(v). Let us assume that this problem has a solution u(x, 1). As in the homogeneous case we obtain that its finite Fourier transform u"(k, t) satisfies for each k E Z the ordinary differential equation
(7.1.12)
+ k2u"(k, t)
aUh(ky at
=
FA@, t)
(t
> 0).
To solve these equations we first of all note that in view of (7.1.11) and Problem 4.1.1 F"(k, t) is a continuous function o f t on (0, co)for each k E Z. In consideration of the initial condition the homogeneous part of (7.1.12) has the solution u;'(k, t) = f ^ ( k ) exp { k2t} (cf. (7.1.6)). Furthermore, by classical methods a particular solution of the nonhomogeneous equation (7.1.12)is given for each k E Z by
(7.1.13)
t)
=
lot
FA&, t 
T)
ekardT,
the integral existing since there is a constant M* for each t > 0 such that for k # 0
Hence, if the equation (7.1.10)has a solution u(x, t) in the above sense, then its finite
FOURIER TRANSFORMS
284
Fourier transform is given by $(k, t ) + &(k, t). In order to represent this solution in terms of the original functions, by Prop. 4.1S(i)
since the series converges absolutely and uniformly in x for each t > 0. Moreover, by Fubini's theorem
giving (cf. Problem 7.1.1)
Theorem 7.1.2. If the source distribution F(x, t ) satisfies (7.1.1 l), then equation (7.1.10) has a unique solution satisfying (7.1.4)(i), (iii)(v). It is given for all x and t > 0 by
7.1.2 Dirichlet's and Neumann's Problem for the Unit Disc
Given a thin plate of a homogeneous and isotropic material in form of a disc of unit radius, letf(x), B Ix I ?r, withf( n) = f ( n )be a constant temperature distribution on the boundary of the disc. What is the temperature distribution in the interior of the disc in the stationary, i.e., timeindependent case? The solution w ( ( , q ) must be independent of the time t and satisfy the twodimensional heat equation which now reduces to the elliptic equation
( P + q2
O),
where ca = ~ / p T, and p being the tension and density of the string, respectively. Let us suppose that both ends of the string are fixed. Then we have the boundary conditions (7.1.30) u(0, t ) = 0, u(n, t ) = 0 ( t > 0). Finally, to describe the motion completely, we need to know the initial position and velocity of the string, i.e. (0 5 x I n),
(7.1.31)
where f and g are given functions. In view of (7.1.30) we assume (7.1.32) f ( 0 ) = f ( n ) = 0, g(0) = g(n) = 0. As in the preceding cases this intuitive introduction of the problem is now completed by the following exact interpretation: (7.1.33) For given functions f, g E X[O, n] satisfying (7.1.32), jind a function u(x, t ) defined on 0 I x I n, t > 0 such that (i) aau(x, t)/axa, aau(x, t ) / W exist for all x E (0, n), t > 0, (Z) u(x, t ) satisjes equation (7.1.29) for all x E (0, n), t > 0, (iii)
lim u(x, t ) = 0,
X.O+
lim u(x, t ) = 0
xn
(t
> O),
it moreover being assumed that the limits lim,,,, au(x, t)/ax, limxrn au(x, t)/ax and limx,o aau(x, t)/ax' exist for each t > 0, (iv) for each t > 0, u(x, t ) and au(x, t)/at belong to X[O, n] as functions of x and
(v) for each t > 0, au(x, t)/ax is absolutely continuous on 0 c x c n as a function of x and aau(x, t)/axa belongs to L'(0, T), (vi) for each t > 0, aau(x, t)/ata E L1(O, n) as a function of x and
TRANSFORM METHODS AND PARTIAL DIFFERENTIAL EQUATIONS
289
Let u(x, t) be a solution of (7.1.33). Since the problem deals with the interval 0 Ix In, one is led to apply a finite halfrange transform, i.e., the finite Fouriersine or cosine transform. By Problem 7.1.8, the transform of a2u(x,t)/ax2would then not only include the boundary values of u(x, t ) but also those of au(x, t)/ax which we actually do not know. Therefore we proceed by extending the functions u(x, t ) , f ( x ) , g(x) to the interval [ ?T, 01, for then we are able to express the nonperiodic problem (7.1.33) as a periodic one. Obviously, the extension is determined by the local behaviour of u(x, t), f ( x ) , g(x) at x = 0. As immediate consequences of the boundary conditions (7.1.33)(iii) it follows that, for each t > 0, u(x, t ) , &(x, t)/ax, &(x, t)/at are continuous functions on 0 I x Ia. Here the derivatives at the end points are of course defined in the onesided sense. Moreover, a2u(x, t)/ax2 and a2u(x, ?)/at2exist and are continuous at x = 0. The essential point is that the continuity guarantees that the differential equation (7.1.29) holds at x = 0, too. Therefore by (7.1.33)(iii)
Since u(0, t ) = 0 and a2u(0, t)/ax2= 0 for each t > 0 (in general au(0, t)/ax # 0), only an odd extension is possible if we desire that au(x, t)/axbe absolutely continuous on [a, n]. Thus we set (7.1.34)
u(x,
t)
u(x,
t),
f(x)
f(X),
g(X)
g(X) (0 I x IT , t > 0).
Then (7.1.30) and (7.1.32) imply u(  n, t ) = u(a,t ) , f ( a) = f (a) and g( 7r) = g(a) for each t > 0. Since au( x, t)/ax = au(x, t)/ax, a2u(x, t)/ax2 =  a2u(x,t)/axz and au(  x, t)/at =  au(x, t)/at, we obtain that u(x, t), au(x, t)/ax and au(x, #)/at are continuous for each t > 0 on a I x I a, moreover au( a, t)/ax = au(a, t)/ax and &(a, ?)/at = &(a, t)/at = 0, and the differential equation (7.1.29) is satisfied for a l l x E (  a , a ) , t > 0. Now our original problem (7.1.33) is solved if we are able to determine a solution of the following problem: (7.1.35) For givenfunctionsf, g E Xan,find afunction u(x, t),defined and periodic in x for all x and t > 0, such that (i) au(x, t)/ax and au(x, #)/at exist for all x and t > 0, aau(x, ?)/axaand a2u(x, t ) / W exist for all x E ( a,a), t > 0, (ii) u(x, t ) satisfies equation (7.1.29) for all x E ( a, a), t > 0, (iii) for each t > 0, au(x, t)/at E X,, as a function of x and
(iv) for each t > 0, u(x, t) E W& as a function of x, I~F.A.
290
FOURIER TRANSFORMS
(v) for each t > 0, a2u(x, t)/at2E L:, and
To solve this problem we take the finite Fourier transform of equation (7.1.29). Then we obtain for k E Z, t > 0 ^(k) = (ik)au"(k, t )
as well as
Thus the finite Fourier transform u"(k, t) of any solution of (7.1.35) satisfies for each k E Z the ordinary differential equation ')
(7.1.37)
+ (ck)%"(k, t ) = 0
which has the general solution
(A(k)4 0 ) + B(k) B(O)t, eickt
uh(kyt ) =
+
,lckt
, k#O k = 0.
To determine A(k) and B(k) we note that limt+,, ,, u"(k, t ) = f " ( k ) which implies
Furthermore, since limt+,,,, [au(o,t)/at]^(k)= g"(k), it follows by (7.1.36) that lim
t+O+
au^(k, t )
=f(k) at
( k E Z).
Therefore, gn(k) =
 ickB(k), {ickA(k)B(O),
k #0 k = 0,
and the finite Fourier transform of any solution u(x, t) of (7.1.35) is given by
By Prop. 4.1.l(i) and Problem 4.1.5
u"(k, t ) =
["
0
+ ct) 2
+f(o
 c')]A(k)+
[+Icf + c
ct
g(0
s)dsIA(k)
29 1
TRANSFORM METHODS AND PARTIAL DIFFERENTIAL EQUATIONS
for all k E Z, t > 0. The uniqueness theorem of the finite Fourier transform then gives for each t > 0 the solution
Thus, if (7.1.35) has a solution u(x, t), then it is of the form (7.1.39). In order to show that the righthand side of (7.1.39) actually is a solution we have to pose certain conditions upon the initial valuesf and g, in contrast to the parabolic and elliptic situation of Sec. 7.1.1 and 7.1.2, respectively. There, the corresponding series (7.1.7) and (7.1.23) always define solutions since they are absolutely and uniformly convergent for any f E XZn.But here the Fourier series of (7.1.39) does not necessarily converge.
Theorem 7.1.5. Let f
E AC:, and g E AC,, be such that f " and g' exist for all x and belong to X,,. Then problem (7.1.35) has a unique solution. It is given for all x and t > 0 by (7.1.39). Moreover, we have
(7.1.40)
u(x, t ) = f A ( 0 )
+ g"(0)t + k2 [f " ( k ) cos ckt + ck1 g"(k) sin ckt #O
1
elkr,
the series being absolutely and unifbrmly convergent. The proof is left to Problem 7.1.3. As an application we obtain the following solution to problem (7.1.33).
Theorem 7.1.6. Let f, g E X[O, 7r] satisfy (7.1.32) and let f and g be extended to the interval  n Ix I0 as odd functions and then to the whole real axis as periodic functions. Suppose that for these extensions f E AC:, and g E AC,, such that f and g' exists for all x and belong to XZn. Then the vibrating string problem (7.1.33) has a unique solution u(x, t ) given by (7.1.39). The representation (7.1.39) holds for all x and t > 0. Moreover, 'I
(7.1.41)
u(x, t ) = 2
5 [LA&)cos ckt + ck1 g,^(k)sin cktI sin kx,
k= 1
the series being absolutely and uniformly convergent. The proof is left to Problem 7.1.3. f,^(k) and g,^(k) denote the kth Fourier sinecoefficients (cf. (1.2.10) and Problem 7.1.8) off and g, respectively. Finally, we shall briefly discuss the problem connected with a vibrating string with both ends at x = 0 and x = 7r free to slide along fixed parallel rails. Again the displacement u(x, t ) satisfies equation (7.1.29) with initial conditions (7.1.31). But the boundary conditions (7.1.30) are now replaced by
(7.1.42)
(1
> 0).
The exact formulation of the problem is very similar to (7.1.33). Indeed, conditions
292
FOURIER TRANSFORMS
(i), (ii), (iv), (v), and (vi) are unchanged but the initial conditionsf, g E X[O, n] need not satisfy (7.1.32), and (iii) must be replaced by
(7.1.43) it being moreover assumed that the limits lim,,,, u(x, t), limx,s u(x, t), lim,,,,, au(x, t)/at, limx+s au(x, t)/at, limx+o+ aau(x, t)/axay and lim,,,, aau(x, t ) / W existfor each t > 0.
As in the case of fixed ends it follows by (7.1.43)that, for each t > 0, u(x, t), au(x, t)/ax and au(x, t)/at are continuous functions on 0 I x I n and that the differential equation (7.1.29) is satisfied at x = 0. Therefore we may again extend the problem to the interval  n I x 5 0. Now this extension has to be even. Then, in particular, au(x, t)/ax is an odd function in correspondence with (7.1.42). Thus the present problem is again a special case of (7.1.39,namely the case of even initial conditions f and g.
Theorem 7.1.7. Let f, g E X[O, n] be extended to the interval  n Ix I 0 as even functions and then to the whole real axis as 2nperiodicfunctions. Suppose that for these extensions f E AC;, and g E ACan such that f” and g‘ exist for all x and belong to Xan. Then the sliding string problem has a unique solution u(x, t ) given by (7.1.39). The representation (7.1.39)holds for all x and t > 0. Moreover, 1 (7.1.44) u(x, t ) = f2 (0) g:(O)t 2 fp(k) cos ckt + g:(k) sin ckt cos kx, k=l ck the series being absolutely and uniformly convergent. The proof is left to Problem 7.1.3.f n ( k ) and g?(k) denote the kth Fourier cosinecoefficients (cf. (1.2.10) and Problem 7.1.8) off and g, respectively.
+
+
2[
1
Problems 1. (i) Show by direct evaluation that u(x, r) as given by (7.1.7) is a solution of (7.1.4). (ii) If the source distribution F ( x , f ) satisfies (7.1.11), show that u(x, 1) as given by (7.1.14)is a solution of (7.1.10)satisfying (7.1.4)(i), (iiiHv). 2. Show that u(x, r) as given by (7.1.23)is a solution of (7.1.17). 3. Complete the proof of Theorems 7.1.57.1.7.
In the following we shall give some further examples of differential, difference and integral equations (or of mixed type) which may be solved by integral transform methods. In any case, the reader is asked to supplement the formal statement by a rigorous interpretation. The references given contain at least the formal construction of solutions. 4. Let f~ X2n.Determine the solution of the firstorder equation (n 5 x I n , r E R)
with initial condition u(x, 0) = f(x) and boundary condition u(  n, t ) = u(n,t), (Hint : WEINBERGER [1, p. 2991) 5. By means of the finite Fourier transform evaluate the solution u(x) E Can of the differentialdifference equations (ii) u’(x + n)  u(x) = sin x. (i) u’(x) + u(x) + u(x + P) = sin3x ,
TRANSFORM METHODS AND PARTIAL DIFFERENTIAL EQUATIONS
293
6. Find the solution u E Xan of the integral equation cos x = 7. Determine the solution u E Xznof u(x)
 T1Jn"n/a
u(x
+ s) ds = [sin XI
8. The finite Fouriercosine [sine] transform off E L'(0, defined on P [N], whose value at k E P [k E N ] is
LYk) =
T
1" o
f ( u ) cos ku du
[f,^(k)
=
2
T
 r2 a
T) is
the function f,^(k) [L(k)],
1" o
f ( u ) sin ku du],
the notation being consistent with (1.2.10) as applied to even [odd] 2rperiodic functions. (i) Let f E L1(O, T ) . Show that if f,^(k) = 0 [AA(k)= 01 for all k E P [k E N], then f ( x ) = 0 for almost all x E (0, r). (Hint: HARDYROGOSINSKI [ l , p. 201) (ii) Let f E L'(0, r) be such that L ' E I'(P) [A" E ll(N)]. Show that for almost all x E (0,T) (iii) Let f,f' be absolutely continuous on [0, TI and f" E L'(0, r). Show that 2 [f"l;(k) = kZLA(k) ; {f'(O)  ( l)kf'(n)> ( k E PI,
[f"lW =
2
 kaf,^(k) + ;k { f (0) (
(kE N).
1Ikf ( 4 1
(Hint: CHURCHILL [2, p. 290 f f ] ) (iv) Let f, g E L'(0, r) andfi, gl[fi,gal be their 2rperiodic extensions which are odd [even] on (r,T). If * denotes the usual convolution of 2rperiodic functions (cf. (0.4.5)), show that  2[fi * gll:(k) = A"(k)g,^(k) ( k E PI, 2[f, * gzlMk) = ff'(k)g,h(k) ( k E PI, 2Va * gzl;(k) = f w g m (k E W, (Hint: CHURCHILL [2, p. 2981) 9. Apply the finite Fouriersine transform in order to solve the following problem: If a is a fixed complex number, determine the solution u(x, y) of (0 < x
01,
for which u(0, y) = 0, U(T, y ) = 1, u(x, 0) = 0, and IIu(0, y)IIxro,nlis bounded for y > 0.(Hint: CHURCHILL [2, p. 3011) 10. The following threedimensional problem can be solved by a repeated use of the finite Fouriersine transform: given a E C, find u(x, y, z ) such that
with boundary conditions u(x, r, z) = u, u(0, y , z) = u(v, y , z) = u(x, 0,z) = u(x, y , 0) = u(x, y , T) = 0. (Hint: TRANTER [I, p. 831) 11. Apply the finite Fouriercosine transform in order to determine the solution u(x, t ) of
(0 < x < T,t > 0)
294
FOURIER TRANSFORMS
with boundary conditions au(0, t)/ax = au(?r, t)/ax = 0 and initial condition [l, p. 841) u(x, 0) = f(x). Here f is a given function of X[O, n].(Hint: TRANTER 12. Apply the finite Fouriersine transform in order to solve the following problem: Let F E X[O, 7r]. Find u(x, t) such that (0 < x c n, r > 0)
with initial conditions u(x, 0) = au(x, O)/at = 0 and boundary conditions u(0, f ) = U(T, r) = aau(O, r ) / W = aau(?r,t)/ax2 = 0. (Hint: CHURCHILL [2, p. 3081) 13. Use the finite Fouriersine transform in order to determine the even solution u of the integral equation sinh(n  Is]) 1x1 (n2  x2). u(x  s) ds = {sgn x} {sgn SI 67r sinh n (Hint: Problem 7.1.8(iv))
7.2 Fourier Transform Method in L1 7.2.1
Diffusion on an Infinite Rod Our first application of the Fourier transform method will be to the classical problem of the flow of heat in an infinite rod with a given initial temperature distribution. Denoting the temperature at the point x E R a t time t > 0 by u(x, t ) and the initial temperature distribution by f (x), the problem is to obtain a solution of the normalized parabolic differential equation ( x E R, t > 0)
(7.2.1)
satisfying the initial condition u(x, 0) = f(4.
(7.2.2)
We shall solve this problem subject to the following conditions: (7.2.3)
Given a function f E L1, we call for a function u(x, t ) , defined for all x E R and t > 0, such that (i) au(x, t)/ax, aau(x, t)/axa, au(x, t ) p t exist for all x E R, t > 0, (ii) u(x, t ) satisfies equation (7.2.1) for all x E R, t > 0, (iii) for each r > 0, u(x, t ) belongs to L1 us a function of x and lim
tOt
114% t )  f (.)I1
1
= 0,
(iv) for each t > 0, u(x, t ) E Wfl as afunction of x, (v) for each t > 0, a+, t)/at E L1 as a function of x and
First of all, the fact that all functions involved belong to L1 implicitly implies that
295
TRANSFORM METHODS AND PARTIAL DIFFERENTIAL EQUATIONS
u(x, t) = 0 certain boundary conditions are already satisfied; for instance, limlxlLm for each t > 0. Indeed, (i) and (iv) give
Since aup, t)/ax E L1 for each t > 0, limIhl.+,u(x + h, t) exists as a finite number which must be zero because u(0, t) E L'. Similarly one shows lim,,,,, au(x, t)/ax = 0. Let us suppose that there exists a solution u(x, t ) of (7.2.3). Denoting, for each t > 0, the Fourier transform of u(x, t) by u^(u, t), we may apply the Fourier transform to the differential equation (7.2.1). Then by (7.2.3)(iv) and Prop. 5.1.14 (u E R, t
> O),
and by (7.2.3)(v) and (5.1.2) (u E R, t > 0).
In particular, u"(u, t ) is a continuous function of t on (0, co) for each u E R. Thus the transform of equation (7.2.1) passes into (7.2.4)
(0 E
R, t > O),
which is really a continuous system with one equation for each value of the parameter u E R and is to be compared with the discrete system of (7.1.5). The latter (ordinary) differential equation may be solved by classical methods giving for each u E R the solution u^(u, t ) = A(u) with constant A(u) independent of t. To determine A(u) we have by (7.2.3)(iii) and (5.1.2) that lim,,,,, u"(u, t) = f"(u) uniformly with respect to u E R. Hence, if our problem (7.2.3) has a solution, then its Fourier transform is given by (7.2.5)
u"(u, t ) = f A ( u ) euat
(u E R, t > 0).
In order to represent it in terms of the original functions, sincef"(u) exp {vat} for each t > 0, we may use Prop. 5.1.10 to deduce (7.2.6)
E L'
1
1 " u(x, t ) = dK ,e'uafA(u)elxudu
for all x and t > 0. Thus if a solution of (7.2.3) exists, it is unique by Prop. 5.1.1 1. The fact that u(x, t) as given by (7.2.6) is actually a solution of (7.2.3) is left to Problem 7.2.1. Using the Parseval formula (5.1.5) and Problem 5.1.2 we may rewrite this solution as (7.2.7)
1
1
u(x, t ) = dG
"
f ( x  s) esa'4tds.
FOURIER TRANSFORMS
296 Therefore
Theorem 7.2.1. Problem (7.2.3) on theflow of heat in an infiniterod has a uniquesolution. It is given by the singular integral (3.1.32) of GaussWeierstrass associated with the initial value f. Let us also examine the related problem with heat sources F(x, t ) distributed along the rod. This phenomenon is described by the nonhomogeneous equation
( x E R, t > 0).
(7.2.8)
Suppose that, for each t > 0, F(x, t ) belongs to L' as a function of x and satisfies (0 < t I l),
IF(0Y t>lll 5
6) (7.2.9) (ii)
lim IIF(0, t
10
+
T)
 F(0, t)II1 = 0
( t > 0).
Then we seek for a function u(x, t ) which solves the problem (7.2.3), equation (7.2.1) being replaced by (7.2.8). As in the homogeneous case the assumption that there exists a solution u(x, t) yields that its Fourier transform ~ " ( u t, ) satisfies for each u E R the ordinary differential equation (7.2.10)
(t
> 0).
In view of the initial condition the homogeneous part of (7.2.10) has the solution u:(u, t ) = fn(u) exp { u2t}. Furthermore, as F"(u, t ) is a continuous function of t on (0,co) uniformly with respect to u E R in view of (7.2.9), a particular solution of the nonhomogeneous equation (7.2.10) is given for each u E R by
(7.2.1 1)
u;(u, t ) =
Iot
P(u, t
T)
e"" dr.
Moreover, by (7.2.9) there exists for each t > 0 a constant M * such that for
tr
#0
Therefore uKu, t ) belongs for each t > 0 to L1 as a function of u. Now Prop. 5.1.10 and Fubini's theorem yield
297
TRANSFORM METHODS AND PARTIAL DIFFERENTIAL EQUATIONS
Thus, introducing the Heaviside function 6(x) of (5.3.3), we may state Theorem 7.2.2. If the source distribution F(x, t ) is defined for all x E R, t > 0 and satisfies (7.2.9), then equation (7.2.8) has a unique solution satisfying (7.2.3)(i), (iiiHv). It is givenfor all x and t > 0 by
(7.2.12)
U(X,
t ) = W ( f ;X ; t )
F(x  s,
t
1 esa'4rS ( 7 )  7)8(t  7 ) {z
7.2.2 Dirichlet's Problem for the HalfPlane Suppose there is a steady state temperature distribution in an infinitely large plate, and the temperature distribution along one edge is known. Do these conditions determine the temperature in the plate? The mathematical formulation of this question is Dirichlet's problem for the upper halfplane. We raise it in the following way:
(7.2.13) Given afunction f E L1, we call for a function u(x, y), defined for all x y > 0, such that (i) aau(x,y)/8xayaau(x,y)/aya exist for all x E R, y > 0, (ii) u(x, y) satisfies for all x E R, y > 0 the equation
E
R and
(7.2.14) (iii) for each y > 0, u(x, y ) E L1 as a function of x, IIu(o, y)ll 5 M uniformly for y > 0 and (7.2.15)
IIu(b7Y )
u+o+
 f(o>IIi= 0,
(iv) for each y > 0, u(x, y ) E WZ,i as a function of x, (v) for each y > 0, au(x, y)/ay and aau(x, y)/aya belong to L1 as functions of x and
By arguments similar to those used in Sec. 7.1.2 and 7.2.1 we obtain for the Fourier transform of any solution u(x, t ) of (7.2.13) (7.2.16) which has as its solution Uh(v7y)
=
A(v) euu + B(v) e"u, v # 0 v = 0. A(0) + B(O)y,
{
298
FOURIER TRANSFORMS
To determine the constants A(u), B(u) we note that Iu"(u, y)l s IIu(0, y)ill 5 M for all u E R, y > 0, and hence by letting y + 00 we obtain that A(u) = 0 for u > 0 and B(u) = 0 for u 5 0. On the other hand, the relation lim,,o+ u"(u, y) = f"(u) implies A(u) + B(u) = f"(u) for all u E R. Summarizing, we have that the Fourier transform of any solution u(x, y) of (7.2.13) is given by ~ " ( u y, ) = f "(u) e1"'v
(7.2.17)
(u E R, y > 0).
Since f ^(u) exp {  IuI y} E L1 as a function of u for each y > 0, we have by Prop. 5.1.10 for every x E R and y > 0 (7.2.18)
or, in view of (5.1.19), (7.2.19)
which is the singular integral of CauchyPoisson as defined by (3.1.38). Thus, any solution of (7.2.13) is given as the CauchyPoisson integral of the boundary value$ Since, by virtue of Problem 7.2.2, the function u(x, y) of (7.2.19) actually is a solution of (7.2.13), we have shown (cf. Sec. 3.1.4)
Theorem 7.2.3. The Dirichlet problem (7.2.13) for the upper halfplane has a unique solution. It is given by the singular integral of CauchyPoisson of the boundary value f. In Sec. 12.4.3 we shall discuss the behaviour of the solution u(x,y) of Dirichlet's problem (7.2.13) on the boundary y = 0. In particular, from assumptions upon the degree of approximation of the boundary valuef conclusions upon the boundary value itself will be drawn.
7.2.3 Motion of an Infinite String The final application of the Fourier transform method is concerned with the problem of the motion of an infinite string with a given initial displacement f and velocity g, u(x, t ) being the displacement at distance x along the string at time t. The precise formulation of the problem is given by: (7.2.20) For given functions f, g E L', find a function u(x, t), defined for all x t > 0, such that (i) aau(x, t)/axa,aua(x, t)/ata, exist for all x E R, t > 0, (ii) u(x, t ) satisfies for all x E R, t > 0 the equation (c > 0) (7.2.2 1)
E
R and
aau(x, t )  ca,aau(x, t ) at2
ax2
(iii) for each t > 0, u(x, t ) and au(x, t)/at belong to L' as functions of x and
299
TRANSFORM METHODS AND PARTIAL DIFFERENTIAL EQUATIONS
(iv) for each t > 0, u(x, t ) E W ~ as I a function of x, (v) for each t > 0, a2u(x, t)/at2 E L1 as afunction of x and
lim T+O
We proceed as in the foregoing sections and assume that there exists a solution u(x, t ) of (7.2.20).Then the Fourier transform uh(u, t) of this solution satisfies for each u E R the ordinary differential equation (7.2.22) which has the general solution
A(v), B(u) being arbitrary complex constants independent of t. To determine these constants we note that by (7.2.20), (iii) and (v), aU*(U, t ) lim  g"(4 lirn u"(u, t ) = f"(u), t+o+
t+O+
at
uniformly with respect to v E R. This implies as in Sec. 7.1.3 that the Fourier transform of any solution u(x, t ) of (7.2.20)is given by (7.2.23)
u^(u, t ) =
1
f"(u) cos cut
1 +g"(u) sin cut, cv
f T )+ g w t ,
v # 0
(t > 0).
v=o
To reconstruct the full solution we may, in view of (5.1.29), rewrite (7.2.23) as
which holds for all u E R, t > 0. Now the uniqueness theorem of the Fourier transform in L' yields that for each t > 0 1
u(x, t ) = 2 I f ( x
1
+ ct) + f ( x  ct)] + 2c
LC* x+ct
(7.2.24) g(s) ds for almost all x E R. In order to show that the righthand side of (7.2.24) actually is a solution, just as in Sec. 7.1.3, we have to pose certain conditions upon the initial values f and g.
Theorem 7.2.4. Let f E W$ and g E WI: such that f "(x)and g'(x) exist for every x E R. Then problem (7.2.20) has a unique solution: it is given by (7.2.24) which holds for all X E R , t > 0. The proof is left to Problem 7.2.3.
300
FOURIER TRANSFORMS
Problems 1. (i) Show by direct evaluation that u(x, t) as given by (7.2.6) is a solution of (7.2.3). (ii) If the source distribution F ( x , 1) satisfies (7.2.9), show that u(x, t) as given by (7.2.12) is a solution of (7.2.8) satisfying (7.2.3)(i), (iiiHv). 2. Show that u(x, y) as given by (7.2.19) is a solution of (7.2.13). 3. Complete the proof of Theorem 7.2.4.
Concerning the following Problems the reader is again asked to supplement the formal statement by a rigorous interpretation. 4. Let f E L1. Find the solution u(0, t) E L1 of the differencedifferential equation u(x
W x , t) + 1, t )  2u(x, t ) + U(X  1, r) = at
which satisfies u(x, 0) = f(x). (Hint: WEINBERGER [l, pp. 326, 4351) 5. Find the solution u E L' of the ordinary differential equation xu"(x)
+ u'(x)  xu(x) = 0
(x E
W).
(x E
W).
(x E
W).
[l, pp. 326, 340, 4351) (Hint: WEINBERGER 6. Find the solution u E La of the integral equation u(x) = e1"'
+ 38
(Hint: TITCHMARSH [6, p. 303 ff]) 7. Determine the solution u E La of the integral equation eX'"
=
m
1
+ ( x  $)a
u(s) ds
(Hint: TITCHMARSH [6, p. 3141) 8. Determine the solution u(0, y ) E L1 of the partial differential equation ( x E R, 0 < y < l),
satisfying u(x, 1) = exp { xa}, au(x, O)/ay = 0. (Hint: WEINBERGER [l, pp. 320,4341) 9. Determine the solution u(0, t ) E L1 of the fourthorder equation a4u(x, t ) aau(x t ) ( x E R, t > O), += ata ax4 which satisfies u(x, 0) = exp {  xa}, au(x, O)/at = 0. (Hint: SNEDDON [3, p. 1291)
7.3 Notes and Remarks For basic results on partial differential equations of secondorder, in particular on the classification of the three types, see e.g. COURANTHILBERT [lII], HELLWIG [I], PETROVSKY [l, p. 451, WEINBERGER [ I , p. 41 ff]. For a useful account of transform methods, particularly the Laplace transform, see DOETSCH [3, p. 339 ff], [4III, p. 13 ffl. Sec. 7.1. The finite Fourier transform method for solving partial differential equations was [2] in 1935. But the method does not seem to have found its first considered by DOETSCH way into the many textbooks on the subject, exceptions being CHURCHILL [2], SNEDDON [l, 21 (see also WEINBERGER [l, p. 1261 and SEELEY [l]). Our approach differs from that of the above authors in that convergence to the initial and boundary values is not considered in the pointwise sense but in the norm of the space in question. This allows a
TRANSFORM METHODS AND PARTIAL DIFFERENTIAL EQUATIONS
30 1
rigorous treatment (including uniqueness), inspired by the study of BOCHNERCHANDRASEKHARAN [l, p. 40 ff] on Fourier transforms. Our definition of a solution of a given differential equation is really raised as an abstract Cauchyproblem. This calls for a solution of the following (see BUTZERBERENS [I, p. 3 ff] and the literature cited there): Given a Banach space X and a linear operator CI whose domain D(U) and range R(U) belong to X and given an element f E X, to find a (vectorvalued) function w(t) = w ( t ; f ) on ( 0 , ~into ) X such that (i) w(t) is continuously differentiable o n (0, co) in the norm, (ii) w(t) E D ( U ) and w’(t) = CIw(t) for each t > 0, (iii) limt40+ IIw(t)  f l l x = 0.
For instance, for the Fourier problem of the ring in CZn,we have w ( f ) = u(0, t ) and [Uw(t)](x)= P u ( x , t)/ax2 with D ( U ) = We,,. Thus condition (i) of the above problem corresponds to (v) of (7.1.4), (ii) to (iv) and (ii), and (iii) to (5). However, in the light of an abstract Cauchy problem, Fourier’s problem in Lg, already shows that it would be more appropriate also to interpret differentiation with respect to the spacevariable x in the normsense. Here we prefer the mixed formulation (7.1.4) which assumes in (i) and (ii) that the differential equation is satisfied in the classical pointwise sense. Similar remarks apply to the other examples treated, e.g. problem (7.1.17) corresponds to an abstract Cauchy problem of second order. Finite Hankel and Legendre transforms have been used in the solution of boundary value problems involving cylinders and spheres of finite radius by SNEDDON [ 1,2] and TRANTER [l], respectively. Legendre transforms have also been applied to problems of potential theory by CHURCHILL [I]. For finite Laplace transform methods see e.g. RNNEY [I], DOETSCH [4III, p. 225 ff]. One disadvantage of transform methods as a whole is that they d o not deliver singular or ‘explosive’ solutions that may be important from a physical point of view (see DOETSCH14111, pp. 3138]). The classical method for solving problems of this section is D. Bernoulli’s method of separation of variables which is treated in every textbook on the subject, e.g. SAGAN[l], SEELEY [l], H. F. DAVIS [l]. It leads to an eigenvalueproblem for which we tacitly assume that the solution is known, since we did not actually discuss the question how to find the suitable transform for a given problem (cf. CHURCHILL [2, 31). [I, p. 21 1 ff], For Fourier’s problem of the ring see the classical accounts in CARSLAW DOETSCH[2], MINAKSHISUNDARAM [ 11, HILLE[2, p. 4021. For nonhomogeneous problems compare e.g. WEINBERGER [ 11, TYCHONOVSAMARSKII [l]. For the (interior) Dirichlet and Neumann problem see e.g. DUFFNAYLOR [l, p. 133 ff], SNEDDON [3], SEELEY[l]. In an analogous fashion the third boundary value problem of potential theory as well as the exterior problems may be considered, The treatment of Sec. 7.1.3 was given by DOETSCH [2] for the differential equation of heat conduction describing the heat transfer in a bar of finite length. The task was, roughly speaking, to describe a typical situation where it is possible to unroll a ‘finite’ problem on the circle group, thus to deal with it as a periodic one. One advantage here is that the vibrating string problem as well as the sliding string problem are particular cases of the periodic problem (7.1.35) (for L%,space see also EDWARDS [II, p. 1361). Normally, the conditions are posed such that one has to apply the finite Fouriersine or cosine transform separately. A further method would be to apply the Laplace transform with respect to the timevariable t (cf. DOET~CH [41). Sec. 7.2. The classical standard reference is TITCHMARSH [6]. The material of the first two
subsections may be found in BOCHNERCHANDRASEKHARAN [l, pp. 4&47], SEELEY[ l , pp. 6596] (see also IVANOV [ 11). The former authors were probably the first to give a really rigorous proof (at least in book form) of these results, using the Fourier transform method. It is to be noted that we have here restricted discussion to the space L1, since a treatment in Lpspace (1 < p 5 2) would involve a study of vectorvalued differential equations (cf.
302
FOURIER TRANSFORMS
HILLEPHILLIPS [I, p. 67 ff]). In this respect the authors cannot follow the reasoning given in BOCHNERCHANDRASEKHARAN [l, p. 1351 in the Lacase. Onedimensional transform methods may also be applied to partial differential equations in three and more independent variables. The procedure reduces an equation in n independent variables x l , xz, . . ., x,, to one in n  1 independent variables xZ, . ., x,, and a parameter u. By a successive use of integral transforms (perhaps of a different type) the given partial differential equation is eventually reduced to an ordinary differential or even an algebraic equation, which may be solved easily. The fact that one might solve linear partial differential equations more effectively by, for example, applying the Fourier transform method depends upon Prop. 5.1.14 which states that the Fourier transform converts differentiation (rtimes) into multiplication with (iuy. On the other hand, for the solution of integral equations of Fourier convolution type (cf. Problems 7.2.6, 7.2.7) Theorem 5.1.3 is essential; it asserts that the Fourier transform converts convolution into pointwise multiplication. But when considered from a distributional point of view, convolution turns out to be the more general process. Various types of differential equations, difference equations, and integral equations are all special cases of convolution equations. For example, if 6 is the ‘delta function’ (defined through ( 6 , d ) = d(O), d E D) and f any distribution (of D’), then f = f * 6, Lf = f * L6, L being any linear differential operator. Thus shifting and differentiation may be written as convolutions, emphasizing the importance of convolution theorems. These aspects are now standard topics of modern treatments on partial differential equations. Fourier transform machinery, in particular its extension to generalized functions, plays an important r6le in these theoretical investigations and not only in the treatment of particular examples as given in this chapter. As a (partial) list of references we can cite SCHWARTZ [l, 31, GELPANDSHILOV [l], LIONS[l], H~RMANDER [2], FRIEDMAN [l, 21, BREMERMANN [l], ZEMANIAN [I, 21, TRBVES[I], SHILOV[l], STAKGOLD [l]. (See also Vol. I1 of this treatise.)
.
Part Ill Hilbert Transforms
It was shown in Sec. 1.2, 1.4, 4.1 that the Fourier series off tof, thus
~,(f; x) =
2 e,(k)f^(k)
E
X,, is 0summable
elk%
k=rn
tends in X,,norm or pointwise to f(x) for p + po, in case {0,(k)} is a convergencefactor satisfying suitable conditions. On the other hand, an application of the &factor to the conjugate Fourier series S  [f’] off gave the expression m
U;(f; x) =
2 k=
0,(k){  i sgn k}f*(k) elkx
m
=
In
1 f ( x  u) [2 2 2n , W
1
do@) sin ku du,
k = l
called the conjugate of Vo(f;x). A particular instance of such an expression is the conjugate of the AbelPoisson integral of (1.2.52). This suggests strongly an examination of the limit of V;(f; x) for p +. po. Indeed, this limit is called the conjugate function off and denoted by f . For 11 Vo(f;0)  f(0)/IXan= 0 to be valid for every f E X,, it was necessary that lim,,,, Bo(k) = 1, k E Z. Therefore, proceeding formally, we have by (1.2.48) lim 2 0PO
2 OD
eo(k)sinku
=
2
k=l
2 sinku = lim W
nm
k=l
and thus lim V;(f; x) = 2n
000
s’,
U
f(x  u ) cot 2 du
+ l)(u/2) [cot 2u  cos(2n sin (42)
304
HILBERT TRANSFORMS
In view of the singularity at u = 0, in order to evaluate the latter limit, one is led to study the Cauchy principal value. Then by the RiemannLebesgue lemma
iff is sufficiently smooth. Thus we formally come to
1 " u f ( x  u) cot 2 du, f "(x)= T n
which is to be understood in the Cauchy principal value sense. As a matter of fact, f " exists finitely for almost all x for any f E X,, (Theorem 9.1.1). It will be shown that the mapping f + f  is a bounded linear transformation of L,2, into itself with 11f " Il a I 11f 112. However, there are functions in ,:L for which f is not integrable, and thus the mapping is not bounded as an operator from L:, into itself. Moreover, there exist (absolutely) continuousfsuch that f " $ L";. On the other hand, the theorem of Marcel Riesz (1927)asserts that for 1 p < 00 there exists a constant Mp 2 1 such that f " E Lg, and 1l . f " IIp I M,ll f [ I p for each f E Lg, (Theorem 9.1.3). Chapter 9 is devoted to a study of the conjugate function, or Hilbert transform as it is often called, on the circle group; Chapter 8 is concerned with the parallel theory of Hilbert transforms on the line group. At this stage we must emphasize that although we speak of a Hilbert transform, it will be useful for us not to regard the entity as a transform but as a function. Indeed, if it will not be possible to characterize certain function classes in terms off, it will often be so in terms of the conjugate function f I. Many authors work with the Hilbert transform on the line as if it were an operational transform, obtain an inversibn formula, etc. ; but this is not our point of view.
=
Hilbert Transforms on the Real Line
8.0 Introduction
Recalling some results of Sec. 3.1 we have noted that for any realfE the integral
LP,
1 Ip < (z = x
00,
+ iy)
+
defines a function H(z) = P ( f ; x ; y) iQ(f;x; y ) which is holomorphic in the upperhalf plane R2*+.Thus it follows that the real part
[ n
m
P ( f ; x ; y) =
m
f(x
 u) du u2 + ya
( x E R, y > 0)
and imaginary part
of H ( z ) are conjugate harmonic functions (in the usual sense of complex function theory). We have seen that P ( f ; x ; y ) tends to the limit f as y +O+ ; the limit of Q ( f ; x; y ) for y + 0 + yields a function which is called the conjugate function off, denoted by f ". If we let y f Of we obtain, formally, (8.0.1)
Im
f"(x) = 1 n
m
f(x
u) U
This convolution integral, however, is not defined even when f is an extremely wellbehaved function. In fact, i f f € LP, 1 Ip < 00, we have
and the second integral exists everywhere and represents a bounded function (by 2eF.A.
HlLBERT TRANSFORMS
306
Hiilder's inequality). The main difficulty lies with the first integral, and it clearly need not exist as a Lebesgue integral at any point. For example, iffis constant and nonzero in a neighbourhood of a point xo, then this integral fails to exist in this neighbourhood of xo. Thus in order to give (8.0.1) a meaning one has to interpret the integral in a suitable way. Definition 8.0.1. The Hilbert transform of a function f E Lp, 1 I p < co,is defined as the Cauchy principal value of the integral (8.0.1) whenever it exists, i.e. f " ( x ) = ( H f ) ( x )is dejned as (8.0.2) at every point x for which this limit exists. Hereby,
For f E LP and any 6 > O,f;(x) is welldefined everywhere as a bounded function. The limit in (8.0.2) will not only be interpreted in the pointwise sense, but also in the norm. In Sec. 8.1 we show that the HilbertStieltjes transform of p E BV exists a.e., treat the existence problem for f E LP, 1 s p < co, and derive fundamental properties of f ",including the Marcel Riesz inequality, the selection partly depending upon their use in our later investigations in approximation theory. It will be important to evaluate the Fourier transform off". Noting that
=
{isgn u}
Som
dy
=
{  i sgn u}d.r/Z,
we have formally (8.0.4) It will be shown that this formula is valid for almost all u if f E Lp, 1 < p I 2 (Theorem 8.1.7, Prop. 8.3.1). If p = 1, it is valid for all u provided f as well a s f " belong to L1 (Prop. 8.3.1). These facts lead the way to characterizations of the classes W[Lp; IuI'], V[Lp; IuI'], 1 5 p I 2, r E N.Indeed (Theorem 8.3.6), f
E W[Lp;
IvI ]
o
f * E AC,,,
and ( f ")'
E
Lp.
Sec. 8.2 is devoted to Hilbert formulae (preparatory to Prop. 8.3.1) and to a detailed study of convergence (pointwise and norm sense) of conjugates of general singular integrals. Furthermore, theorems of Privalovtype are considered. Sec. 8.3 also deals with summation of allied integrals as well as with a theorem of Hille and Tamarkin (Theorem 8.3.8) on the normconvergence of the Fourier inversion integral.
HILBERT TRANSFORMS O N THE REAL LINE
307
8.1 Existence of the Transform
8.1.1 Existence Almost Everywhere We are first concerned with the existence almost everywhere of the Hilbert transform off E Lp, 1 Ip < oc). In fact, we establish the existence of the HilbertStieltjes transform.
Definition 8.1.1. The HilbertStieltjes transform p* of a function p E BV is dejned by the Cauchy principal value
at each x E R for which the limit exists. To show that the HilbertStieltjes transform exists almost everywhere we need the following lemmata. Lemma 8.1.2. Let dk > 0, a,
E
R, 1 5 k In, and
(8.1.2)
Then n
n
(8.1.3)
and this set consists precisely of 2n intervals. Proof. Since g ( a k  ) = co, g ( a k + ) = +co and g'(x) < 0 for all x # f&, 1 s k In, 1 I k i n  1, therearepreciselynpointsmksuchthatg(mk)= Mwithak c mk < and a, < m,,. The set where g ( x ) > M therefore consists of the intervals (ak, mk) and has total length
But the numbers mk are the roots of the equation
z= "'
k1
and hence by a multiplication with
dk
M
x  ak (x

ak)
we have
or If we compare the coefficients of the last equation with those of
nC=l(x  mk) = 0, then
If we substitute into (8.1.4) and combine this with the corresponding result for the set where g(x) <  M , (8.1.3) follows.
308
HILBERT TRANSFORMS
Lemma 8.1.3. Let p E BV be realvalued.I f ( & intervals such that
 6 k , x& + 6k)y 1
I
k In, are disjoint
(8.1.5)
then
2Ll 6,
I
1/%(16/M)IIpIIev.
Proof. First we observe that the integrals (8.1.5) are welldefined and bounded by GL'llpllsv, 1 5 k I n. Hence they may be approximated arbitrarily closely by finite RiemannStieltjes sums, if the norm of the subdivision is sufficiently small. Thus let uj, 1 Ij 5 m, be a finite subdivision including the points x k  8 k , x k , x k 8 k , 1 Ik In, such that, according to (8.1.5), for each k, 1 I k 5 n,
+
(8.1.6)
holds for y =
xk,
where the set z k of indices omitted is defined by
Now suppose that p is monotonely increasing. Then the left member of (8.1.6) is a decreasing function of y for x k  6, < y < X k 6,. Hence, if the sum is positive (negative), the inequality (8.1.6) holds for X k  6, < y 5 X k ( X k 5 y < X k &). For every such y one of the following inequalities is therefore satisfied:
+
+
Since p ( ~ , + ~ )p(u,) 1 0, we may apply Lemma 8.1.2, and summing over k it follows that
If p is of bounded variation, we may by Jordan's theorem decompose p into the difference of two increasing functions, p = p1  pa, such that llpllev = IlpiIInv Ilp,aIIsv. Then the hypothesis (8.1.5) means that
+
Thus one of the two terms in the curly brackets must in the absolute value be greater than M/2, and since p1and pa are now increasing, we may apply our foregoing considerations in order to obtain the result.
Lemma 8.1.4. The Hilbert transform of any step f i c t i o n vanishing outside of compact sets exists except at ajinite number of points. In particular, the Hilbert transform of the ] except at the two points a and b and is given by characteristicfunction K [ Q , ~exists (8.1.8) Indeed, according to Def. 8.0.1, (8.1.8) follows by an elementary calculation.
309
HILBERT TRANSFORMS ON THE REAL LINE
Theorem 8.1.5. The HilbertStieltjes transform (8.1.1) of p E BV exists almost euerywhere. Moreover, for every M > 0, meas{x 1 I P W l > MI 5
(8.1.9)
128
3 11~11BV.
Proof. Let p be realvalued. We first prove the existence a.e. of p”(x). By Cauchy’s criterion, it is sufficient to show that, given e > 0, for every x except in a set of measure less than e (8.1.10) for all sufficiently small 6 and 6’, 0 < 6‘ < 6. Let p(x) = pAC(x) psing(x)be the Lebesgue decomposition of p into its absolutely continuous part pAC and its singular part pslnS,and let pAC(x) = J?m g(u) du with g E L’. Since the step functions which vanish outside of compact sets are dense in L’, we may, to given el > 0, choose a function h such that 11 g  hll < e’. Thus, if pl(x) = j? h(u) du, then IlpAc  plllBV< el. Furthermore, psingcan be approximated to within e’ by a singular function p2 with variation confined to a closed set of measure zero, that is, which is constant on the intervals of an open set B whose complement has measure zero: IIpBing p2[IBv < e’. Thus, taking e’ = e2/3841/%, we set p = p1 pa + p3 with llpgIIBv< e2/1922/%. Let IE, be the set of x for which the inequality
+
+
(8.1 .I 1) fails to hold for all sufficiently small 6 and
a’, 0
0 such that for all 6 and 6‘ with 0 < 6’ < 6 < 6, (8.1.1 1) holds for pl, p2, and p3, and hence (8.1.10) holds as was to be shown.
310
HILBERT TRANSFORMS
The relation (8.1.9) now follows immediately from Lemma 8.1.3.In fact, if x is such that p ~ ( x exists ) and lp”(x)l > M, then
(8.I. 13) for sufficiently small 6 > 0. Again Vitali’s theorem gives a sequence of disjoint intervak(x,  Sk,x, + 6,)satisfying(8.1.13)whichcovers the Set of x with Ip”(x)l > M up to a set of measure zero. Therefore 16 meas (x I Ip(x)l > M > 0) 5 2 2 6, I 2.zIlpllSv. k
If p is complexvalued, then the above considerations apply to the real and imaginary part of p, respectively. Thus Theorem 8.1.5 is completely established.

Theorem 8.1.6. The Hilbert transformf of afunction f E LP, 1 I p < everywhere.
03,
exists almost
Proof. The casep = 1 follows by Theorem 8.1.5 since the Hilbert transformf” may
be regarded as the HilbertStieltjes transform p” of the absolutely continuous function p ( ~ ) = ~ X _ ~ f ( u ) d u1. I du = Il + 12. LAX) =; xu x  u Ixulrd IxUl2d But 1 lim la = f o d u
1
1
a+o+
71
J
lul
2
xu
2a
for all x E [a, a], and since flE L1, lim,,o+ Il exists almost everywhere. Thus the Hilbert transform f exists for almost all x E [a, a], and since a was arbitrary, the theorem is established. 8.1.2
Existence in LaNorm
So far we have discussed the pointwise limit in (8.0.2) and thus the pointwise existence of the Hilbert transform f of functions YE LP, 1 I p < 03. We shall now turn to the question of mean convergence in (8.0.2). This will be part of the general problem of carrying over certain properties of the original function f to its Hilbert transformf”. In particular, we shall show that f E LP, 1 < p < 00, implies f E Lp. We proceed with the case p = 2. Let f E L2 and set


(8.1.14) (0, elsewhere, (8.1.15)
fd:,,(x)
=
is
?T
m
m
f(x
 u)kdev(u)du = ;
s
asiulrn
f ( x U
du.
HILBERT TRANSFORMS ON THE REAL LINE
31 1
Then, since kd,nE L' n L2 for all 0 < 6 < 71 < coy it follows by Prop. 0.2.1 and 0.2.2 that fayn(x)exists everywhere as a function in Co n La. Furthermore, (8.1.16)
lim hyn(x) = f 6  ( x ) 
n+
(9
By Theorem 5.2.12 we conclude [fdTnl^(u) = ~ ~ f n ( v ) I k 6 , n l A ( v ) Since
we have (8.1.17) and Ik&,(u)I I M with constant M independent of u, 6 , q (see also Problem 1.2.10). Therefore = { i sgn v}f"(u) a.e.,
and since I[fa;n]A(u)l I M If"(u)l is bounded by an L2function, we may apply Lebesgue's dominated convergence theorem to deduce
Since {  i sgn u}f"(u)
E
L2, by Plancherel's theorem there exists g E La such that
g"(v) = {  i sgn v}f"(v) a.e. Therefore by the Parseval equation
(8.1.1 9)
11 [h7n1A(0>  {  i sgn '>f"('> 1 2
= IlfdYn
 gl12.
Thus (8.1.18) implies l.i.m.6~o+,n.mf6yn(x) = g(x). Moreover, by (8.1.16) and Theorem 8.1.6 it follows that limd+o+, ,,. &(x) =f"(x) a.e. Therefore g(x) = f(x) a.e. in view of Prop. 0.1.10.Since [I2 = 11 g"1I2 = I/{  i sgn o } f A ( ~ ) I I 2 = llfl12 by (5.2.5), we may summarize our results in the following
[If

Theorem 8.1.7. For f E L2 the Hilbert transformf exists almost everywhere, belongs to L2, and satisfies
(8.1.20)
Ilf112
=
Ilfllz.
Moreover,
(8.1.21)
lim
a+o+
llf"  h  l l z
= 0,
and the Fourier transform o f f " is giuen by
(8.1.22)
Lf"]"(u) = {  i sgn u}f"(v) a.e.
3 12
HILBERT TRANSFORMS
It only remains to show (8.1.21). In view of Fatou’s lemma we have by (8.1.16), (8.1.18), and (8.1.19) that
1l.t f”lli I liminf Ilfd:, f“lli = 41) n m
(8 +O+)Y
proving (8.1.21). In particular, fd” E La with Ilfd” 112 I M 11 f \la for every 8 > 0, E La. Thus the mapping f +.f “ is a bounded linear transformation of La into La which preserves norms. It is actually onto as follows by (8.1.22).
f
8.1.3 Existence in LPNorm, 1 < p < 00 To this end we apply the theorem of Marcinkiewicz (cf. Sec. 4.4) with R1 = R2 = R and pL1= pa = Lebesgue measure on R. The operator Twhich assigns to each simple function h its Hilbert transform h“, (8.1.23)
(Th)(x)= h”(x) = lim d*O+
l /IuI@r6? )
du
?T
is welldefined by Theorem 8.1.6 as a linear operator on So, which satisfies (8.1.24) by (8.1.9), and furthermore (8.1.25) by (8.1.20), since h E So, implies h E LQ, 1 I q I 00. Thus T is of weak type (1 ; 1) and of strong type (2; 2) on So, (with constants MI = 128 and Ma = 1). The theorem of Marcinkiewiczthen implies that T is of strong type (p; p), 1 c p < 2, on So,, and that llThllp I MpIlhllpfor all h E So, with constant M p depending only uponp. Hence Proposition 8.1.8. Let 1 < p c 2. Zfh E So,, then h“ E Lp and
llhllp I MPIlhllPY
(8.1.26)
the constant M p being independent of h. Moreover, Proposition 8.1.9. The operator T as dejined on So, by (8.1.23) has a unique extension as a bounded linear transformation of L P into L P , 1 c p < co,such that
(8.1.27) holdsfor all f
IKfllP
E
MPllfllP
Lp with constant M p independent off:
Proof. By Theorem 8.1.7 and Prop. 8.1.8 the operator T as defined on So, by (8.1.23) is of strong type ( p ; 1 c p I2, satisfying (8.1.27). To show that T is of strong type
HILBERT TRANSFORMS ON THE REAL LINE
313
( p ;p ) on So, for 2 c p < CQ, let h, g E So, and 6 > 0. Then we have by Fubini's theorem (8.1.28)
hd"(x)g(x)dx
/mm
=

h(x)gd"(x)dx.
Since h, g E L2, we obtain by (8.1.21) and Holder's inequality (8.1.29) J
Jw
m
for all h , g Soo. ~ Now, let 2 c p < CQ. Then T g E Lp' with l\Tg\Ip,5 Mp,llgllp,by Prop. 8.1.8. Therefore, by (8.1.29) and Holder's inequality,
Hence Th defines a bounded linear functional on the dense subset Soo of Lp' which may therefore be extended to a bounded linear functional on all of LP' (having the same bound). The F. Riesz representation theorem then implies Th E Lp and
IIThIIP 5 MPKPl~ll~IlP
(8.1.30)
for all h E Soo. Thus T defines a bounded linear transformation of Soo into Lp for all 1 < p c 00. Therefore it can be extended to a bounded linear transformation on all of Lp into L P satisfying (8.1.25), (8.1.26), and (8.1.30), respectively. So far we have defined the Hilbert transformf off E LPy 1 < p c 00, as a function which by Theorem 8.1.6 exists almost everywhere. On the other hand, by Prop. 8.1.9 we have assigned to each f E Lp, 1 c p c CQ, a uniquely determined function Tf E Lp which satisfies (8.1.27) and coincides withfl for all f of the subset So, by (8.1.23). In the following we shall show that Tf = f  for all f E Lp. To this end we need two elementary lemmata. Lemma 8.1.10. For f
E
LP,
1 4p c my
(8.1.31)
In particular, for a l l y > 0 (8.1.32) Proof. Firstly. all terms exist everywhere as continuous functions by Prop. 0.2.1. We have fory > 0 (8.1.33)
7r
Irn m
f(x 
U
U)
3
4
du
+4+
HILBERT TRANSFORMS
3 14
say. Given > 0, choose 6 > 0 such that Ilf(0 + u)  f(o)IIp < Prop. 0.1.9). If y < 6, then by the HolderMinkowski inequality
8
for all IuI I 6 (see
If we now fix 6, we may take y so small that
proving (8.1.31). For (8.1.32) we note that llZlllp
I(2/n)llf\lp and
1\12
+
Z311p
5 (l/~)llfllp.
Let us recall (see (3.1.46)) the notation (8.1.34)
of the conjugate of the CauchyPoisson integral o f f . I f f E Lp, 1 5 p < to, then Q ( f ; x; y ) E C by Prop. 0.2.1, since d8/2q(x) = x/(l + 2)(cf. (3.1.47)) belongs to L', 1 < r I to. But note that q I# L1. If, as usual, P(S; x; y ) denotes the CauchyPoisson integral (3.1.38) off, then
Lemma 8.1.11. For f
E
Lp,
(8.1.35)
1 < p < to, we havefor all x
P(?'f; x; Y ) =
In particular, Q ( f ;
0 ;
E
R, y > 0
Hf;x; Y ) .
y ) ,b E Lp and
(8.1.36) IIQ(f;o;Y)Ilp 5 M P l l f l l P , Ilb"I1p 5 (1 + M?J)IIfIIp uniformlyfor y > 0, 8 > 0, respectively, the constant Mp being independent o f f . Proof. Since 1/(1
+ x 2 ) E L1 n L" and
(8.1.37)
by Problem 8.1.2, we may argue as in the proof of (8.1.29) to obtain (8.1.38)
for every h E Soo. Thus P ( h  ; x ; y ) = Q(h; x ; y ) for all x E R, y > 0 and h E Soo. To establish (8.1.35) for all f E Lp, 1 < p < co, let {fn} c Soo be a sequence such that limn \ I f  fnllp = 0. Then P(Tfn;x ; y ) = Q(fn; x ; y ) by (8.1.38). Since it follows by (8.1.27) that limn+mIIT(f  f n ) I p = 0, we have by Holder's inequality and (8.1.23) for each x E R and y > 0 that IP(Tf; X ; Y )  P(f,";X ; Y ) [ 5 Y"' IIP(o)II~*IIT~f," IIp = 4 1 ) (n a), +

I Q < ~ ; X ; Y ) Q ( f n ; x ; Y ) I 5 Y"'
IIP"(o)IIP,IIffnIIp
=
O(l)
(n+m).
Combining the results we obtain (8.1.35). Since forfE Lp, 1 < p < Q),Tf E Lp and therefore P(Tf; x ; y ) E L P for all y > 0, we have Q ( f ; x ; y ) E Lp for all y > 0, and by (8.1.32) f6 E LP for all 6 > 0. Finally (8.1.27), (8.1.32), and (8.1.35) imply (8.1.36).
Let us mention that we will soon generalize the results o f the last two lemmata considerably.
315
HILBERT TRANSFORMS ON THE REAL LINE
Theorem 8.1.12. For f E Lp, 1 < p < co, the Hilbert transform as defined by (8.0.2) cxists almost ererywhere, belongs to Lp and satisfies
Ilf
(8.1.39)

IIP
5 MPllf IIP
with some constant Mp independent o f f . Moreover, (8.1.40)
lim
dO+
11s  h  l l p
= 0,
i.e., the limit (8.0.2) exists not only pointwise a.e. but also in the mean of order p .
Thus for 1 < p < 00 the Hilbert transform defines a bounded linear mapping of Lp into Lp. (8.1.39) is often referred to as the Marcel Riesz inequality for Hilbert transforms.

Proof. Let 1 < p < oc). By Theorem 8.1.6 the Hilbert transform f off E Lp exists almost everywhere, in fact limd+o+f ; ( x ) = f ” ( x ) a.e. According to Prop. 8.1.9, to each f E Lp we may assign a uniquely determined function Tf E L P such that (8.1.27) holds. We now show that (8.1.41)
lim
dO+
11 Tf  fa” \ I p
= 0.
Indeedsf,“ E Lp for all S > 0 by Lemma 8.1.11, and by (8.1.35) llTf  f b 
Ilp
5 II(Tf)(o) P(TA 0 ; S)llp
+ II Q ( f ;
0 ;
6)  fa”(~)IIp,
which tends to zero as S + 0 + by (3.1.41), (8.1.31). Now it follows as an application of Prop. 0.1.10 that (8.1.42)
(Tf)(x) = f “ ( x ) a.e.
for allfe Lp, 1 < p < co. In virtue of Prop. 8.1.9 and of (8.1.41) the proofis complete. Problems 1. Letf,gELP,1 I p < co.Show that
+
+
(i) [uf Pg](x) = &(x) /3g(x) a.e. for every a,P E C, (ii) iff is an even function, then f is odd, (iii) [ f l  ( x ) = f”(x)a.e., (iv) [ f ( o h)l“(x) = f ” ( x h) a.e. for each h E R, (v) [pf(po)l(x) = pf“(px) a.e. for each p > 0. 2. By means of (8.0.2) evaluate the following Hilbert transforms: (i) [ K [ , , ~ , ]  ( X ) = ( l / d log (Ix  bl/lx  a [ )for all x f a, b, (ii) if p(x) = dG(l/(l+ x2)), then p”(x) = d F ( x / ( l + 2))(=q(x)) for all x E R. (For a further method of evaluating Hilbert transforms see Problem 8.3.2.) 3. Give examples of functions f E L1 for which f does not belong to L’. (Hint: Use e.g. f ( x ) = 1/(1 x2) or TITCHMARSH 16,p. 1431) 4. LetfE LP, 1 I p < m.Iffbelongs to Lip (C; a) for someac > O,showthatf”(x)exists
+
+
for all x E R.

+

HILBERT TRANSFORMS
316
5. (i) Show that g(u) = (eIhu  l)/lul belongs to La and g"(u) = 4/2/rrlOgIu/(h
(ii) Let f E La. Show that
+ u)l.
(Hint: Use Theorem 8.1.7, Problem 5.2.9, Prop. 5.2.13; see TITCHMARSH [6, p. 1211)
8.2 Hilbert Formulae, Conjugates of Singular Integrals, Iterated HiIbert Transforms 8.2.1
Hilbert Formulae
During the course of the preceding proof we have shown some useful relations which we now generalize. Proposition 8.2.1. Let f
E LP,
1
0 there exists S > 0 such that according to the continuity off in L1norm
Just as for uZ(x) it follows by f * Furthermore, in view of (8.1.9) meas { x
E
L1 that limllzll+,Il(f"
I ILf* gl(x)  G ( 4 >
> 01 5
* g)(o) 
128 M IIU*g)(.) 
u;(o)II1
= 0.
~Z(O)IIl,
which tends to zero as llZll 3 0 for each M > 0. Therefore, by Prop. 0.1.10, there exists a sequence {Z,}, lim,+m llZ,ll = 0, such that
Lf* g](x) = j lim uz;(x) = (f" * g)(x) a.e., + establishing (8.2.3) for p = 1 and g E Coo. If g E L1 only, let {g,} c Coosuch that llg  gnlIl = 0 (Problem 3.1.2 (iii)). Then Lf* gn]"(x)= (f"* g,)(x) a.e. for eachn E N. Sincef" E L', we havef " * g E L1 and IlV * g )  (f"* &)Ill 0
(8.2.6)
[P(A
0;
~ ) l " ( x )=
Q ( f ; x; Y)
=P(f
;x; V ) a.e.,
which holds for all f E LP, 1 < p < CQ, extending (8.1.35). In case p = 1 only the first equation of (8.2.6) holds in general. Proposition 8.2.4. Let f E L1 be such that f (8.2.7) (8.2.8)

E
L1. Then& E L1for all S > 0 and
IlfJIll 5 I l f 111 + Ilf 111, lim Ilf
dO+
&Ill
=
0.
Conjugates of Singular Integrals: 1 < p < 00 Relation (8.2.6) on the singular integral of CauchyPoisson again calls our attention to the convergence problem for general singular integrals which we have already
8.2.2
HILBERT TRANSFORMS ON THE REAL LINE
319
studied in some detail in Chapter 3. In Sec. 3.1 and 3.2 we in particular considered the convergence of the integral (8.2.9) towardsf€ LP, 1 I p < co, as p + m. We saw that if x
E
NL1, then
for all f E Lp. If x furthermore is even, positive, and monotonely decreasing on [0, m), then limp+ J c f ; x ; p) = f( x ) a.e. The problem now is to establish similar results for the Hilbert transform f " . In particular, since the explicit evaluation of the Hilbert transform f " of a given f~ Lp, 1 I p < m, by means of definition (8.0.2) is often troublesome, we are interested in singular integrals of type (8.2.9) involving the original function f which nevertheless approximatef" in the mean and pointwise a.e. The Hilbert formulae so far established suggest that for x in (8.2.9) one takes the Hilbert transform x " , and thus that one considers the integral (8.2.10) As an example, for the integral P ( f ; x ; y ) of CauchyPoisson x(x) = p ( x ) , ~ " ( x = ) = Q ( f ; x ; y ) , the conjugate of the CauchyPoisson integral. Generally, we call (8.2.10) the conjugate of the singular integral J ( f ; x ; p), which by definition is not to be mistaken for the Hilbert transform [ J c f ; 0 ; p)](x) of J ( f ; x ; p). But in our future results the hypotheses upon x are so strong that J " ( f ; x ; p) = [ J ( S ; 0 ; p)]"(x) a.e. for every f E LP, 1 Ip < 00.
p"(x) = q(x), and P " ( f ; x ; y )
Proposition 8.2.5. Let 1 < p
0, [px(po)]"(x) = px"(px) a.e. by Problem 8.1.1, an application of (8.2.5) gives J " ( f ; x ; p) J ( f "; x ; p ) . Therefore (8.2.1 1) follows by Theorem 3.1.6 as applied to f.If y, is even, positive, and monotonely decreasing on [0, m), then, by Prop. 3.2.2, limp+mJ(f ";x ; p) = f " ( x ) at all Dpoints o f f " and hence a.e., which establishes (8.2.12).
320
HILBERT TRANSFORMS
8.2.3 Conjugates of Singular Integrals: p = 1 Since the first assumption of the last proposition upon the kernel is usually satisfied in the applications, the problem posed above for the Hilbert transform f" may be considered as solved for 1 < p < a.It remains to treat the case p = 1. Since J'E L1 does not necessarily implyf E L' and since, in particular, the assumption x E L1 n L" does not ensure x" E L1, we cannot use the results for the original function f as in the case 1 < p < a.Therefore we can hardly expect to prove the counterpart of Prop. 8.2.5 for p = 1. But we shall carry over the weaker results of Problem 8.2.2.

Proposition 8.2.6. Let f
E
L' and x E L1 n L" be an even function.
If 0 as defined by
(8.2.13)
belongs to Ll(1, m), then
00, we have x" E Lq, and therefore the expressions occurring in (8.2.14) are meaningful. Since x is even, it follows by Problem 8.1.1 that x" is odd, and thus as in (8.1.33)
Proof. First of all, since x E Lq for every 1 c q
0 there exists 8 > 0 such that (Y(r)(< ~t for all 0 < t s 8. We now fix this 6 and choose p so large that p8 > 1. Then in the notation of (8.2.15) and by partial integration
since
X I
is monotonely increasing on [0, 11. Regarding 12,we have
=
7r
[O(pS)Y(S)  @(l)Y(l/p)]
+
'Sd
7r
l/P
Y ( u ) d [  8(pu)]
3
la'
+ I:.
Since 8 is monotonely decreasing on [l, a),it follows for x > 2 that X
/x:2
8 ( u ) d u 2 8 ( x ) 2'
and since the lefthand side tends to zero as x + 00, we obtain lim xO(x) = 0.
(8.2.20)
x+ m
Therefore
1121 I7r [(p8)@(p8) +
@(I)] 5 2@(1) e
for all p sufficiently large. Furthermore,
Finally we deduce
which by (8.2.20) tends to zero as p + 00, establishing (8.2.17). Since limd+,,+ fd"(x) = f " ( x ) a.e. by Theorem 8.1.6, relation (8.2.18) follows. We observe that the proofs of the last two propositions may of course be carried over to the case 1 < p < a, as well.
As an example we again treat the singular integral P ( f ; x; y ) of CauchyPoisson. In view of Problem 8.2.3 we have as a generalization of Lemma 8.1.10 2 1F.
A.
HILBERT TRANSFORMS
322
Corollary 8.2.8. The conjugate of the singular integral of CauchyPoisson o f f E Lp, 1 I p < 00, satisjes:
(8.2.21)
lim
y0
+
QM x; y) = f  ( x )
a.e.,
For 1 < p < 00 we furthermore have
which remains valid for p = 1 under the additional assumptionf
 E L1.
Let us conclude with the following result of Privalovtype which indicates that, in general, the operation of taking Hilbert transforms leaves invariant (norm) smoothness properties of the original function f (cf. also Problems 8.2.58.2.7 and Prop. 8.3.7).
Proposition 8.2.9. Let f f E Lip (LP; a).t

E
Lp, 1
5
p c
00,
and 0 c a < 1. Then f
E Lip (Lp;a )
implies
Proof. For 1 < p < co the proof follows immediately by the fact that the operation of taking Hilbert transform is a bounded linear transformation of Lp into LP. Indeed,
Ilf"(0 + It)  f"(0)IIP 5 MPIIf(0 + It)  f ( 0 ) I I P . Let p = 1. Since f exists almost everywhere by Theorem 8.1.6, we have
Now the hypothesis implies that [ Q ( f ;x ; y )  f " ( x ) ] belongs to L1 and$
where M is the Lipschitz constant. Furthermore, since q(x) E W:, it follows by Problem 3.1.6 that
and therefore
t For p = 1 we indeed use the following slight generalization of the concept of Lipschitz classes: g belongs to Lip (Lp; a) if [g(o + It)  d o ) ] E LP for every h E R and there exists a constant M with IIg(0 h)  g(o)IIp 5 M lhla. Thus not each term of the difference g(x h)  g(x) need belong to Lp. $ In fact, the following inequality gives a direct approximation theorem for the integral Q ( f ;x ; y), stating thatfe Lip (Lp; a) implies y )  f"(n)IIP = O ( p ) .
+
+
IIQv, 0 ;
HILBERT TRANSFORMS ON THE REAL LINE
323
proving the assertion. 8.2.4 Iterated Hilbert Transforms
Thus far we have assigned to a functionf its Hilbert transform. Iff€ Lp, 1 c p < co, then one of our results implies that f " E Lp. We may therefore take the Hilbert transE Lp. Our next proposition states that the iterated Hilbert form off to obtain If"]" transform If"]" is equal to the original functionf apart from a minus sign. Proposition 8.2.10. Let 1 < p c 00 and f E Lp. Then Lf"]"(x)=  f ( x ) a.e. This remains valid for p = 1 if one furthermore assumes that f E L'.

Proof. If 1 < p < co, then in virtue of the Hilbert formulae (8.2.1), (8.2.2) we obtain for every g E Lp'
This implies I f " ] " ( x )= f ( x ) , i.e. (8.2.24) which may be regarded as an inversion formula for the Hilbert transform. Concerning p = 1, the CauchyPoisson kernel p(x) satisfies [ p  ]  ( x ) = p(x). This implies by Cor. 8.2.8, Problem 8.2.1, Cor. 3.2.3 that If"]"(x) = lim Q ( f  ; x ; y ) = lim u+o+
=
 lim
y0+
P ( f ;x ; y) = f(x)
Y+O+
for almost all x E R, and Prop. 8.2.10 is completely established. Thus we have in particular shown that the mapping f +f " is a bounded linear transformation of LP onto Lp for 1 c p < co. Moreover, if we employ the notation Hf = f " , H ' f = H ( H r  ' f ) , then (8.2.25) for every f
E
Lp, 1 < p c a,and r E M.
324
HILBERT TRANSFORMS
Problems 1. Letf,f* E L1 a n d gE LPfor some 1 < p < 00. Show that [f*gl" = f " * g = f *g". p)  f17D(o))(p = 0, 2. Let f~ LP, 1 < p < 03, and x E NL1 n L". Show limp+" IIJ"(f; and if x is furthermore even, positive, and monotonely decreasing on [0, a),then limpam[ J " ( f ;x ; p )  f&(x)] = 0 a.e. 3. If p(x) denotes the CauchyPoisson kernel, then 0 ;
1
8(x)
=
+
~ ( i xaj
Show that p(x) satisfies the assumptions of Prop. 8.2.58.2.7. 4. (i) Letfe L1, x E L1 n L" be an even function and x" be positive on (0, m), monotonely increasing on [0, 11. If 8 is defined by (8.2.13), assume that there is a majorant @*: l@(x)I I@*(x) on [1, co) such that 8* is monotonely decreasing and integrable on [ l , 03). Show that at each point x for which I f ( x + u)  f ( x  u)l du = o(r), rt 0 + , one has limp+" [ J " ( f ;x; p )  f;ip(x)] = 0. In particular, limo+ J " ( f ; x ; p) = f " ( x ) a.e. (ii) State and prove the counterpart of Cor. 8.2.8 for the singular integral of Fejkr. (Hint: It follows by Problem 8.3.2 that
5. Let 1 < p < and 0 < a I1. Show thatfe Lip (LP; a) if and only iff" E Lip (Lp; a). 6. (i) Show that f E Lip (Ll; 1 ) implies f" E Lip* (L1; 1). (Hint: Use the method of proof of Prop. 8.2.9; cf. OGIEVECKI~BO~CUN [ l ] ,BUTZERTREBELS [2, p. 211) (ii) Show furthermore that f E Lip* (L1; 1) impliesf" E Lip* (Ll; 1) (see also BUTZERBERENS [ l , p. 2521, BUTZERSCHERER [l, p. 1461). 7 . Let 1 Ip c my0 < a < 1, and f~ Lp n Lip (C; a). Show that f" exists everywhere [6, p. 1451) and belongs to Lip (C; a). (Hint: TITCHMARSH 8. LetfeLP, 1 s p < 03. Show that Q,
(Hint: Compare the proof of Prop. 8.2.6; see also the literature given in Problem 3.4.6 (ii))
8.3

Fourier Transforms of Hilbert Transforms
8.3.1 Signum Rule We begin with the proof of relation (8.0.4), the importance of which we already emphasized in the introduction to this chapter.
Proposition 8.3.1. Let f E lp,1 < p I2. Then the Fourier transform of the Hilbert transformf" is given by
(8.3.1)
[f "]"(v) = {  i sgn v}f"(v)
a.e.
For p = 1 this remains valid and then holds everywhere under the additional assumption
f

E
L1.
325
HILBERT TRANSFORMS O N THE REAL LINE
Proof. The casep = 2 is already given by (8. I .22). LetfE Lp, 1 I p < 2. The assumptions imply that (8.2.3) holds for every g E L' n L". Since f * g E Lp n C by Prop. 0.2.1, 0.2.2 and hence f * g E La, Lf* g]' E L2, we have by the convolution theorem and (8.1.22) Lf"lA(u)gA(4=
Lf * g l W
={
=
"f * gl"I"(u) = {  i s m
 i sgn u)f"(u)g"(u).
w *g l A ( 4
Thus, if g is such that g"(u) # 0 (e.g. g ( x ) = exp { x2}), (8.3.1) follows. Although the next proposition is an easy consequence of the Hilbert formulae, it will be of some use in the actual evaluation of Hilbert transforms. Proposition 8.3.2. Let f E L1 n Lp, 1 < p I 2, be an even function. Then
f"(4 = fWY
(8.3.2)
If"]"(u) = f ; ( u )
(8.3.3)
a.e.
Proof. In view of Problem 5.1.10 we only need to prove (8.3.3). Let h E Lp. On the one hand, by (8.3.1), (5.2.18), and (8.2.1) m
[ f ( u ) {  i sgn u)h"(u) du = [ Jm
m
f(u)[h"]"(u)du
=
[
m
If"]"(u)h(u) du.
Jm
Jm
On the other hand, since
(8.3.4)
[{  i sgn olf(o>lA(0) =
we have
jmm f(u){  i sgn u}h"(u) du
=

which implies (8.3.3) by Prop. 0.8.1 1. 8.3.2 Summation of Allied Integrals The preceding results may be regarded as preparatory to the summability of the allied integral (8.3.5) In Chapter 5 we assigned to each f E LP, 1 I p 5 2, the Fourier transformf^, and the problem then was to recapture the original functionffrom the values of its Fourier transform f". The formal inversion was given by (5.1.6). But since the Fourier inversion integral does in general not exist as an ordinary Lebesgue integral, it was summed up via (5.1.9). Concerning the allied integral (8.3.5) we expect in view of (8.3.1) that it converges to the Hilbert transformf".
HILBERT TRANSFORMS
326
In fact, if 1 < p 5 2, then Theorems 5.2.14 and 5.2.15 apply here, too, and introducing (8.3.6)
j
u cf; X; p ) =
8( :){
 i sgn u}f ^(u) elxudu
m
(cf. (5.2.20)), we have
Proposition 8.3.3. Let f E L p , 1 < p I 2. Thenfor a 9factor the 9means (8.3.6) of the allied integral exist for all x E R, belong to L p , and satisfr (8.3.7)
II~Wo;P ) l I P
~ P l l ~ l l l l l f l l P
(P > 01,
(8.3.8) If 0 furthermore satisfies the conditions of Theorem 5.2.15 (or Problem 5.2.4), then
(8.3.9)
lim VY.; x ; p ) = f " ( x ) a.e. D m
According to (8.3.1) and the Parseval formula (5.2.18) we have (8.3.10) P
=
"
d~ j 
mf
x  u)P(pu) du, (
and the proof follows since f " E Lp. The last argument does not apply to the case p = 1. Here we have
Proposition 8.3.4. Let f E L1 and assume for a 9factor that O(x) = (l/x) fl2 [ P ] " ( x ) belongs to L1(l, a).Then (8.3.11)

lim
P.
IIWf;";P )  f G l ( o ) I l 1

= 0.
I f all further hypotheses of Prop. 8.2.7 (or Problem 8.2.4) are satisjied (with x
= e"),
then (8.3.12)
lim V  ( f ; x; p )
D."
=f
" ( x ) a.e.
Proof. Since 9 E L1 n L", we obtain by (5.1.5), (8.3.3), and (8.3.4) (8.3.13) and the proof follows by Prop. 8.2.6, 8.2.7, and Problem 8.2.4, respectively. In particular, in view of (5.2.23) and (8.2.10), the notation (8.3.6) is justified by (8.3.13). To evaluate [PI for important examples of 9factors (such as those of Ceshro, Abel or Gauss, cf. (5.1.8)) one may use Prop. 8.3.2 or Problem 8.3.2. Then we have
HILBERT TRANSFORMS ON THE REAL LINE
327
Corollary 8.3.5. Let f E Lp, 1 5 p 5 2. The allied integral is Ceshro and Abel summable to f " ( x ) for almost all x E R, i.e. (8.3.14) (8.3.15)
,'\I zJp 1 p (1  ~ ) {  i s g n u } f " ( u ) e i x u d v= f " ( x ) lim Yo+
dzr
a.e.,
eY'u'{ i sgn u}f "(u) etxudu = f " ( x ) a.e. OD
The proof is left to Problem 8.3.1.
8.3.3 Fourier Transforms of Derivatives of Hilbert Transforms, the Classes (W)[P, (V")?
We now investigate the Fourier transform of derivatives of the Hilbert transform E Lp, 1 I p I 2, in Sec. 5.1.3,5.2.5, and 5.3.3). The definitions of the classes W;Ccw,, V&(R)in Sec. 3.1 and 5.3.3 suggest the introduction of (r E N, 1 Ip < 00)
f
 (as considered for the original functionf
(8.3.16)
(W)[P = {f {{ f
(8.3.17) (V)lp =
E
L1 I f If
E LP

= E
46 a.e., 46 E ACf;',
#k)
WLP}
I f = +a.e., +EAC;;~, {fELP I f EVlp}
E
L1, 1 Ik Ir} (1 < P < a),
qYkkL1, 1 Ik Ir  1, c$('~)EBV) (1 < p < 00).
By Theorem 8.1.6, f E Lp, 1 I p < 00, implies that f  exists almost everywhere. Therefore the above definitions are meaningful. They are particularly easy for 1 < p < GO, since then f belongs to L p by Theorem 8.1.12. But for p = 1, f " does not necessarily belong to L' so that the relevant function 4 is not assumed to be an element of L' (cf. footnote to Prop. 8.2.9). The following theorem gives equivalent characterizations of (V)[P for 1 Ip 5 2, whereas those of (W)p are left to Problem 8.3.3.

Theorem 8.3.6. For f
E
Lp, 1 5 p I2, the following assertions are equivalent:
(i) f E (V)[P, (ii) f E V[Lp; {i s g n u}(iv>'], (iii) there exists p E BV i f p = 1 and g E LP if 1 < p I 2 such that
the integrals existing in the sense of Theorem 5.3.15 (apart possibly from thejirst one in casep = 1). Proof. Let 1 < p 5 2. I f f satisfies (i), then f " E V?, and therefore by Theorem 5.2.211. there exists g E LP such that (iv)"f"]"(u) = g"(u) a.e. Since p > 1, we have [f "]"(v) = {  i sgn u}f ^(u) by Prop. 8.3.1, and thus f E V[LP; {  i sgn u}(iu>'].On the
t Note that Vp and W:P are equal for 1
< p 5 2 by definition.
328
HILBERT TRANSFORMS
other hand, iff belongs to the latter class, then {  i sgn o}(iu>.f"(u) = g"(u), g E Lp, implies (iu)'Lf"]^(u) = gA(u) by Theorem 8.1.12 and Prop. 8.3.1. Thus (iii) and (i) follow by Theorem 5.2.21 as applied tof". Let p = 1. I f f € (V)p, then as in the proof of Prop. 5.3.13
Since the righthand side belongs to L1 by the convolution theorem (cf. Prop. 5.3.3, Lemma 5.3.4), d;f E L', and since obviously d ' , f ~L', it follows by Prop. 8.3.1 that [d;f"]^(u)
=
{  i sgn u}(etu  lyfA(u).
Therefore by (5.3.8) and (8.3.18)
i.e. f E V[L'; {  i sgn u}(iu>.]. Conversely, let there exist p E BV such that {  i sgn u}(iu~f"(u)= p"(u) for all u E R. To prove that (ii) implies (iii) (and thus (i)) we consider three cases. Let r = 1. Then by (8.3.14), (5.1.15), and Lemma 5.3.4 for each h E R
= p(x
+ h)  p(x)
a.e.
To show thatf"(x) = p(x) a.e., we integrate the latter equation with respect to x over [0, y ] , thus consider
s,' MX)  f " ( x ) ]dx for any y > 0. Nowf"
E
= f,'p(x
+ h) dx  / h u + h f  ( x )dx
L2 since
Hence f E L2 by Plancherel's theorem and f E L2 by Theorem 8.1.7. Therefore = 0. Moreover, limb,  p(x + h) = 0, since p is normalized. This implies 1irnh+  m j: p(x h) dx = 0 by Lebesgue's dominated convergence theorem. Thusf(x) = p(x) a.e., in other wordsf(x) = JZ dp(ul) a.e. Secondly, let r be odd, i.e. r = 2n + 1, n E N. Then lul'f^(u) = ( l)"p"(u), and it follows by Theorem 6.3.14 that there exists g E L1such that IuI f"(u) = g^(u). Therefore (iu)"g"(u) = p"(u), and g E V2,: by Theorem 5.3.15. In particular,
h,+  m ~ ~ ' h f " ( xdx )
+
329
HILBERT TRANSFORMS ON THE REAL LINE
s?
Moreover, IuJf"(u) = g"(u) impliesf"(x) = g(ul) du, a.e. by the first case, which proves (iii). Finally, let r be even, i.e. r = 2n, n E N. Then {  i sgn u}lul'f^(u) = ( l)"p"(u). iu)' E [L']" (cf. (5.1.34)), it follows that Since ( 1
+
{isgn u} (iu)2" + ( i ~ ) ~ f"(u) "+'
UI
1
+
+ iu
E
[BV]",
{  i sgn u} (iu)aYA(4 [L'IA. 1 iu
+
Therefore I 2n '( 1 iu)  tf"(u) E [BV]", and since ( 1 + iu) tf"(u) 6.3.14 implies 1uJ2"'(1+ io)'f"(u) E [L']". Furthermore, since +
lu12"tfA(u)
= Iu12n1 ( 1
+ iu)lf"(u)
 {isgnu}lu12"(1
E
[L1]", Theorem
+ iu)!f"(u),
we have Iv12"'f"(u) E [L']". Thus, letg E L' be such that lulr'f"(u) = ( ly'g"(u). with ~ #k) E L1 for 1 5 k Ir  1 Then by the previous (odd) case there exists 4 E A C ~ G such thatf"(x) = +(x) and #'l)(x) = g(x) a.e. Moreover,
However, {i sgn u } J u l g"(u) = p"(u), that is, (iu)g"(u) = p"(u) which implies g(u,J = dp(u,) a.e. by Theorem 5.3.15. Thus, (iii) is completely established, proving the theorem. Let us conclude with a brief account of the problem whether the operations of taking Hilbert transform and differentiation may be interchanged, i.e., under what conditions the relation (f")'= (f')"is valid.
pi'
Proposition 8.3.7. Let f E LP, 1 < p c 00. Then f E W:P if and only i f f " E W ~ Pand ; iff($ = $(x) a.e. with 4 E AC,,,, 4' E Lp a n d f " ( x ) = $(x) a.e. with $ E AClOo #' E Lp, then (4')"(x) = $'(x) a.e. P f(x) = $(x) a.e. with 4 E ACloc, 4' E Lp, then Proof. Let 1 < p 5 2. I f f E W ~and iu fA(zl) = [4']"(u) by Prop. 5.2.19. Sincef", (6')" E LP by Theorem 8.1.12, it follows P Theorem 5.2.21, and by Prop. 8.3.1 that iulf"]"(u) = [(4')"]"(0). Thusf" E W ~by iff "(x) = $(x) a.e. with $ E AC1,,, $' E LP, then $'(x) = (+')"(x) a.e. The converse assertion follows by the same reasons. If 2 < p < 00, then the proof is obtained by a duality argument which however is left to Problem 8.3.5.
8.3.4 NormConvergence of the Fourier Inversion Integral We shall finally give an application of the theory of Hilbert transforms to the theory of Fourier transforms of functionsf€ LP, 1 < p I2. We shall show that the inversion formula (5.2.37) already established for p = 2 also holds for 1 < p < 2. This is the given by famous result of E. HILLEand J. D. TAMARKIN Theorem 8.3.8. Iff
(8.3.19)
E
Lp, I < p I2, then
330
HILBERT TRANSFORMS
Proof. By (5.2.18) we have (cf. Problem 5.1.5)
where the righthand side exists by Holder's inequality as an ordinary Lebesgue integral for every x E R. Hence by Theorem 8.1.6 S ( f ; x ; p ) = sin px
[f(o)
cos po]"(x)  cos px
sin po]"(x) a.e.
[f(o)
By (8.1.39) this implies ( P > 0).
IISM 0 ; P ) l l P 5 2MPIlfIIP Given E > 0, there exists a functionfi
E
Soo such that f = fi
+ fa with Ilfall, < + ( 2 + ~1) e,~
E,
and since
IlW; P )  f(o)IIp 5 I l ~ ( f i ; 0 ; p)  fXo)IIp it suffices to prove (8.3.19) for functionsfi E Soo. Moreover, it suffices to consider the case fi = K[a,b]. For such fl we have 0 ;
(8.3.20) Let us choose A such that A > max { l a ! , Ibl}. Then, if x > A, by the second mean value theorem (cf. Problem 1.2.10), we obtain for some with ( x  b)p < 5 (x  a)p
which implies IS(fl; x ; p)I
which tends to zero as p Thus it remains to show
< 2/7r(x  b)p. Therefore

(8.3.21)
Similarly SI, IS(fl; x ; p)  fi(x)Ipdx = ~ ( l for ) p
03.
ISM; x ; p )
/;A
 h ( X ) l P dx = o(1)
(P
+
03.
a).
But it follows from (8.3.20) (cf. Problems 1.2.10(i), 5.1.2(iv)) that S(fl; x ; p ) is uniformly bounded in x and p and tends tofi(x) for x # a, b as p + 03. Therefore (8.3.21) follows by Lebesgue's dominated convergence theorem, and (8.3.19) is established.
Problems 1. Prove Cor. 8.3.5. 2. (i) Show that for f, f"
E
L1
1
{  i sgn u}f"(u) elxudu a.e. d2m  m (ii) As an application of (8.3.22) (or Prop. 8.3.2) evaluate the Hilbert transform of the kernel of CauchyPoisson (cf. Problem 8.1.2), Fejer (cf. Problem 8.2.4), Weierstrass, Jackson etc. (see also GOLINSKII[2]for further examples). (iii) Show that iffe Lp, 1 < p 5 2, then
(8.3.22)
f"(x) =
Y
pu  sinpu Ua
1 du = 
d!G 
(1 
F)
{isgn v}f"(u)e'xudu.
HILBERT TRANSFORMS ON THE REAL LINE
331
3. Show that for f E Lp, 1 5 p I2, the following assertions are equivalent : (i) fE(W*)[p, (ii) f t z W[LP;{isgnu}(iv)'], (iii) there exists g E LP such that
f "(x)
=
1" :1 m
dul
dua . . . du,, / : i l g ( u , ) du, a.e.,
the integrals existing in the sense of Theorem 5.2.21 (apart possibly from the first one in casep = 1). 4. State and prove corresponding results for the classes W[Lp; lulr], V[Lp; [ul'], r E N, (Hint: Distinguish the cases r even, odd) 5. (i) Complete the proof of Prop. 8.3.7. (Hint: HILLE[3]; see also Sec. 10.6, 13.2) (ii) State and prove the counterpart of Prop. 8.3.7 for the rth derivative (see also BUTZERTREBELS [2, p. 191). (iii) Under suitable stronger hypotheses state and prove corresponding results in L1space. (Hint: BUTZERTREBELS [2, p. 44 ff]) 6. (i) Let f E L'. Show that f" E L1 necessarily implies J?a f(u) du = 0. (ii) Show that the set { f L'~ I f " E L'} is not dense in L'. (Hint: KOBER111)
8.4 Notes and Remarks Although many of the results on the Hilbert transform presented in this chapter are more or less standard, they are somewhat scattered in the literature. The main references are TITCHMARSH [6], ZYGMUND [7], HEWITT[l], WEISS[l], and the literature cited there. We again emphasize that TITCHMARSH, for example, regarded the Hilbert transform as a transform rather than a function, in contrast to our point of view. The Hilbert transform will be required here to solve equations of type 101 f"(u) = gA(v), our final aim in this chapter; in subsequent chapters this will be important for our investigations on saturation theory. Sec. 8.1. The proof of Theorem 8.1.5, which not only gives the existence a.e. of the HilbertStieltjes transform but also that it is of weak type, is due to LOOMIS [l]. Whereas the existence a.e. of p c ( x ) of p E BV is connected with the name of S. POLLARD [l], the weak type property (8.1.9) is often referred to as a KOLMOGOROV bound for p c . The present treatment studies the Hilbert transform along strictly real variable lines, an aspect for which we also refer to HARDY [l], COSSAR [11, and the fundamental (higher dimensional) investigations of CALDER~NZYGMUND [1,2], C A L D E R ~[2], N ZYGMUND [6]. (See also DUNFORDSCHWARTZ [lII, pp. 104410731.) There are other proofs for the PLESSNER result on the existence off for f E L1 which depend on complex function theory (see e.g. TITCHMARSH [6, p. 1321) or on a reduction to the (periodic) Hilbert transform off E L,: (cf. e.g. ZYGMUND [7II, p. 242 ff]). There is a further proof by STEINWEISS [l] to the effect that the Hilbert transform of f t z LP, 1 Ip < a,is of restricted weak type ( p , p ) , 1 5 p < a, (see also [l, p. 240 ffl). BUTZERBERENS Theorem 8.1.7 is standard, e.g. TITCHMARSH [6, p. 121 ff], WEISS[l, p. 49 ffl. The Lptheory for 1 < p < a, was developed by M. RIESZ[l] (cf. e.g. TITCHMARSH [6, pp. 1321431, HEWITT[l, p. 195 ff]). The present proof rests upon the theorem of MARCINKIEWICZ (cf. also WEISS[l, p. 711). In our approach, in which the Hilbert transform f" o f f E Lp, 1 < p < 00, was first defined as a pointwise limit through (8.0.2), it must of course be shown that the extension of T i n (8.1.23) to all of LP as given by the theorem of Marcinkiewicz coincides with f . Our proof leading to Theorem 8.1.12 was suggested by work of WEISS[l, pp. 4 6 5 6 , 721. For Lemma 8.1.10 compare also TITCHMARSH [6, p. 1241.

332
HILBERT TRANSFORMS
Theorem 8.1.12 breaks down for p = 1. Indeed, as we have seen, i f f € L1, the Hilbert transform f " need not be integrable, even locally. However, A. KOLMOGOROV has shown that i f f € L', then If(x)la is locally integrable for every 0 < a < 1 (for extensions see KOBER[l], KOIZUMI [ l , 21, and the literature cited there). There are many investigations concerning the class of functions f for which 1 f(x)l(l + log+ If(x)l) belongs to L1. For this class one can prove the local integrability o f f " (compare ZYGMUND [5, p. 1501, KOBER [2], CALDER~NZYGMUND [l]). In view of the fact that we wish to have the property (8.0.4) at our disposal, we commence with f " E L'. The Hilbert transform can also be defined for more general classes of functions. Following ACHIESER [2, p. 1591, let Wa be the classt of all measurable functions f for which f(x)/(l 1x1) E L2. (For instance, every function belonging to L&, La or C is contained in Wa.) Let 77 be the L2Fourier transform off(x)/(x  i). Then by (5.2.38)
+
(a)
d G f"
f(x) = (x  i) 1.i.m. D+m
u ) etxudu.
D'(
In view of (8.3.22) and the theorem of Plancherel a generalized Hilbert transform may be introduced through
j"/isgn
f"(x) = (x  i) 1.i.m. P+m
d%
u}q(u) elxudu.
It then follows by the Parseval equation that
[2, p. 1601 shows that the above definition is essentially Iff belongs to L& or L2, ACHIESER equal to the usual one. On the other hand, since C = Wa, the definition may be used to associate a Hilbert transform with every continuous bounded function. Then an analysis quite similar to that in LP, 1 I p < co, may be developed. Further generalizations are due to KOIZUMI [I, 21. He introduces a Hilbert transform for f~ Wa through
f"(x) = Moreover, he considers Hilbert transforms on classes of functions f for which If(x)lP/(l [xiu)E L' for somep 2 1, a > 0. There is also a theory of discrete Hilbert transforms. Let {ak}&',,. be a twoway infinite sequence of numbers. Then the Hilbert transform of {ak}is defined by
+
where the prime indicates that the index k = n is omitted. A classical result asserts that \la" 112 I n llalla for every a E la. The result was extended by M. RIESZwho showed that ]la"\Ip I Mp llallPfor every a E Ip, 1 < p < co. See ZYGMUND [a] and, for further generalizations, KOIZUMI [l, 21. Hilbert transforms in connection with distribution theory may be found in BELTRAMIWOHLERS 11, 21, G~~TTINGER [l], HORVATH [l], LAUWERIER [l], NEWCOMB [l], TILLMANN [ l , 21. See also Vol. I1 of this treatise. Sec. 8.2. For the Hilbert formulae of Prop. 8.2.1 see e.g. Titchmarsh [6, p. 1381. The proofs are similar to the arguments used for the Parseval formulae for LPFourier transforms (Prop. 5.2.13). Prop. 8.2.2 is preliminary to one of our main results, namely Prop. 8.3.1. The proof presented is due to A. P. C A L D E Rand ~ N was communicated to the authors by G. WEISSin Nov. 1965. For a paper largely concerned with the set {YEL' I f ,. L') we
t We recall that the class Wa plays an important rBle in the generalized harmonic analysis of BOCHNERWIENER, see e.g. WIENER[2, p. 138 ff], BOCHNER [7, p. 138 ff].
HILBERT TRANSFORMS ON THE REAL LINE
333
refer to KOBER[I]. For conditions sufficient to ensure f * E L' see also KONIG[I]. Prop. 8.2.58.2.7 in this general form do not appear to be stated elsewhere; they are reminiscent of work on singular integrals (compare Sec. 1.4 and 3.2) and are suggested by some remarks of HARDYROGOSINSKI [l, p. 981 in the periodic case. Of course, the applications given to the integrals of CauchyPoisson (Cor. 8.2.8) and Fejtr (Problem 8.2.4) are wellknown and contained in all texts cited here. For the Privalovtype result of Prop. 8.2.9 we refer to BUTZERTREBELS [2, p. 201 and the literature cited in Problems 8.2.58.2.7. For the second footnote to Prop. 8.2.9 compare also BUTZER[4]. For Prop. 8.2.10 see e.g. TITCHMARSH [6, p. 1321 or WEISS[ I , p. 491. Sec. 8.3. Concerning Prop. 8.3.1 for 1 < p < 2 see HILLE[3]. The case p = 1 is the
difficult one; see the notes to Sec. 8.2 above. This material is also covered in BUTZERTREBELS [2, p. 21 ff] and BUTZERBERENS [ I , Sec. 4.21. There is considerable literature on allied integrals. TITCHMARSH [6, p. 1471 e.g. shows that the allied integral is (C, a) summable tof"(x) a.e. for every oc > 0. Thus in particular Cor. 8.3.5 is wellknown. GOLINSKII [ I , 21 defines a further transform through the allied integral (8.3.5) and compares it with the traditional definition (8.0.2). In particular, he discusses applications to Prop. 8.3.2. For Theorem 8.3.8, due to HILLETAMARKIN [I], see also TITCHMARSH [6, p. 1481 and HEWITT[l, p. 1991. Concerning Theorem 8.3.6 in the more difficult case p = 1, see BUTZERTREBELS [2, p. 411; first results are due to HILLE[3], COOPER [2]. This calls to mind the results of BOCHNERCHANDRASEKHARAN of Sec. 5.1.3. For extensions of Theorem 8.3.6 to Lpspaces, 2 < p < 03, we refer to the treatment of Sec. 10.6 and 13.2. In our future investigations the classes W[Lp; IuI"], V[Lp; Iul"], 1 i p 5 2, a > 0, shall play the key rde. Since these are defined in terms of the Fourier transform off, the problem is to find equivalent characterizations in terms of the original function f. If r is an even positive integer, the results of Theorems 5.1.16, 5.2.21, 5.3.15 will do. If r is odd, we may apply Theorem 8.3.6. For extensions to fractional a > 0 we refer to Chapter 11. Prop. 8.3.7 is due to HILLE[3]; see also TITCHMARSH [4] and BUTZERTREBELS [2].
Hilbert Transforms of Periodic Functions
9.0 Introduction
This chapter is concerned with the periodic version of the theory presented in the preceding chapter. We commence with our definition of the conjugate function, thus with the correct interpretation of
1’
(9.0.1)
f “ ( x ) = 2x * f ( x
 u ) cot U2 du
considered in the informal approach of the introduction to this Part. Definition 9.0.1. The Hilbert transformt (or conjugate function) of a function f E X,, is dejined as the Cauchy principal value of the integral (9.0.1) whenever it exists, i.e. f  ( x ) = ( H f ) ( x )is defined as
(9.0.2)
f “(x) = PV[L 2rr
1’’
f(x
 u ) cot 2 du = d +lim f;(x) O+
]
at every point x for which this limit exists. Hereby,
(9.0.3) f f ( X ) = 2rr
1 dslulsn
f(x
U  u)cot2du
1
= 1
Zrr
, [ f ( x  u)
 f ( x + u)]cotph4. U
d
Again, J,“(x) is welldefined everywhere as a bounded periodic function for every f E X,, and 0 < S < rr. The limit in (9.0.2) will not only be interpreted in the pointwise sense, but also in the norm. Iff E Lp, 1 5 p c 03, the Hilbert transform f ” was defined as the convolution off with the ‘kernel’ { @ x  l } , the integral being taken in the Cauchy principal value
t The Hilbert transform (8.0.1) o f f € LP, 1 5 p < 00, as well as the Hilbert transform (9.0.1) o f f € Xznare denoted by the same symbol f  and, generally, by the same name. But there is no danger of confusion since the class of functions under consideration specifies the meaning o f f ” .
HILBERT TRANSFORMS OF PERIODIC FUNCTIONS
335
sense (Def. 8.0.1). Recalling Sec. 3.1.2, we used results on meromorphic functions in order to convert the periodic singular integral of FejCr into its nonperiodic version, thus to unroll results on the circle group onto the line (and conversely). The expansion?. m 1 z l 1 p o t 2 = Z k =2’ a, (9.0.4) (z = x + iy)
+
{n
of the periodic ‘Hilbertkernel’ (cot (42)) strongly suggests that many of the fundamental properties concerning the Hilbert transform on the circle group may be obtained by reduction to those on the line group. This program will be outlined in Sec. 9.1.1, concerned with the existence of the transform. But to evaluate the Fourier transform of the conjugate function, we already make use of different methods, in particular using the fact that the exponentials {exp (ikx)}, k E Z, form a fundamental set in X2z.This has the disadvantage of being somewhat unsystematic, but it has the advantage of avoiding duplication of proofs and of exhibiting the simple structure of harmonic analysis on the circle group. While Sec. 9.1.2 is reserved to Hilbert formulae, Sec. 9.2 deals with the convergence problem of conjugates of singular integrals, the treatment being parallel to Sec. 8.2.2, 8.2.3. Finally, Sec. 9.3 is concerned with summation of allied series and normconvergence of Fourier series as well as with characterizationsof the classes W[X,,; {  i sgn k}(ik)’], V[X,,; {  isgn k}(ik)’],r E N. The latter result enables one to characterize the classes W[X,,; lkIr], V[X,,; lkIr] which will be of interest in our later investigations in saturation theory.
9.1
Existence and Basic Properties
9.1.1 Existence We fist prove a fundamental result due originally to N. LUSINand I. I. PRIVALOV. Theorem 9.1.1. The Hilbert transform f
 o f f E X,
exists almost everywhere.
Proof. According to (9.0.4) we have for 1x1 IT
To each f E X, we define fi on R by h ( x ) = K [  ~ , , ~ , , ( x ) ~ ( xThen ). $1 E L1 and f ( x  u ) = h ( x  u) for all 1x1, IuI 5 T.By the definition of& (cf. (9.0.3)) we have that the limit of
t This wellknown Laurent series expansion may be found in any text on complex function theory, cf. e.g. HILLE[4, p. 2611. The prime indicates that the value k = 0 is omitted. Note that the series converges absolutely and uniformly in every compact set of the complex plane not containing any of the points Zvk, k EZ,k # 0.
336
HILBERT TRANSFORMS
(9.1.2) h"(x) = n
s
u)du
for 6 3 O+ exists for some 1x1 I n if and only if Il(x) converges, since Iz(x) is everywhere convergent by (9.1.1). Moreover
(9.1.3)
IhW
 Il(X)l
I(43)llf
Ill.
On the other hand, if we replace the intervals 6
I IuI I n in Il(x) by 6 I IuI < co, we obtain V;];(x) (cf. (8.0.3)). It follows for 1x1 I n that I,(x) converges as 6 3 O+ if and only if f;(x) exists. Moreover,
Collecting the results we see that for 1x1 In the Hilbert transform f "(x) off E X,, exists if and only if the Hilbert transform f ; ( x ) offi E L' exists, which, according to Theorem 8.1.6,is the case for almost all x E R. Thus, sinceh" is periodic, f "(x)exists as a periodic function for almost all x. Proposition 9.1.2.
Iff E Lin, then there exists a constant M such that
(9.1.5)
meas {x E [n, 7 4 I
M I f "(41> Y > 01 I Ilf Y
Ill.
Proof. We may suppose that 11f [I1 = 1. If again fl = ~ [  , ~ , , ~then , f , by (8.1.9)there exists a constant A such that (9.I .6)
A
meas{x I If1(x)l > Y > 01 I  Ilfilll. Y
On the other hand, by (9.1.3)and (9.1.4)
(9.1.7)
If
"(x)  f X X ) l
I2
Ilf
1 1 1
(x E [
=, 73).
Therefore we see that for all y > 4 the set of points x E [ n, T]for which If (x)l > y is contained in the set of points x E R for which Ifl"(x)l > y / 2 , and so by (9.1.6)has measure not exceeding 4A/y. This gives (9.1.5)with constant 4A provided y > 4. If we choose M = max {4A, 8n}, then M y  l 2 2n for all 0 < y I 4, and (9.1.5) is trivially satisfied.
Theorem 9.1.3. For f E LZn, 1 < p < there exists a constant M p such that
00,
the Hilbert transformf
(9.1.8)
Ilh
IIP
5 MPIlf IIP
(9.1.9)
Ilf 
IIP
5
MPIlf IIP'
 belongs to LZ,,
and
(0 < 6 < n),
HILBERT TRANSFORMS OF PERIODIC FUNCTIONS
337
Proof. Iffi is defined as above, we obtain by (9.1.3), (9.1.4), and (8.1.36) that for any 0fELip*(Lh,; 1)Thus the problem of characterizing the subclass of functions f E X,, for which IIun(f;  f(0)IIXon= O(nl) is still unsolved, even in Lg,, 1 < p < a.For the solution the reader is referred to Chapter 12. 0)
Problems 1. Prove Cor. 9.3.4. 2. State and prove counterparts of Problem 8.3.2 for periodic functions. 3. Show that for f E Xan the following assertions are equivalent: (i) f E (W &on, (ii) f E WXa, ;  i sgn kI(ikY1, (iii) there exist g E Xan and constants ajE C, 0 I j Ir  1, such that
4. State and prove corresponding results for the classes W[Xan; lkl'], V[Xan; lkl'], r E N. (Hint: Distinguish the cases r even, odd) 5. State and prove counterparts of Prop. 8.3.7 and Problem 8.3.5 for periodic functions. 6. Let f E Lg,, 1 c p < 00. Show that there is a constant A, such that for the nth partial sum of the Fourier series off: IISn(f;o)IIp I A , ~ ~ f ~ ~ , . 7. Let F(x) be 2aperiodic and the indefinite integral of a function f E LP, 1 5 p c a. Show that IlS,,(F;0 )  F(o)II, 5 A n  l IIS,,(f;  f(o)IIp if p > 1, and in case p = 1 IIS,,(F;0)  F(o)IIp 5 A(1 log n)nl IIS,(f; 0)  f(o)[, (Hint: QUADE[l, p. 5421)
+
0)
HILBERT TRANSFORMS OF PERIODIC FUNCTIONS
353
9.4 Notes and Remarks Most of the notes and remarks to Chapter 8 also apply mutatis mutandis to Hilbert transforms of periodic functions. The main general references to the subject are ZYGMUND [5, 71, HARDYROGOSINSKI [ll, BARI[I], HEWITT[l], WEISS[2], EDWARDS [ll, KATZNELSON [I]. The present choice of material has essentially been dictated by the condition that the topics in question once again exhibit the common structure of harmonic analysis on the line and circle group, and that they lead readily to our main result, namely Theorem 9.3.5. Sec. 9.1. For the reduction of the basic properties of Hilbert transforms of periodic func
tions from those of Chapter 8 we have followed CALDER~NZYGMUND [2], a paper mainly interested in extensions to the several variable case. For a given 'kernel' k(x) on the line group one associates a periodic 'kernel' k*(x) defined through m
(9.4.1)
k*(x) = k(x)
+ 2'
I=m
{k(x
+ 27rj)  k(27rj)I.
The procedure assumes that k possesses certain properties that are transferable to k*. Thus, for example, (9.0.4) is the case k ( x ) = ( I / x ) of (9,4.1). In Sec. 11.5 we shall discuss the situation when k(x) is equal to I x I a  ' , {sgn X } I X I ~  ~ , and K ~ , , , ~ ) ( X ) X ~respectively. ~, In case k E L' we refer to the parallel treatment in Sec. 3.1.2. Other proofs for the existence of the conjugate function may be given by complex methods (see e.g. ZYGMUND [7, Chapter VII], KATZNELSON [I, p. 62 ff]) or by a separate application of the theorem of Marcinkiewicz (see e.g. EDWARDS [III, p. 168 ff]). According to Problem 9.1.3, the Hilbert transform is not a bounded linear transformation of X2, into X2, if Xznis C2, or Liff. However, one can show that f E L b implies I f "(x)lRE L:, for every 0 < a < 1 . Moreover, if If(x)l log+ If(x)l E Li,, then f * E Liff; see ZYGMUND [5, p. 1501 and the literature cited there (for further extensions cf. also O"EIL [l]). Hilbert formulae in this or another form are contained in all texts cited. Sec. 9.2. The same comments as to Sec. 8.2.2, 8.2.3 apply here. See also the references
given in the Problems. Prop. 9.2.7 is due to TABERSKI [3] who considers asymptotic relations (of higher order) for summation processes of Fourier series. He also discusses the asymptotic behaviour of the quantity sup llI;(fi  f(0)llcan, where the supremum is to be taken over all f E Lip (Czff;a), 0 < a I 1, with Lipschitz constant M I 1. For the application to the AbelPoisson integral we refer to the earlier result of SZ.NAGY[4] (see also NATANSON [3], TIMAN[l]). The asymptotic behaviour of the singular integral of Fejtr is considered by ZAMANSKY [ I ] (cf. also SUNOUCHI [lII]). Problem 9.2.6 establishes the Bernstein inequality for the conjugate of a trigonometric polynomial and plays an important r61e in the theory of best approximation. Indeed, given a certain behaviour of the best approximation En(X2,;f) off as R > co, we have deduced smoothness properties off in Sec. 2.3. It is also possible to deduce those for the conjugate f . The interested reader is referred to BARISTECKIN [l], S T E ~ K I[2], N BUTZERNESSEL [l], and in particular to TIMAN [2, p. 389 ff] and the literature cited there. For corresponding Jacksontype results see TIMAN[2, p. 315 ff]. 0)
Sec. 9.3. Conjugate Fourier series are standard topics of all texts on trigonometric series.
In particular, Cor. 9.3.4 is wellknown (cf. the references given to Problems 9.2.1, 9.2.2). Concerning (C, a) summation of conjugate Fourier series, see ZYGMUND [I]. The informal approach as given in the introduction to this Part may find a rigorous interpretation in the setting of distribution theory as, for example, in the modern treatment of EDWARDS [III, p. 86 ff]. For the characterizations of the classes (W)s2,, ( V * x 2 , see also GORLICH [I], BUTZERGORLICH [I]. These results pave the way for equivalent descriptions of the classes W[X,,; lkl'], V[Xzff;lkl'], r E N, in terms of the original functions or their conjugates. These classes play an intrinsic rale in our later investigations on satiiration theory. 23F.A.
354
HILBERT TRANSFORMS
In this section, as throughout the volume, the order of performing the conjugate operation and differentiation is always fixed. For conditions which ensure interchange (and its correct interpretation) compare Prop. 8.3.7, Problem 8.3.5 (see also BUTZERGORLICH [l I). Theorem 9.3.6 is the periodic version of the famous result of HILLETAMARKIN [l], compare also ZYGMUND [71, p. 2661, HEWI~T 11, p. 1401. For the Parseval formula (9.3.22) see e.g. ZYGMUND[71, p. 2671, Hewrrr [l, p. 1421, for Prop. 9.3.8 and Cor. 9.3.9, 9.3.10 QUADE[l] and the literature cited there.
Part IV Characterization of Certain Function Classes
The function classes that are to be characterized in this Part are those that arise in connection with saturation theorya central theme of the later chapters. These classes are actually Favard (or saturation) classes associated with convolution integrals, thus precisely those functions which furnish the best possible (optimal) order of approximation by a given convolution integral. The function classes are considered both for functions defined on the circle group (the 2aperiodic case) as well as on the line group. But the classes are also of independent interest. Indeed, our study (in the integral case) may be motivated as follows. In Chapter 2 we characterized the class Wztn, r E N, 0 < a < 1, in terms of the best approximation E,(C,,;f). In particular,fE Wztnif and only if E,(C,,; f) = O(nra) for 0 < a < 1, whereas for a = 1 no simple characterization in terms of En(C,,;f) was possible. On the otherhand,fE Wz&implies Ilf(‘)(o + h)  f(r)(~)I(Con = o(h) in case c( > 1, and thus f is a constant. From this point of view a = 1 may be regarded as an extremal index for the Wgclass. We recall that to bypass the difficulties in the characterization of the function class { f C,, ~ I E,(C,,;f) = O(n.’)} we introduced the Zygmund class *Wgn (but then a = 2 is an extremal index). Here, on the other hand, we shall characterize the original class Wttn for a = 1 not in terms of the order of best approximation but in terms of other properties upon$ In particular, f E W2.& if and only if the Riemann derivative (cf. Sec. 5.1.4) of order (r + 1) offexists in Cannorm. In Sec. 2.42.5 we considered equivalence theorems for particular convolution integrals in the case of ‘nonoptimal’ approximation. Thus for the integral of FejCr Ilu,(f; = O(n=) for 0 < a < 1 if and only i f f € Lip (X,,; a). In this equivalence theorem the order of approximation is ‘nonoptimal’. In case it is ‘optimal’ we recall that Ilu,(f; 0)  f(o)IIxan = o(nl) implies f is constantwe speak of a saturation theorem, and theorems of this type are one object of Part V. For the given example, the optimal order is O ( n  l ) and this order is reached precisely by those functions which make up the saturation class. Such classes are to be studied here. Saturation theorems may also be viewed as extremal cases of the corresponding 0)
f(0)IIXan
356
CHARACTERIZAHON OF CERTAIN FUNCTION CLASSES
results on nonoptimal approximation in the sense that a = 1 is the extremal index for Fejdr's integral. The classes are also to be distinguished according to whether they are associated with smallo or large0 approximation theorems. In the case of periodic functions this gives rise to the classes W[X,,;
M)]= {fE
and {fE
{
XZ,
I rcl(klf^(k) = g w , g E XZYJ
cz, I $ ( ~ l f W= g^(k), g E
LiJI I $(~lf^(k)= P W , P E BV,,} Lg, I wwA(k) = g w , g E L%,) (1 < P < m), respectively, where 4 is a given function on Z. (Actually, we would like these classes to be defined in terms of the functions f directly, instead of through their Fourier transforms.) The instance that h,t is some power of k is the most interesting one. Chapter 10 is concerned with the case that the power a is integral, Chapter 11 with fractional a. V[X,,;
+(Wl =
{fE
(fE
10 Characterization in the Integral Case
10.0 Introduction
The classes W[X,,; (ik)'] or Wk,,, (cf. Theorem 4.1.10) are connected with various generalizations of the classical rth derivative. In Sec. 5.1.4 we have shown that the concept of a Riemann derivative of order r of a function f at some point xo is more general than that of the Peano derivative (off at xo), which in turn is more general than that of the Taylor derivative and this again being more general than the (ordinary) derivative of order r off at xo. We shall first prove the (apparently surprising) fact that all four definitions of a derivative of order r off are equivalent to another provided limits are taken in the strong topology and not in the pointwise sense. In particular, a function f E X,, will belong to Wk,,, (we recall its pointwise definition in Sec. 1.1.2) if and only iff has an rth Peano derivative in X,,norm. We shall also consider ordinary and generalized derivatives in the weak sense, thus in the weak topology. We shall, for example, prove the (seemingly perplexing) fact that the weak Riemann derivative of order r offexists if and only if its strong (ordinary) derivative of order r exists. Then both are equal. The classes V[X,,; (ik)'] or Vk,, (cf. Theorem 4.3.13) are connected with generalizaThus tions of some significant theorems of G. H. HARDYand J. E. LITTLEWOOD. f~ Xzn belongs to Vg,, if and only if for the rth Riemann difference ~ ~ A ~ f = ~~x2,, O(lh(')(h + 0) and in turn if and only if the quotient h'Aifis weak* convergent for h+0. Results of the type just discussed are also considered for the conjugate function f" o f f € Xzz. This enables one to give characterizations of the classes V[X2,; lkl'], reN. The W and Vclasses are themselves connected through the important concept of relative completion. Indeed, the completion of W[X,,; $(k)] relative to X,, is equivalent to V[X,,; $(k)] for an arbitrary function $ on Z. The counterparts of these results are also discussed for functions defined on the real line R. At this stage let us just mention that we modify the corresponding proofs at a number of points. In the case of the circle group we use Fourier transform methods together with weak* compactness arguments for C,, as well as Lg, for all 1 Ip < 00.
358
CHARACTERIZATION OF CERTAIN FUNCTION CLASSES
For the line group we employ Fourier transform methods plus the representation p I 2. However, for theorems of Sec. 6.1.2 for functions in LP, p restricted to 1 I p > 2 we must turn to dual arguments. Sec. 10.1 is devoted to a discussion on generalized derivatives as well as to characterizations of the classes Wka,,. Sec. 10.2 and 10.3 are reserved for characterizations of the classes Vs,,,, and (V”);ca,,, respectively. In Sec. 10.4 fundamental properties associated with the concept of relative completion are given. Sec. 10.5 is concerned p I 2, and Sec. 10.6 with the casep > 2. with the case of functions defined in Lp, 1 I
10.1 Generalized Derivatives, Characterization of the Classes WL,,,
In this section we introduce the (finite) Fourier transform method in its simplest form as applied to the Riemann derivative, for example. We recall Sec. 5.1.4 for the pointwise definition of this derivative as well as for its basic properties. From Problems 5.1.1 1,5.1.12 it follows that the existence of the ordinary derivative of order r of some function at some point xo implies the existence of the Riemann derivative of order r at xo. But the converse assertion is not necessarily true. But in replacing the pointwise limit in (5.1.42) by the limit in norm of the spaces X,, we obtain a type of generalized derivative which in some sense has lost some of those special properties which distinguishes it from the ordinary derivative. We shall prove for example that the existence of the rth Riemann derivative in the norm of Czn is equivalent to the existence of the continuous rth ordinary derivative. 10.1.1 Riemann Derivatives in X,,Norm For a function f defined in a neighbourhood of a point xo the rth Riemann derivative at xo with respect to the rth onesided difference (10.1 .l)
A i f ( x ) = k  0 ( l);k(L)f ( x
+ kh)
(h
E
w
is defined by (10.1.2)
f M(xo) = lim h+O
1
h
A; f(xo),
in case this limit exists. Throughout this chapter Riemann derivatives are defined via the difference (10.1.1) unless otherwise stated. We shall also simply refer to Ai f as the rth difference unless we explicitly distinguish between central, forward (h > 0) or backward (h < 0) differences (cf. (5.1.41)). We prefer the definition via the difference (10.1.1) since for r = 1 it reduces to the first ordinary derivative and may therefore be considered as a more genuine generalization of the concept of an rth derivative than (5.1.42). Definition 10.1.1. Iffor f E Can there exists Df‘Y. Can such that (10.1.3)
359
CHARACTERIZATION IN THE INTEGRAL CASE
then D r y is called the rth uniform Riemann derivative o f f . Similarly, i f f and Drlf are functions belonging to LZny 1 Ip < 03, and if(10.1.3) holds with the Cannormreplaced by the L5,norm, then we speak of DSIfas the rth Riemann derivative off in the mean of order p . Both derivatives will also be called rth Riemann derivatives in Xannorm or strong Riemann derivatives? of order r. Obviously, the existence of the pointwise Riemann derivative at every xo does not necessarily imply that of the strong Riemann derivative. But the existence of the rth uniform Riemann derivative DrIf of a functionf E Canalso implies that of the pointwise limit (10.1.2) at every xo, whereas this is not so for the Riemann derivative in the mean of order p. Although this reveals that there is a difference between the uniform Riemann derivative and the Riemann derivative in the mean of order p , both do have properties in common which will be illustrated by the following results. As an introductory example of the problems which may be treated by the (finite) Fourier transform method we consider the relationship between the continuous second ordinary derivative off E Can and the second uniform Riemann derivative of f. Our first result states that these two concepts are equivalent.
Proposition 10.1.2. The function f E Can has a continuous second ordinary derivative i f and only if it has a second uniform Riemann derivative. In this event, (DiaIf)(x)= f (2)(x)for all x.
Proof. (i) Supposef ( 2 )exists and belongs to Can. Then by the mean value theorem f(x
+ 2h)  2 f ( ~+ h) + f ( X ) = hzf@)(x+ 28h)
(0
2 we have
Theorem 10.1.3. Thefunction f E Canhas a continuous rth ordinary derivative ifand only i f it has an rth uniform Riemann derivative. In this case, all Riemann derivatives of lower order exist in C,,norm and (DijIf)(x) = f( j ) ( x )for every x, j = 1,2, . . .,r. The proof of this theorem which is essentially that of Prop. 10.1.2is left to Problem 10.1.1. As a consequence, Theorem 10.1.3 contains the following sufficient condition for f E Canto have a continuous rth ordinary derivative. Corollary 10.1.4. Iff and g both belong to Can,and iffor some integer r > 0 1 lim  A; f ( x ) = g(x) h0 h' uniformly for all x , then the rth (ordinary) deritiative fcr) exists, is continuous, and f(')(x) = g(x)for all x. In particular, g(x) = 0 impliesf ( x ) = const. The latter assertion as formulated with respect to the central difference (cf. Problem 10.1.2) is related to the wellknown lemma of H. A. Schwarz (see Problem 10.1.3). For periodic functions it states that iff E Czn and lim f ( x + h)  2fW + f ( x  h) = o h+O ha for each fixed x E R, then f is a constant on R. Thus Problem 10.1.2 in particular asserts that the lemma of Schwarz is valid under the stronger assumption that (10.1.4) holds uniformly. But Problem 10.1.2 holds for general r E N whereas the lemma of Schwarz fails to hold for r > 2. For further comments we refer to Sec. 10.7. On the other hand, Theorem 10.1.3 may be regarded as a further characterization of the class W& (cf. (1.1.16)). Indeed (10.1.4)
Corollary 10.1.5. Necessary and suficient for f E Can to belong to WLanis that f has an rth uniform Riemann derivative.
It is this latter aspect which may be carried over to Riemann derivatives in the mean of order p. In fact
Theorem 10.1.6. A function f E Lg,, 1 I p < 00, belongs to Wig, if and only ifit has an rth Riemann derivative in the mean of order p. In this case, all strong Riemann derivatives of lower order exist as well. Proof. (i) Let f E Wig,, 1 I p < 00, i.e., there exists 9 E ACI,;' such that f = a.e. a nd #'I E Lg,. Then integration by parts (rtimes) yields
+
(10.1.5)
A; f ( x )
=
loh... f," P(x+
u1
+ .  . + u,) du, . . . du,
a.e.,
CHARACTERIZATION IN THE INTEGRAL CASE
361
and hence by the HolderMinkowski inequality
which tends to zero as h + 0 in view of the continuity in the mean of #r). (ii) Conversely, we have by Prop. 4.1.1(i) that
[A; f]"(k) = (elhk l)y"(k), and hence ri
lim
h0
1"
1; 1 (k) A;f
=
(ik)'f"(k)
( k E Z).
The existence of DrIfimplies that
for each k E Z and so (ik)lf "(k) = [ D~If]"(k),k E H, which according to Prop. 4.1.9 shows that f E W:,.,. As an immediate consequence of the particular instance that DrIfexists and vanishes a.e. we note Corollary 10.1.7. lffor f E Lb,, 1 I p c co,
Il&f
IIP
=
(h * 01,
o(lhl')
then f = const a.e. Let us mention that in Definition 10.1.1 there was no restriction upon how h approaches zero. It may easily be seen that the preceding results also remain valid for the (seemingly more general) concept of a Riemann derivative obtained by restricting h to positive values and considering the lim inf as h + O+ (cf. Problem 10.1.7). 10.1.2 Strong Peano Derivatives
Another concept of a generalized derivative for which similar results may be established is the Peano derivative (cf. (5.1.45) for its pointwise definition). Extending the pointwise definition (5.1.44), an rth Peano direrence o f f E X,, is an expression of the form (10.1.6) where /k, 1 1 the methods of proof of HARDYand LITTLEWOOD [21, p. 599; 3, p. 6191 may be applied) 2. Show that if f~ Vl,,,, then for the central Riemann difference IIzkfIIxan = O(lhl'), h 3 0. Conversely, if there is a nullsequence {h,) such that lim inf,, ~ ~ @ , , f ~ ~< x z ,a, , then f~ VSan. 3. Show that f~ Xznbelongs to VSa, if and only if for the Taylor difference 11 Vifilx,,, = O(lh['), h + 0, providedf('l) exists and belongs to Xzn. 4. Show thatf E X z n belongs to V:,if and only if Iln~)(.f; = 0(1), n a. 5. The fact thatfE V[X,,; (ik)'] implies (IAkf((Xan= O(Ihlr), h  t 0, may be shown directly by using the uniqueness theorem and the fact that, e.g., for Xzn = C,, and k # 0 0)jlxan
f
6. Give another proof of the implication (ii) => (i) of Theorem 10.2.2 by considering the limit h + O of the quantity u,,(A'J; x) and using Problem 10.2.4. (Hint: See BUTZER[lo, p. 2941) 7. Show that if there is an rth Peano difference off E Xan such that I1OMllxa,= O(lhl'), h  t 0, then Dir"fexists. (Hint: cf. Lemma 5.1.24) 8. Let f E Can.Show that the following statements are equivalent: Ilh'ArhfIlCan = O(1)
(i)
(ii) there exists g E Lzn such that for every s E Lin
9. Let f E Lin. Show that the following statements are equivalent: (i) there exists an rth Peano difference offsuch that Ilh'oL.fllL:, = W) (h + 01, (ii) there exists an rth Peano difference off and a function p E BVzn such that
for every s E C2,.
371
CHARACTERIZATION IN THE INTEGRAL CASE
10.3 Characterization of the Classes ( V ");can
We establish results similar to those of the preceding section for the Hilbert transform f" o f f € X2,. Thus we are interested in some further characterizations of the classes (V"yx2nas introduced in (9.3.18).
Proposition 10.3.1. Let f E Xan.Thefollowing assertions are equivalent: (0 (ii)
f E
IIALf
(V>b,,
IIXan
=
'(lhl')
(h += 0).

Proof. (i) * (ii). By definition, the assumption f E (V"yxz,, means that f E Vkzn, and thus (ii) follows by the implication (i) s (ii) of Theorem 10.2.2 withfreplaced byf. (ii) 3 (i). For X, = LE,, 1 < p < 00, Theorem 9.1.3 implies f " E Lg,. Hence we may apply Theorem 10.2.2to the Hilbert transformf" in order to show thatf' E Vkzn. By definition this is (i). For the cases X, = Can, L:n we first observe that we do not explicitly assume f E X., This entails that we cannot apply Theorem 10.2.2 directly but have to proceed via Theorem 9.3.5. Thus let X,, = C,,. Then by Theorem 9 . 1 . 3 , f ~C,, yieldsf" E Lln for each 1 I q < 00. By the assumption Ilh'A;fIlm = O(1) and weak* compactness for L,", we deduce (cf. proof of Theorem 10.2.2)that there exists g E L,", such that (ik)"f"]"(k) = g"(k), k E H. Using (9.1.16) it follows t h a t f e (V")'c,, by Theorem 9.3.5. When X2, = Li, we first recall that, by Theorem 9.1.1, f~ Li, only says that the Hilbert transform f" exists almost everywhere. But condition (ii) assumes that at least the difference A; f " belongs to Liz, and since A;f" = (ALf)" a.e. and A i f ~Liz, we have by (9.2.24) (k E Z). [A;f"]^(k) = {  i sgn k}[A;f]"(k) Again as in the proof of Theorem 10.2.2 we may conclude the existence of a nullsequence {h,} and p E BV,, so that A
~ " ( k=) jlim  m [;A;J]
(k)
=
which by Theorem 9.3.5 ensuresf€ (V")[;,.
{isgnk} flim +m
[iA;,f]
h
(k)
372
CHARACTERIZATION OF CERTAIN FUNCTION CLASSES
We recall that in this section the order in forming the operation of Hilbert transforms and differentiation is always fixed since we do not investigate here the problem of commutativity of these operations, i.e., under what conditions the relation (f")")= (f"))"is valid (cf. Problem 9.3.5). Furthermore, let us mention that results similar to those of Prop. 10.3.1 also hold for central Riemann (5.1.41), Peano (10.1.6) and Taylor differences (10.1.14), respectively. We leave them to Problems 10.3.110.3.3 as they may easily be inferred from the results of this and the preceding section. Problem 10.3.5 contains a collection of characterizations of the classes (V");czn. We compile the analogous results concerning the classes (W")k,, in Problem 10.3.6. Problems = O(Ihl'), h + 0. 1. Show that f e Xznbelongs to (V"K2, if and only if 1lh.J2. Show that f E X2,, belongs to (V")ka,, if and only if there exists an rth Peano differencet off" such that 1 O l f  llXan = O(lhlr), h f 0. 3. Show that f E X,, belongs to (V"Xa, if and only if IIVM" ]Ixan = O(lh13, h + 0,pro
vided Cf")('l) exists and belongs to X2,.
4. Show that f~ X2, belongs to (V");,,, if and only if ll(o;cf; ~ ) ) ( ~ )=l l0(1), ~ ~ ~n 4 03. 5. Letfe Xzn. Show that the following statements are equivalent: (9 fe(V")xa,, (3f€V[X,,;{isgnk}(ik)'l, (ii9 Il(u;cf;~))(~)llX~~ = 0(1), n + a , (iv) Il%f IIXan= O(lhlr), h  t 0, (v) I I A i f " llxan = O(lh13,
(vi) there exists an rth Peano difference off" such that 1 OM" ~IX~,, = OClhl'), (vii) [IVif" l X l n = 0(lhlr), h f 0, provided Cf")('l) exists and belongs to Xan. 6. Let f E X2,. Show that the following statements are equivalent: 0) f e W#,,, (ii) f E W[Xm; {  i sgn M(iW1, (iii) there exists g E Xa, such that limb,,, IlhrAlff  gllxa,= O t , (iv) there exists an rth Peano difference of f" and a function g E X2, such that limh0
Ilh'ou" gIIxz,=
0,
II(0,"cf; ON(')  (0; cf; o))(') 1I xan = o(lh m,n + a. L e t f E LZ,, 1 < p < co. Show that IlAUfll,= O(lhlr) if and only if (v)
7. IlAlf" 1, = OClhl?. 8. Let f E LZ,, 1 < p < co. Show that there is an rth Peano difference of f such that 11 Olfll, = O(lhlr) if and only if there is an rth Peano difference off" such that 1 Oif" I!, = O(Ih1'). 9. State and prove characterizations of the classes W[Xg,; lkl'] and V[Xzn; lkl'], r E N.
(Hint: Use the characterizations of, e.g., Vka,,and (V")San; see also Sec. 11.5)
t The definition of
Peano differencesof the Hilbert transform f" of functions f belonging to assumes a slight generalization of our definition (10.1.6). Indeed, we assigned Peano differences to functionsfe La,, say, and defined 0;fas any expression of the form(l0.1.6), where 1, E La,, 1 Ik Ir  1. Replacing f by its Hilbert transform f" it is not necessarily true that f" E LI,. Nevertheless, we call Can or Ll, tacitly
an rth Peano difference of f,if 1, E Li,, 1 Ik 5 r  1, and at least [f"(x + h)  f  ( x ) ] E Liz for every h E R. Corresponding modifications have to be made for the Peano (and Riemann) derivative of the Hilbert transform f o f f € L1,. The same remarks are necessary in Cz,space, whereas in L!, 1 < p < to, there is no difference with the definitions involving since then f" E LE, by Theorem 9.1.3. g may be referred to as the rth strong Riemann derivative off. Here we again tacitly assume that at least ALf" belongs to Xan.
373
CHARACTERIZATION IN THE INTEGRAL CASE
10.4 Relative Completion
So far we have established characterizations of the classes W[X2,; I,@)] and V[X2,; $(k)] (for their definition cf. (4.1.24), (4.3.19)) separately for particular functions $. Now we shall briefly discuss how these types of classes are connected, the relevant notion being that of relative completion. Let X be an arbitrary Banach space endowed with norm II~IIx. A linear manifold Y of X is called a Banach subspace of X if it is a Banach space (with norm Ilolly) continuously embedded in X, in other words, if there exists a constant c such that 1 f (1' I c l l f l l ~ for allfE Y, or the identity considered as a mapping of Y into X is a bounded linear operator of Y into X. Without loss of generality we may assume in the sequel that Y is normalized (relative to X), i.e.
llfllx 5 llflly (fE Y). (10.4.1) Definition 10.4.1. Let X be a Banach space and Y a normalized Banach subspace of X. The completion of Y relative to X, denoted by,'y is the set of those elements f E X for which there is a sequence {fn}FS c Y and a constant M > 0 such that )If n IIY I M for all n and limn+m11f  fnllx = 0. In other words,'y is the set of all elements f E X which are contained in the closure in X of some bounded sphere in Y, i.e. (10.4.2) Here
mxdenotes the closure of Sy(p) in Xnorm (cf. Sec. 0.7).
Proposition 10.4.2. If X is a Banach space and Y a normalized Banach subspace of X, then is a normalized Banach subspace of X under the norm
v'
(10.4.3)
Ilfllyx = i n f b
'0 If€SY(P)'>
Proof. The fact that ?' is a linear manifold and I[f I(?x is homogeneous and satisfies the triangular inequality follows immediately from the observation that if f l E Sy(p1)' and f 2 E Sy(p2)', then alh + a2f2E Sy(lall p1 + la2[p2)'. Furthermore, in view of (10.4.1) we have llfllx I 11f (Ip. Hence (10.4.3) defines a norm, and is a normalized (normed linear) subspace of X. It remains to show that ?' is complete. Let {f n } be a Cauchy sequence in 0'. It is, a fortiori, a Cauchy sequence in X with limit f, say. Certainly, J EPx. For any e > 0 we can choose an n(e) such that f m  f n E Su(E>' for all m, n > n(e). Hence also f m  YE Sq', and thus f is the limit of {f,,} in the ?'metric.
v'
Proposition 10.4.3. If Y is reflexive, then the Banach spaces Y and ?' are equal with equal norms. For a proof as well as for further general properties of Yx we refer to the references given in Sec. 10.7.
374
CHARACTERIZATION OF CERTAIN FUNCTION CLASSES
We now turn to a discussion of the relation between the classes W[X,,; #(k)] and V[X,,; $(k)], illustrating the usefulness of the above concepts. In what follows, # is an arbitrary complexvalued function on Z. Proposition 10.4.4. (a) The class (10.4.4)
W[X2,; #(k)]
=
{ f X2, ~ I #(klf"(k)
= g"(k)7 g E
X2A
becomes a normalized Banach subspace of X,, under the norm (10.4.5)
IIfIIWIXan;&(k)I = IISIIX2n
I IIgIIX2n.
(b) The set Tn of (trigonometric)polynomials is dense in W[X,,; #(k)] (with respect to norm (10.4.5)).
Proof. (a) Obviously, W[X,,; $(k)] is a linear manifold of X,, on which a normalized norm is defined by (10.4.5). It remains to show that W[X,,; #(k)] is complete. 0 as Let {fn} be a Cauchy sequence in W[X,n; $(k)], i.e. 11fm  fnIIW[X2n;14(k)] m, n + co. If g, E X,, is such that +(k)f,"(k) = g;(k), k E Z, the former limit means IIfm fnIIx,, + IIgm  gnllX,,+ 0 as m, n+ 00. Thus {fnl and { g n l are Cauch~ sequences in X,,. The completeness of X,, now ensures the existence of two functionsf,gE X,, such that fn and gn tend to f and g in X,,norm, respectively. By (4.1.2) this implies that lim,+mft(k) = f"(k) and gc(k) = g"(k) for every k E Z. Therefore $(k)f"(k) = g"(k), i.e.fE W[X,,; $(k)]. (b) First of all, T n C W[X2,; $(k)]for every n. For, iff(x) = ,, c k exp {ikx}, then $(k)f^(k) = g"(k) with g given by g(x) =  n #(k)ck exp {ikx}. Therefore, let f~ W[X2,; $(k)] and g E X,, be such that #(k)f"(k) = gA(k), k E Z. For the Fejkr means off we have $(k)[a,(f; o)]"(k) = [un(g; 0)lA(k), and therefore f
xc=
x:=
Ilan(f;
1' .f(o)llW[X2,;14(k)l
which tends to zero as n
f
= Ilan(f;
+ Il'n(g;
'1 f(o)llX,,
 g(0)IIXzn,
co by Cor. 1.2.4. This completes the proof.
Proposition 10.4.5. The class
I #(k)f"(k) I #(k)f%) I#(k)fW
(10.4.6)
= g w ,g E = ~"(k),P E
L,",l BVZJ
= g w ,g E
LInl (1 < P < a).
becomes a normalized Banach subspace of Xan under the norm
IlfIIXIVan;@(k)]
=
i
+ llgllL&,
x2,
=
IIfIILa, IIPIIBVI~Y2', IlfIIq, + llgllL& x,,
=
IlfIIC2,
(10.4.7)
C2n
= L%,,
(1 < P
(ii) => (iii) follow by routine arguments we only prove (iii) * (i). For this purpose, we consider the FejCr means of h'ALf, obtaining by (3.1.17) IlALfIlP
=
Thus the hypothesis and the Parseval formulae (5.1.5) and (5.2.18) yield
On the other hand, by Lebesgue's dominated convergence theorem we have
for each x E R and p > 0. Hence by Fatou's lemma
so that Theorems 6.1.7 and 5.3.15 give f E VLp. In conjunction with Cor. 10.2.3 and Prop. 10.2.4 let us state Proposition 10.5.5. Let f
(ii) for p
= 1:
E LP,
1
5 p I2.
The following assertions are equivalent:
there exists p E BV such that for each s E Co f m
1
for 1 < p I 2: there exists g E Lp such thar
r m
CHARACTERIZATION OF CERTAIN FUNCTION CLASSES
380
We only indicate the proof of (ii) * (i) for p = 1. Taking s as in (10.5.6) we have as in the proof of Prop. 10.5.3
Therefore by Problem 3.1.9 uniformly for all p > 0
By Theorem 6.1.7 this implies the existence of Y E BV such that (iu)'f''(u) = v"(u) for all v E 08 (in fact, p and Y are equal). Assertion (i) follows by Prop. 5.3.14 and Theorem 10.5.4.
Theorem 10.5.6. Let f E LP, 1 I p 5 2. The following statements are equivalent: (i) f belongs to (V")[P, (ii) there exists an rth Peano difference? of the Hilbert transform f" such that
IIO i f  IIP = O(lhl?
(h 01, (h f 0). f
IlAJIf IIP = O(lhl')
(iii)
Proof. If 1 < p I2, then f E L P implies f"E L P by Theorem 8.1.12, and therefore the proof follows by Theorem 10.5.4 with f replaced by f". Thus it remains to prove the case p = 1 for which f E L1 a priori only implies the existence of f " almost everywhere (cf. Theorem 8.1.6). Let f E (V)[i, i.e., there exists E ACIG~with +(k) E L1, 1 I k Ir  1, +(rl) E BV such that f" = 4 a.e. By partial integration we have (cf. (10.2.9)) r  1 hk (10.5.7) f"(X + h)  f"(x) = 2 T # ~ ) ( x ) k=ik.
+
Since the righthand side of (10.5.7) belongs to L', Lf"(x function of L1 and therefore by (5.3.7)
s sup IUI L
Ihl
I]+('l)(o
+ h)  f "(41represents a
+ u)  C p r  1 ) (o)lllr Ihlr'
I r ~ ~ # ( r  lhl'. l)~~~v

Thus there exists an rth Peano difference of the Hilbert transform f " such that (ii) is satisfied. The proof of (ii) (iii) again follows by (lO.l.lO), and so in particular AJIf"E .'L Finally, let f be such that (iii) is satisfied. Since ALf E L1 by hypothesis, and since obviously AJIf E L1, we have by Prop. 8.3.1 that
[A; f ]"(u) = {  i sgn v}[A; f ]"(v) = {  i sgn u}(eihu 1)'f "(v) for all u E R. Since
t Cf. footnote to Prop. 8.2.9 and Problem 10.5.6.
CHARACTERIZATION IN THE INTEGRAL CASE
381
it follows as in the proof of Theorem 10.5.4 that 1
 !){isgnu}(+)

efhu
r
Ill
f^(o)eioudv = O(1)
(p
> O,h+0)
and
1 1 JIp~ I
(1
 !$){isgn
u}(iu)'f^(u)eioUdv111 = O(1)
(p
> 0).
Theorem 6.1.7 shows that there exists p E BV such that {  i sgn v}(iv)'f^(u) =p"(v), which in view of Theorem 8.3.6 implies f E (V)p. As in the periodic case, we may generally interpret the Vclasses as relative completions of the relevant Wclasses. We state
Proposition 10.5.7. Let f function defined on R.
E
Lp, 1 I p I 2, and $ be an arbitrary complexvalued
(a) The class WWP;W)l = { f E LP I
(10.5.8)
*(4f^(v)
= g w , g E LP)
becomes a normalized Banach subspace of Lp under the norm (10.5.9)
llfIIW[LP;@(u)I
=
11111P
+ IlSllP.
(b) The class
becomes a normalized Banach subspace of
LP
under the norm
(10.5.1 1)
(c) A function f belongs to V[Lp; +(u)] W[Lp; $(u)] relative to Lp.
if and only i f f belongs to the completion of
We finally formulate some characterizations of the classes V[LP; lulrJ. Corollary 10.5.8. euen (odd):
Let f
E

Lp, 1 5 p 5 2. The following assertions are equivalent for r
1~1'1, (ii) f~ W[Lp;
LP
, (iii) f E V[P((V)[p), O(lhlr), htO,un&r theadditionalassumptionrhatf(rl)(Cf)(rl)) exists and f ( k )( ( f  ) ( k ) ) , 1 5 k 5 r  1, belong to Lp in case r is euen (odd), (v) there exists an rth Peano diference such that 1 (11 I p ) = O(Ihl'), (vi) I I W I P 0. Therefore, the function Lf(x)  G,(x)] is locally integrable and satisfies AnLf G,](x) = 0 a.e. Thus Lemma 10.6.1 implies the representation (10.6.4) which may equally weU be expressed by saying that there exist constants ak E @, 0 S k S r  1, such that for almost all x E R
If, in particular, f E Clw, then Lf(x)  G,(x)] E Cleo, and (10.6.4). or (10.6.8) holds everywhere by Lemma 10.6.1.
Lemma 10.6.3. If for f rhen f
E
E X(R)
there exisrs g E X(R) such rhar (10.6.3) holds for each h > 0,
Wha.
Proof. Let X(R) = LP, 1 I p < w. Then by Lemma 10.6.2 and in particular by the representation (10.6.8) there is a function 4 such that f = 4 a.e. with 4 E A C L ' and +cr) = g a.e. In view of definition (3.1.43) it only remains to show that #k) E Lp, 1 i k < r. Integration by parts gives (10.6.9)
A:,
*
.A:,$(X)
=
Iohl . johr+ g(X
~1
+
*
+ Ur) dul
a
a
a
dllr
for each h k 2 0, 1 I k I r, and all x E R. We now show via mathematical induction that all derivatives +(k), 1 s k I r, belong to LP, if (10.6.9) holds for E AC&l n Lp and g E Lp. For r = 1 the assertion is trivial. Assuming now that it is valid for r  1, then, putting (10.6.10)
+h,<x)
= +(x f hr)
 +(XI,
gh,(x) =
lohr+ g(X
Ur)
dur,
the hypothesis (10.6.9) and the assumption as applied to +h, E AC:G~ n LP and ghrE Lp imply that +g)(x) = t+(k'(x + h,)  +("(x)] E Lp, 1 Ik i r  1, and +&')(x) = gh,(X) for each fixed h, 2 0 and all x E R. We shall show that I]+;, IIp is a continuous function of h,. Since and g belong to Lp, we obtain by Prop. 0.1.9and 0.1.7 that ll+h,llp and ll+&l)/lp = Il&?h,IIp are continuous functions of h, on [O, r], t > 0,respectively. Since
+
+ t ,  ~ x + t)  +f,3)'<x)  r+t,a)(x) = and +h,(X
+ t )  +h,(X)
r+t,a)(x)
s'
(r
 u>+g,l)(x + u) du
+ h,)  +,(XI,we have (cf. (10.63, (10.6.6)) = 9f3'(~ + hr)  4 f  3 ) ( ~) Jdul J" du g(x + u + u,) du, = +dx
IOhr
and, setting t = 1,
Therefore it follows that
II+~,YI~ 5 114r3~~ + h,)  + Y  Y ~ ) I +I ~hr o sSUP iigU,iip. u1s1
But since 4f3) E Lp, the righthand side is a continuous function of h, on [0, b] for each b > 0, and therefore ~ l ~ f ,  a ) ~ ~is p one, too. Applying this argument successively, it follows that ~ ~ &is ~a continuous ~ p function of h, on [0, b]. Now
and therefore
385
CHARACTERIZATION IN THE INTEGRAL CASE
which is bounded since /l~&,ll~ is continuous on 0 I h, < 1. Thus we obtain that Now we may divide (10.6.9)by h, and take the limit h, + O + . Then
+' E
Lp.
As 4'. g E L", the induction hypothesis implies that all derivatives # k ) , 1 I k I r, belong to LP. If X(R) = C, then an application of Lemma 10.6.2shows that f E AC;' andf")(x) = g(x) for all x E R. Moreover, it follows similarly thatf(k)E C, 1 I k I r, and thus f~ W;
Proposition 10.6.4. Let f E X(R). The following statements are equivalent:
(i) f belongs to Wx,,, (ii) there exists g E X(R) such that
s
m
(10.6.1 1)
for every
+
Proof. Let X(R)
E
m
f(u)+'"(u)
du = ( 1)'
m
m
CL,,.
=
1
C. If J E W; and
E CLo, then
g(U)+(u) du
successive integration by parts yields
 w
+
since has compact support, i.e., (10.6.11) is satisfied with g holds for all elements of CL0, then (10.6.12) ( l).sm f(x)#')(x  u1 
*
 u,) dx =
 w
= f('). Conversely, if (10.6.11) g(x
+ u1 + . . . + u,W(x) dx,
since with d(0) E cL0 also +(o  u1  . . .  u,) E cLo for all uk E R, 1 I k I r. By Fubini's theorem, integration of (10.6.12) over uk E [O, h], 1 I k 5 r, yields after an obvious change of variables
But Problem 3.1.2(iv) implies that A~(x= )
Joh
. . . Joh B(X + U I +
* * *
+ Ur) dUl+..dur
for each h > 0 and all x E R. Hence it follows by Lemma 10.6.3 that f~ WE. After evident modifications the above arguments also give Prop. 10.6.4 for the spaces
X(R) = LP, 1 I p < a.
We are now in a position to restate Theorem 10.5.1 for all spaces X(R).
Theorem 10.6.5. Let f E X(R). The folZowing statements are equivalent: (i) f belongs to WL,,,, (ii) f has an rth Peano derivative D,")f in X(R)norm, (iii) f has an rth Riemann derivative D r y in X(R)norm.
Proof. The proof of the implications (i) * (ii) 3 (iii) follows by the standard argument (see proof of Theorem 10.5.1). Thus let (iii) be satisfied. For any E C;o and h E R we define
+
25F.A.
386
CHARACTERIZATION OF CERTAIN FUNCTION CLASSES
Obviously, (Ih(f) is a bounded linear functional on X(R) for each 4 E Coo and h E R. We have (10.6.14)
Since by the mean value theorem
it follows by (10.6.13) that for everyfE X(W) lim
(10.6.15)
h0
Ah(
f) = ( I>'
lm
f(u)+(r)(u) du.
m
On the other hand, the existence of the rth Riemann derivative DPlf in X(R)norm implies by (10.6.14) that rw
Therefore
Cm f(u)+")(u) du = ( 1)' 1" (DB1f)(u)+(u)du
(10.6.17)
Jm
J03
+
for every E Cbo, and (i) follows by Prop. 10.6.4. Concerning the extension of Theorem 10.5.4 to all X(R)spaces we need the following analog of Prop. 10.6.4 for the class Vk. Proposition 10.6.6. Let f E C . The following statements are equbalent:
(i) f belongs to VL, (ii) there exists g E L" such that (10.6.18)
f(u)+(I)(u)du = ( 1>' OD
g(u)+(u) du
for every # E Cb.,
We leave the proof to Problem 10.6.9. Theorem 10.6.7. Let f E X(R). The following statements are equivalent: (i) f belongs to V;CW), (ii) there exists an rth Peano direrence off such that (iii) Proof. The proof of the implications (i) 3 (ii) * (iii) is again standard. In view of Theorem 10.5.4 only the spaces C and LP, 2 < p < co, need to be considered. Let (iii) be satisfied and X(R) = Lp, 2 < p < 00. Again we consider the bounded
387
CHARACTERIZATION IN THE INTEGRAL CASE
linear functionals /fh(f) of (10.6.13) to deduce (10.6.14) and (10.6.15). The weak* compactness for LP implies that there exists a nullsequence {hf}and g E Lp such that (10.6.19)
[h;'A~J(u)]s(u)du
lim
g(u)s(u) du
=
t00
for every s E Lp', in particular for every (10.6.20)
lim
/fh,(f)
j+
and we have that for every
+
E C;,.
Therefore
Sm
s(uM(4 du,
=
m
E Cl,,
S_mmf(u)+(')(u)du = ( 1)'
(10.6.21)
g(u)+(u)du.
Then (i) follows by Prop. 10.6.4 since Vl. = W> for 1 < p < co by definition. If X = C, then (iii) states that Ilh'A;lfllQ = O(1) as h  t 0. Therefore the weak* compactness for L" may be used to deduce the existence of a nullsequence {h,} and of g E L" such that (10.6.19) holds for all s E L1. Thus (10.6.20) and (10.6.21) are again valid, and (i) follows by Prop. 10.6.6. Thus we have extended the results of Theorems 10.5.1 and 10.5.4 to all spaces X(R). By the way, we have found a further characterization of WLm, by the condition (10.6.11) and of V: and V[P, 1 < p < 00, by (10.6.18) and (10.6.11), respectively. The still missing characterization of this kind for V[l is given by Proposition 10.6.8. Let f E L1. The following statements are equivalent: (i) f belongs to Vp, (ii) there exists p E BV such that (10.6.22) .for every
+
Jm
Jm
E
C;,.

Proof. The proof of (i) (ii) follows as usual by partial integration. Conversely, if (10.6.22) is satisfied, then by partial integration
Replacing r$(x) by +(x  u1 Prop. 10.6.4 that ( 1)'
Sm a
~(x)+"'(x
 UI  .
*
*
   ur),
uk E
 Ur) dx =
R, 1 5 k 5 r, we obtain as in the proof of

P(X
+ UI +
* * *
+
U~,>+'(X
 ur) dx
and
 p(x
+ u1 +    + u ,  ~ ) dul } . . . du,,]r$(x) dx.
388
CHARACTERIZATION OF CERTAIN FUNCTION CLASSES
Again Problem 3.1.2(iv) implies that for each h E R (10.6.23)
Aif(x)
=
jOh.. Ioh + h +
+ + U ,  ~ ) }du, . . . d ~ ,  a.e. ~  p ( x + u1 +
{P(X
UI
+
* * *
Ur1)
But since I[p(o + h)  p(o)JI1 IIhl llpllsvfor each h E 08 and p E BV by (5.3.7), it follows by (10.6.23) that IIWIII
5
lhlr1 J1
. . . jol + h)  p(o)111 dul. I I ~ ( O
.*
dUr1 5
I I ~ II~I'. I~~
An application of Theorem 10.6.7 now yields f E V[l. For another proof of Prop. 10.6.8 which is independent of Theorem 10.6.7 and thus of 10.5.4, we refer to Problem 10.6.12. Problems 1. Show that i f f € Lp, 1 Ip < co, and A7,f(x) = 0 a.e. for each h E R,then f(x) = 0 a.e. State and prove the analogous assertion in Cspace. 2. Show that i f f E Lfw and J?m f(x@(l)(x)dx = 0 for every 9 E C60, then there exists E P,l such thatf(x) = prl(X) a.e. 3. Let doE Cto be such that JZm +,,(u) du = 1. Show that every 4 E C;,, may be decomposed uniquely as $(x) = M40(x) + ~ ( x ) ,where 7 is the derivative of a function belonging to Ct0 and the constant M is given by M = JYrn $(u) du. 4. Show that for $ E C & there exists #EC:~ such that 4 ( x ) = $'(x) if and only if j:m#(u) du = 0. 5. For each t > 0 let P ,  ~ ( X ; f) be an algebraic polynomial of degree r  1 at most. Suppose that limt,o + p ,  l ( x ; t ) exists almost everywhere. Then this limit is again an algebraic polynomial p r  l ( x ) of degree r  1 at most and limt+o+P ,  ~ ( X1 ;) = prUl(x)for all x E R. 6. Show that G, as defined by (10.6.5) satisfies (10.6.6), (10.6.7), and (10.6.9) with G, in place of 4. 7. Show that the converses of Lemmata 10.6.110.6.3 are valid as well. 8. Let f , g belong to X(R) and satisfy (10.6.9) with f in place of 4. Show that the kth strong derivative Dik'f of f exists for each 1 Ik 5 r and DF'f = g. (Hint: BERENS PI) 9. Prove Prop. 10.6.6. 10. If for f E LfOc there is p E BV,,, such that
for each h > 0, show that there existsp,~E Pr1 such that f ( x ) = pr  i(x)
lou'
+ jo8 dui
duz
. . . dur  1 /our1
dp(Ur) a.e.
11. If for f e L1 there exists p E BV such that (10.6.24) holds for each h > 0, show that f E v:1. 12. Give a proof of Prop. 10.6.8 which is independent of Theorem 10.5.4. (Hint: Use the preceding two Problems) 13. Let f E Lp, 2 c p < co. Show that the following statements are equivalent: (i) f E (V"):P, (ii) there exists an rth Peano difference of the Hilbert transform f" such that IIO'hf" (Ip = O(lhlr), (iii) I[A'hf"1, = O((h[').
CHARACTERIZATION IN THE INTEGRAL CASE
389
14. Show that the following statements are equivalent for J E C: (i) Ilh'A:Jilc = 0(1), (ii) there exists g E L" such that for every s E L1 m
El
s(u)[h'ALf(u)] du =
rm
s(u)g(u) du.
15. Show that the following statements are equivalent for f E Lp, 1 < p < 00: (i) IIh'AV[Ip = 0(1), (ii) there exists g E Lp such that limh,o Ilh'AM  gllp = 0. 16. Let f E X(W). Show that the following statements are equivalent: (9 f belongs to w;,a,. (ii) there exists g E X(W) such that for every E C$o
+
/mm
f(uW')(u) du
= (  1)'
Imm g(uM(u) du,
(iii) f h a s an rth strong (weak) derivative D!"f(Dg'f), (iv) f has an rth strong (weak) Taylor derivative DFf ( D z f ) provided that f('l) exists and f(")E X(W) for each 1 5 k Ir  1, (v) f has an rth strong (weak) Peano derivative D?) f ( D z ' f ) , (vi) fhas an rth strong (weak) Riemann derivative D ~ l f ( D E 1 f ) . 17. Formulate and prove results corresponding to Problem 16 for the class V:(R). 18. Show that the following assertions are equivalent forfrz LP, 1 Ip < a): (i) f(x) = 0 a.e., (ii) there exists an rth Peano difference of f such that IIOVfllp= ~ ( l h l ' ) , (iii) IIAVfllp= o(lh1'). State and prove corresponding results in Cspace. 19. Letfe Lp, 1 Ip < 00. Show that if Dil'fexists and vanishes a.e., thenf(x) = 0 a.e. State and prove corresponding results in Cspace. 20. (i) Show that the set of allfE L1 which satisfy (10.6.22) becomes a normalized Banach subspace of L1 under the norm I l f l l , ~ + Jlpllsv. Show that this Banach space and the Banach space V;l are equal with equal norms. State and prove corresponding results for arbitrary X(R)spaces. (Hint: Sec. 10.4, Problem 10.5.10, Prop. 10.6.4, 10.6.6, 10.6.8) (ii) Show that the Banach spaces V:(R) and Lip, (X(R); r) are equal with equal norms. (Hint: Sec. 10.4, Problem 10.5.10, Theorem 10.6.7)
10.7 Notes and Remarks The material of this chapter is largely based upon BUTZER[8,10]. In these papers, a first systematic use of Fourier transform methods was given in connection with Riemann and Taylor derivatives and with theorems of HardyLittlewoodtype. For a parallel treatment of some of these topics using a very different approach, namely semigroup theory, see BUTZERBERENS [ l , pp. 9294, 1061111. Both the Fourier transform as well as the semigroup approach to the material in question allow a general and unified presentation of the proofs as well as of the results. Sec. 10.1. For general comments concerning Riemann and Peano derivatives we first recall Sec. 5.1.4. For Theorems 10.1.3 and 10.1.6 see BUTZER[lo]. These are related to a A. MARCHAUD, C. DE LAVALLBE number of classical results including those of A. DENJOY, POUSSIN,W. T. REID,S. SAKS,S. VERBLUNSKY. For these as well as Schwarz' lemma we [l, p. 107 fl and the references cited there (cf. also VERBLUNSKY refer to BUTZERBERENS [l], KEMPERMAN [l]). For extensions of Schwarz' lemma of Problem 10.1.3 compare also WHITNEY[ 1, 21. Peano derivatives from the point of view of Theorems 10.1.9, 10.1.10 were first considered in GORLICHNESSEL[ 11. For elementary properties concerning strong derivatives see ASPLUNDBUNGART [ l , p. 4561. For an Lg,version of Cor. 10.1.15 for r = 1 see EDWARDS[lI, pp. 1291301.
390
CHARACTERIZATION OF CERTAIN FUNCTION CLASSES
Sec. 10.2. Concerning the classical results of TITCHMARSH [2] and HARDYLITTLEWOOD [21, 31 we refer to the comments given in BUTZERBERENS [l, p. 1481 as well as the papers listed there. The weak* compactness theorem which plays an important r61e in the proofs was suggested in a written communication to the firstnamed author by KAREL DE LEEUW in 1959 in connection with the integral of de La Vallie Poussin; it has previously been used by SuNoucHi [lII] in solving related questions (cf. Sec. 12.1). But it may be replaced by a different argument, see BUTZER 181 and Problem 10.2.6.Theorem 10.2.5 is due to GORLICHNESSEL 111. Sec. 10.3. Most of this material is to be found in BUTZERG~RLICH 111. The proof of Prop. 10.3.1, (ii) = (i), is due to H. BERENS (unpublished). Sec. 10.4. The concept of relative completion in the form of an explicit definition was introduced by GAGLIARDO [l] in his study on interpolation spaces. For Prop. 10.4.2, a proof of Prop. 10.4.3 and further properties connected with this concept see ARONSZAJNGAGLIARDO [l]. This concept is stronger than that of closure in X yet weaker than compactness in Y, and fits in precisely in our study on the connection between W[X2n;#(k)l and V[X2,; 4(k)]. In Part V (Sec. 13.4) it will be seen that the notion of relative completion is especially adaptable to the phenomenon of saturation. The idea of using the concept of relative completion to study such problems goes back to BERENS (see his Habilitationsschrift [31, also the remarks in BUTZERBERENS [l, p. 2231). For a separate interpretation (employing a language of the theory of partial differential equations) see H. S SHAPIRO [4, Sec. 3.51, where Y is the set of strong solutions of a certain problem and Yx that of weak solutions.The Friedrich’s mollifier technique used there enables one to prove theorems about these weak solutions by operating with the strong solutions and passing to a limit. The proofs of Prop. 10.4.4, 10.4.5 are straightforward, see also BUTZERGORLICH 12, p. 381 fl, BERENS 13, p. 751. For the proof of Theorem 10.4.6see also BERENS 13, p. 761.
Sec. 10.5. Theresults are largely taken from BUTZER 18,101, GORLICHNESSEL [l], BUTZER
NESSEL121. The proof of Theorem 10.5.6, (iii) = (i), is due to H. BERENS; see BUTZERGORLICH[l, p. 411. For Prop. 10.5.7 see also BERENS [3, p. 75 f]. We emphasize that the results of BOCHNERCHANDRASEKHARAN (cf. Sec. 5.1 ., 5.2) influenced us. Sec. 10.6. For the proofs of Lemmata 10.6.1, 10.6.2 see BUTZERKOZAKIEWICZ [l], also
LOOMIS 13, p. 1761, WIENER Ill. These are connected with the problem of defining a primitive of a distribution (cf. SCHWARTZ [lI, p. 51 ff], ZEMANIAN 11, p. 681) and may be compared with a fundamental lemma in the calculus of variation (cf. COURANTHILBERT [lI, p. 2011). Problem 10.6.8, thus in particular Lemma 10.6.3, is due to BERENS [2] who, more generally, treated the problem of defining the rth infinitesimal generator of a semigroup by rth Peano or rth Riemann differences (cf. Sec. 13.4.2). For Prop. 10.6.4 see S . GOLDBERG [I, p. 1801. Concerning the connection of (10.6.11) (and of (10.6.18), (10.6.22)) with the definition of distributional derivatives we mention the Notes and Remarks to Sec. 1.1 and 3.1. Furthermore, the classof functions f satisfying (10.6.11) may be regarded asa genuine extension of V[Lp; (io)’], W[Lp;(io)’] to Lp, 2 < p < 03 (see also (13.1.4) and Sec. 13.5)). For the remaining results see GORLICHNESSEL [l] and the literature cited there (compare also with H. s. SHAPIRO [4, Sec. 3.51). The results there are also treated in the setting of SCHWARTZ’ distribution theory; for a thorough discussion of various definitions of a (ordinary) derivative in connection with distribution theory see also LOOMIS [3, p. 164 ff]. For pointwise and normconvergence of distributions we also refer to BELTRAMI [l]. For related results concerning functions defined on an open interval (a, b) we mention BUTZERKOZAKIEWICZ 111 and GORLICHNESSEL [l].
Characterization in the Fractional Case
11.O Introduction
In Chapter 10 the class W[Lp; IuIr], 1 5 p I 2, was characterized in terms of differentiability properties uponf (r even) or f (r odd). The question arises whether for fractional a > 0 the class W[LP; Iul"] is connected with a derivative off of fractional order a. This will be shown to be the case. It is opportune to define fractional differentiation through integration of fractional order. There are at least two such definitions. If [ L l f ] ( x )is the integral o f f over (a, x), and [ L a f ] ( x )the integral of [ L a  l f ] ( ~ )over (a, x), a = 2, 3 , . . ., then

(11.0.1)
(x > a).
Here a is finite or a,in which case one assumes that the growth off at infinity will and B. RIEMANN took (11.0.1) as a not cause the integrals to diverge. J. LIOUVILLE definition of [ L a f ] ( x )for every a > 0. This integral of fractional order possesses the property [L,(L,f)](x) = [ L a t s f ] ( x )(a,,!I > 0). If a > 0 is arbitrary and n > a (n E N) one could define a (pointwise) derivative off of fractional order a by ( d / d ~ ) ' " L ,  a f ] ( ~ ) * However, we shall not follow this approach to the definition any further since it is connected (cf. Problems 11.1.2, 11.2.7) with theclass W[Lp; (iu)"]instead of W[Lp; Iula] which is our aim. For this reason we shall consider another definition, introduced by MARCELRIESZ and more convenient for our purpose. For 0 < a c 1 set
wherefis such that no difficulties arise in the calculations involved. This integral of fractional order also has the property [R,(R,f)](x) = [ R a t s f ] ( x ) (a > 0, ,!I > 0, a ,!I < l), and we could define a pointwise derivative of fractional order a,0 < a < 1, by (d/dx)[Rl a f ] ( x ) . But this definition would be associated with the class W[LP;{i sgn u}lula] (cf. Prop. 11.2.3). However, if we replacefby its Hilbert transform in the latter definition, then it would be associated with the class W[LP; )via]. There is
+
392
CHARACTERIZATION OF CERTAIN FUNCTION CLASSES
one important drawback here: one must suppose thatfbelongs to L1 n L2 instead of only to L P for some 1 Ip I 2. Since [ R 1  J " ] ( x ) = [R,"_.f](x) i f f E L1 n La, and the conjugate of the Riesz integral is also meaningful forfE L', one would be led to define the fractional derivative by (d/dx)[R;,f](x). However, for 0 < a 5 l/p', [R;J](x) does not necessarily exist (cf. Problem 11.2.3). Noting that
(1 1.0.2)
[R;afl(x
+ h)  [R;afl(x) = f * m i ' , l  u ( ~ )
where
then the expression on the right in (1 1.0.2) exists as a function in Lp for each 1 Ip < co and 0 c a c 1 (cf. Lemma 11.2.2). At last we come to the desired definition of a (strong) derivative of order a which for want of a suitable name we call the srrong Riesz deriuatiue D f ' f of order a, 0 c a c 1. It is the limit in Lpnorm of h  l ( f * mh.l,J as h + O if it exists. If a > 0 is arbitrary, then the ath strong Riesz derivative off€ Lp is defined successively (cf. Def. 11.2.8), noting that DjlY = (f)'(at least formally). This fractional derivative has the desired properties. Indeed, iff E Lp, 1 I p I2, and a > 0, then f E W[Lp;IuI"] if and only if Dryexists. Moreover [Dt'f]^(u)= Iu/af^(u). In distinction to Chapter 10, if a > 0 is arbitrary we shall characterize the class W[Lp; lula] in terms of
(0 c
(11.0.3)
a
c 2j; E > 0),
which may be interpreted as a fractional (Riemann) difference quotient offof order a. Our fundamental result is the following (cf. Theorems 11.2.611.2.9, 11.3.7, 11.5.111.5.3, Problem 11.5.2). Theorem 11.0.1. Let f E Lp, 1 equivalent:
(9 f E W L P ;
0 arbitrary. The following assertions are
IuIal,
(ii) there exists g E LP such that
where j E N is chosen so that 0 c: a c 2j, and C,,aj is a constant defined by (1 1.3.14),
(iii) DF'f exists and equals g, (iv) there exists g E LP so that f o r 0 < a < 1,p = 1 : 1cps2: a 2 1,p = 1 : for
f(4 = [R,gl(x) a.e. RaIv*gE LPandf(x) = Ra/pK&/p,g)l(x) a.e. f belongs to L1, R6g belongs to L', and
CHARACTERIZATION IN THE FRACTIONAL CASE
where a = 1.I transforms.
+ B, 0 I fl
l(x) = [ L r + afl(x).
396
CHARACTERIZATION OF CERTAIN FUNCTION CLASSES
Thus for a = XI the fundamental formula (1 1.1.3) is valid for all values of a and fl for which Re (@),Re (a 8) > r. If a is finite, this formula holds for all these values of a and /I, except for those values of fl which are negative integers. In the obvious case that a is zero or a negative integer, the formula still remains valid. Let us finally give a further formula which is useful for the process of analytic continuation to be discussed below and which is close to the Hadamard approach to the subject. If we set
+
Iff")(x) exists and is continuous, the latter integral obviously converges for a >  r, and the righthand side of (11.1.9) defines the analytic continuation for all these values. [Lf](x) is a holomorphic function of Q since (k + a)P(a) # 0 throughout. In particular for a =  k we have ( k + a ) k ! r ( a )= ( l)k (cf. Problem 11.1.3) from which we again derive (1 1.I .lo)
[Lofl(x) = f(x),
tLrfl(x)
= f'"(x).
For the latter arguments to be valid it is not necessary that the derivativef") exists at x in the classical sense. It suffices that there exists an algebraic polynomial p(u) such that [f(u)  p(u)](x remains bounded as u + x. For example, it is sufficient that the r th Peano derivativef(')(x) exists, 11.1.2 Integral of M. Riesz
In his monumental treatise [2],M. RIE~Z introduced a modified form of an integral of fractional order. For 0 < a c 1 he considers
(11.1.11) the functionfbeing assumed to be so wellbehaved that no difficulties arise in the calculations that are based upon the definition. We shall show that the constant M(a), which only depends on Q and plays the r61e of r ( a ) in the case of the RiemannLiouville integral, may be determined in such a way that the fundamental formula
(11.1.12) is satisfied.
[Ra(R,f)l(x) =
[ R a+
,fW
(a
> 0,fl > 0 , a
+ /I < 1)
397
CHARACTERIZATION IN THE FRACTIONAL CASE
First we show that we may evaluate M(a) such that (1 1.1.12) is valid for the particular function f ( x ) = exp {ix}. Indeed, for 0 c a c I
Thus if we set
then
(1 1.1.13)
[ R , eiG](x)= efX,
and the result follows. Note that, since ( 1 1.1.14)
(0 c
[ R , efu0](x)= IuI  a efUx
a
< 1)
for all u # 0, we may connect the integral of M. Riesz with the theory of Fourier transforms (cf. Lemma 11.2.2). To evaluate M ( a ) defined above we use Problem 11.1.1 and obtain M(a) = 2r(a) cos ( 4 2 ) which holds for 0 < a < 1 to ensure the convergence of the integral. Thus, if we set
(11.1.15)
&(a)
=
the Riesz integral of order
2r(u) cos C(
lTa J
2
A,(a)
=
2r(a) sin
lTa 9
2
o f f takes on the form
(0 c
(1 1.1.16)
a
< I).
To examine the fundamental formula (11.1.12) let us moreover suppose that f is such that all integrals which occur in the following are absolutely convergent. We have
Again by the substitution u = ( x  t ) u
then (11.1.17)
+ t we obtain
398
CHARACTERIZATION OF CERTAIN FUNCTION CLASSES
Obviously, A(a, can show that
m only depends on a and fl and exists for a > 0, fl > 0, a + p < 1. If we
(1 1.1.18) then (11.1.17) reduces to the fundamental formula (11.1.12) which would thereby be established. To verify (1 1.1.18) we have
=
IOm + + (1
t)altBl dr
+ l o m tUl(l + r)Bl
B(p, a)
dr,
where B(a,B) is the Beta function (cf. Problem 11.1.3). Substituting t follows that A(a,p) = =
IO1 (1
u)uBuBl du
B(p, 1  a  p)
+ B(a,B) +
'I
=
u/(l  u), it
(1  u)"BuUl du
+ B(a,m + B(a, 1  a  j?),
and thus in view of Problem 11.1.3
which in view of (11.1.15) proves (11.1.18). As for the RiemannLiouville integral (11.1.2) we recall that the functionf has to satisfy certain regularity requirements, in particular at infinity, in order that the integral (1 1.1.1 1) converges. If e.g. f E Coo,then (1 1.1.16) exists and represents a holomorphic function of a for all complex a with Re (a) > 0 with the exception of a = 1 + 2k, k E P (cf. (1 1.1.15)). At these points [R.fJ(x) has poles of order one. We now proceed by discussing the case a = 1. Then the integral in (1 1.1.16) is independent of x , but the factor l/Ac(a)is infinite. In the particular case that J'? f ( u ) du = 0, we may then define (1 1.1.19)
Thus [RlfJ(x) is defined as a logarithmic potential. In the general case that J?' f ( u ) du # 0 we may always write m
C1
+ co + 1
Ck(a

k=l
in the neighbourhood of a = 1 for any fixed value of x. Here, c1 =
1
1 " ; f (u) dlf
1)"
CHARACTERIZATION IN THE FRACTIONAL CASE
399
and, except for an additive constant, co is equal to the logarithmic potential
+
We note that for a = 1 2k, k E N, we obtain kernels of the form Ix  ulaklog ( l / l x  u [ ) by an analogous procedure. Concerning the connection with differentiation, we have for a
> 0,
a # 1
+ 2k,
k E P, that
and thus (see also (1 1.1.13))
(1 1.1.20)
holds under suitable? regularity conditions on J We finally turn to the problem of extending the definition of [ R a f ] ( x )to a I0. As for the RiemannLiouville integral, under certain conditions upon f this extension is constructed by analytic continuation. For a = 0 we may consider a limiting process. Supposing that f is continuous at x we obtain by the same arguments as applied to (11.1.6) (1 1.1.21)
[ R o f l ( x ) = alim [Rafl(x) = f ( x ) . 0+
Concerning the case a < 0, let us assume that the function f is 2rtimes continuously differentiable and thatfand its derivatives behave at infinity in such a way that the integrals occurring in the following are absolutely convergent and that the integrations by parts are justified. Then we obtain, first of all for a > 0, a # 1 2k, k E P, that
+
and generally, (11.1.22)
[&fl(x)
= ( 1 Y [ R a + ~ r f " T ( x ) .
Thus [ R a f l ( x ) possesses an analytic continuation to all values of a with Re(ct) > 2r. Furthermore, by the continuity of fear) at x (which is not necessary for the validity of ( 1 1.1.22)) the extension to CY =  2r follows as above from the limit
[ R  2 t f l ( x )=
lim
aZr+
[ R a f J ( x )= ( lYf(2r)(x).
An analogous formula of course holds for all values k c r, k s N. The fundamental
+
t We observe that the property da[[R,+,fl(x)/dxa= [R,f](x) may also be deduced if we replace &(a) by exp{ina/2}Ac(a).In a certain sense this value is more suitable than that used by us since it gives [R,et0](x)= i"erx.The present choice is justified by the fact that A,(a) is real for a real.
400
CHARACTERIZATION OF CERTAIN FUNCTION CLASSES
formula (11.1.12) may also be readily extended to the domain a > 2r, a + /3 > 2r.
p
> 2r,
Problems 1. Show that for 0 < a < 1 * cos u
du =
SO
s
m
0
Iom
r(a)cos na2 ?
du = r ( a ) sin
71oL 9
2
du = r(a)lxl  a exp d  a
(Hint: Compare with the definition of the gamma function given in Problem 1 1.1.3; see also BOCHNER[7,p. 511, ACHIESER [2, p. 1121) 2. Let a = a and 0 < a c 1. Show that? (i) [L.,eiU"](x) = (iu)  aeiux
(ii) [Laek0](x)= kaekx 3. The gamma and beta functions are defined for Re (z), Re (a),Re (8) > 0 by
respectively. Show that (i) r(l z) = zF(z), (ii) r(z)P(l  z)
+
=
71
sinl (nz),
Furthermore, the gamma function is a meromorphic function with poles of orderoneatz=O, 1,  2 , . . . . S h o w t h a t ( j + x ) T ( x ) # O f o r a l l x E R , j E N a n d limx  I (j x ) r ( x ) = ( l)'/j!. (Hint: ERDBLYI[l I, p. 1 ff])
+
11.2 Characterizations of the Classes W[Lp; 101
a],
V[Lp; IvI a], I
Ip
s 2
11.2.1 Derivatives of Fractional Order The aim of this subsection is to prepare the way for characterizations of the classes W[Lp; Iula] and V[Lp; Ivl"] for fractional a, in particular in terms of certain smoothness conditions upon the original function$ To this end we consider the Riesz integral of order 1  a,0 < a < 1, (1 1.2.1) In comparison with the preceding section we now try to derive some properties of (11.2.1) for functionsfe Lp, 1 I p I 2. We first restrict the discussion to functionsf in L1 n La and show that it is possible to define a pointwise derivative of fractional
t (iv)" is defined by (it.)"
= IuI"
exp {i(na/2) sgn u). the principal branch.
CHARACTERIZATION IN THE FRACTIONAL CASE
40 1
order. Then we remove this restriction to obtain suitable definitions of fractional differentiation in Lpnorm. Lemma 11.2.1. Let f E L' n L2 and 0 < a < 1. Then [ R 1  J ] ( x ) , [ R 1  J  ] ( x ) exist almost everywhere and are locally absolutely integrable. Furthermore (1 1.2.2)
(11.2.3)
lim
Ihl
Proof. Let y &(x)
=
E
IOU
[R1J"](x
+ h) dx = 0
R be fixed. We consider the function
IOU
Ix  uI
du
=
1 {sgn(y  x )  l y  XI^^ + s g n x . I ~ l l  ~ } , 1a
which obviously is continuous (as a function of x ) for 0 < a < 1. Since #,(x) = O( 1x1 ") as 1x1 00, we have $, E C,. Moreover, $v E L9 for all q with aq > 1. SincefE L' n La implies f~ Lp for 1 I p I 2 (Sec. O.l), by Theorem 8.1.12f" E LP for all 1 < p I 2. Let g be equal to f orf". For given a, 0 < a c 1, we may choose p, 1 < p < 2, such that ap' > 1. Then g$, E L' by Holder's inequality. Thus the iterated integral f
converges absolutely, and by Fubini's theorem the integral
as well. Thus [R,,g](x) exists almost everywhere and Rl,g
E
L:oc. Moreover
andsince Il&(o  h)IIp, = ~ ~ # , ~ 5~ pMuniformlyforh , E Randlimlhl,, $,(u  h) = 0, (1 1.2.2) and (1 1.2.3) follow by Prop. 0.1.11. In terms of the Riesz integral (11.2.1) of order 1  a we may define a pointwise derivative offof order a, 0 < a < 1, as lim [Rl  u f I(x
(1 1.2.4)
It0
+ h)  [Rl .m> h
for all x E R for which the limit exists. In order to consider the difference quotient the following lemma will be fundamental. Lemma 11.2.2. Let 0
1. Then f E W[Lp; lul"] ifand only f has an crth strong Riesz derivative.
if
Proof. Let f E W[LP; IuIa], i.e. lul.f"(v) = gA(u), g E LP. By Theorem 6.3.14 we have lul"f^(v) E [LP]" for each 1 I k I[a] so that a successive application of Theorem 11.2.7 yields the existence of DikY for each 1 Ik I [a]. This proves the case for a = [a]. For fractional a we have I ~ l "  [ ~ ~ [ D ! [ ~ ~ '= f ]g"(u). " ( u ) By Theorem 11.2.6 the assertion follows, and Df'f = g. Conversely, iff has an ath strong Riesz derivative, then by definition Djk'fexists for 1 Ik s [a] and thus Ivl'Cf"(v) = [D!"f]"(v),proving the assertion for 01 = [a]. In the fractional case we apply Theorem 11.2.6 and obtain [D!"'~]''(u) = IuI~"[D!["')~]"(u)=
IU~"~"(U).
Let us observe that D!"X k E N, is equal to the 2kth strong derivative of ( 1)"f since the classes W[Lp; ( l)klulak]and Wt: are equivalent by Theorem 5.2.21. We turn to characterizations of the classes V[LP; Ivla].
Theorem 11.2.10. Let 0 < equiualent:
a

< 1. For f E Lp, 1 Ip
I2,
the following assertions are
LP
(0 f (iii)
(ii) f E W[Lp; Iula]
E WLP; IUI"1,
IIf * % l  a l l p
=
O(lhl)
, (h f 0).
Proof. First of all we observe that we have to establish Theorem 11.2.10 only for p = 1; for in the reflexive case 1 < p I 2 the classes V[LP; Ivla] and W[LP; Ivla] are
equivalent by definition. In particular, Theorem 11.2.6 then shows that (iii) may be strengthened to the strong convergence of the difference quotient h  l ( f * m;, as h + 0. Furthermore, the equivalence of (i) and (ii) is given by Prop. 10.5.7. (ii)
(iii). Let f E W[L'; IulQ]], Then there exists a sequence {fn}E W[LP; Iula]
t Theintegral case ( D ( " f = f"', at least formally) would suggest the notation (d/dx H)% where Hdenotes the operationof taking Hilbert transforms. Formally, [(d/dx H)']^(u) = [to{  i sgn u}y = IuJ' for r E N. Note that the existence of Dl"fimp1ies that of DkYfor each 1 5 k 5 [a].
CHARACTERIZATION IN THE FRACTIONAL CASE
I(
[I1
407
+
such that f ,  f = 0 and IIfnlll IIg,lJ1 I M uniformly for all n where g, E L' is such that luly:(v) = g;(u). We have by (11.2.15) for each n
( A * m;  ~ x =)
lo'' + g,(x
u) du as.,
and therefore 1) f , * m; lalll I Ihl llgnlll IM Ihl uniformly for all n. Since m;lcr E L' by Lemma 11.2.2, limnm 1 f , * m;laII1 = 11f * m;lolll, and thus (iii) is established. (iii) * (i). Since by Lemma 11.2.2
which tends to lul.f"(v) as h f 0, (i) follows as in the proof of (iii) s (i) of Theorem 10.5.4. For a = 1 we refer to Theorem 10.5.6 and have (cf. (11.2.13))

Theorem 11.2.11. Let f E Lp, 1 5 p I2. The following assertions are equivalent:
(0 f E WP;101I, (iii) Ilf + h)  f "(0
L'
SEW L P ; I4 1 = O( Ihl 1
(ii) "(0)llP
(h +0).
The counterpart of Theorem 11.2.9 reads as

Theorem 11.2.12. Let a > 1 and f equivalent:
E
Lp, 1 Ip I 2. The following assertions are L'
(0 f
E V[LP;
IulOl,
(ii) f~ W[Lp; [u1"]
,
(iii) for a # [a](a = [a])the [a]th ([a  11th) strong Riesz derivative off exists and a
z
[an:
IIDPV m;ral+lcriiP = o ( i w h)  (DlralJ'f)"(o)[[p = O(lh1)
a = [ a ] : ll(Djc"ll'f)"(.
+
(h t 01, ( h  t 0).
The proof is left to Problem 11.2.6. For a = 2k, k E N, we again refer to Sec. 10.5, observing that the classes V[LP; ( 1)kIu12k] and V:p" are equivalent by Theorem 5.3.15. Note that condition (iii) ofthe1asttheoremfore.g.a = 2reducesto Il(D!')f)"(o h)  (D!'lf)"(o)IIP = O(lhl), which converts formally (since (D6"f)" = (f"')" = f ') into 11 f ' ( 0 h)  f'(o)llp = O( lhl). The latter assertion is rigorous in view of Problem 10.5.3.
+
+
Problems 1. Let f E L1 n La and 0 < a < 1. Show that [R;.f](x) exists almost everywhere, is locally absolutely integrable, and lim,,,,+~ Ji [R,"_.f](x h) dx = 0 for each y E R. 2. Let f E Lp, 1 Ip < co, and 0 < a < 1. Show that [R1JI(x), [R,"_.f ](x) exist almost
+
everywhere and are locally absolutely integrable for l/p' < a < 1. Furthermore, show that (11.2.2) is valid as well. 3. Let p > 9 > 1. Show thatf(x) = [Ixl(l + llog 1x1 I)a]l'p is an example of a function which belongs to LP, but for which [Rafl(x)diverges on a set of positive measure for l/p 5 a < 1.
408
CHARACTERIZATION OF CERTAIN FUNCTION CLASSES
4. Show that (11.2.12)holds everywhere i f f e Lip(Co;m for some fl > 0. 5. Let f E L1 n La and 0 c a < 1. Show that [Rr, f ] ( x ) = [R1,fl"(x) a.e. 6. Prove Theorem 11.2.12. 7. (i) Let 0 < a < 1, h e R, and7 be defined by
Show that AAv(x) E L1 and (cf. Problem 11.1.2) [A:qlA(u) = (elhu l)(iu),. (ii) Let (I in (11.1.2) be co and 0 < a c 1. I f f € L1 n La and there exists g e Lp, 1 Ip 5 2, such that (iu)"fA(u) = gA(u), then [L1 . f ] ( x ) = fi g(u) du a.e. 8. Let [R, dp](x), [R; dp](x) be defined for p E BV and 0 < a < 1 by
Show that [ R , dp](x), [R," dp](x) exist almost everywhere, belong to L,: and satisfy (11.2.2). 9. L e t f € L P , 1 I p 12. (i) I f 0 < a < 1 and m;lallF = o(lhl), show t h a t f = 0 a.e. (ii) If the ath strong Riesz derivative exists for a > 0 and vanishes, show that f = 0 almost everywhere. (iii) Show that the existence of the ath strong Riesz derivative off implies that the Pth strong Riesz derivativeoffexists for all 0 < fl 5 a and [D!8'flA(u)= Iu18f"(u). (iv) Show that f e V[LP; Iul"] implies the existence of the flth strong Riesz derivative of f for every 0 < fl < a. 10. Let f E Lp, 1 5 p 5 2, and 0 < a < 1. Show that f E W[LP; lula] if and only if there exists g E Lp such that for every s E Lp'
[If*
(f
* mKh'  u ) ( x )
s(x) dx =
11. Let f~ L' and 0 < a < 1. Show that f E VIL1; Iul"] if and only if there exists p E BV such that for every s E Co
12. LetfE Lp, 1 I p 5 2,anda > 0.Show thatfE V[Lp; lul"] if and only if IIu(~)C~; 0 ; p)llp= 0(1),p t co. (Hint: For 0 < a < 1 the pointwise derivativef'") off is defined by the limit h'{[R,"_,f](x + h)  [R;,f](x)} as h + 0 if it exists. For a = 1 we set f")(x) = lirnh,o h  l { f " ( x h)  f " ( x ) } if the limit exists. The definition may be extended to a > 1 successively as usual. Show that
+
13. (i) Let 4 B C.,: Show that 4 has strong Riesz derivatives in Lp, 1 I p I 2, of every order a > 0 and [DF)4lAE L1. (ii) Let pa, q, be defined for a > 0 by p.(x) = d2/.rrF(l a) Re ((1  i x )  l  a } , q,(x) = dGF(1 + a) Im ((1  i x )  l  a } . Show that p:(u) = lula exp {  I u l } , q:(u) = {  i sgn u}lul" exp { I u I } .
+
CHARACTERIZATION IN THE FRACTIONAL CASE
409
11.3 The Operators Rp)on Lp, 1 Ip I 2
11.3.1 Characterizations
We now turn to characterizations of the classes W[Lp; 101"] and V[LP; IuIa] in terms of the integrals (1 1.3.1)
(O
0, R?) is a bounded linear operator of X(W) into X(W) satisfying (1 1.3.3) In this section we suppose throughout that x satisfies the following conditions : (11.3.4)
(i) x belongs to C n NL1, is even, and x E WE n W ~ I , (ii) m(y; a) and m k " ; a) arefinite, where
m h ; a) = ( l / m J : m lul"lx(u)l du, (iii) [x"]"
E
L1.
The kernels of GaussWeierstrass and Jacksonde La VaMe Poussin are examples satisfying (11.3.4) (cf. Problem 11.3.1). In the following, Jcf; x ; p) always denotes the singular integral (3.1.8) having kernel {px(px)}. Lemma 11.3.1. I f x satisjes (11.3.4), then f E Lip* (X(R);
a), 0
0, x E R since lual Ix"(u)l = 1[X"]"(v)l E L1. Therefore the integral
41 1
CHARACTERIZATION IN THE FRACTIONAL CASE
converges absolutely, and Fubini's theorem yields (11.3.9)
Sm
Sm "(!) '
u  ( ~ ~ ' ) J ( Lx ;Ep~) du ~ = Ac(a) 
dzr
0
 w x
P
vI.f"(u) efxudv.
Here we may use (cf. Problem 11.1.1)
Thus, as
we have
Since rnh;a )
, 0).
D is a dense subset of the Banach space W[Lp;lvla]. Indeed, f by the convolution theorem that
Ivl*[J(f;
0;
f)lW = x~(;)lvlYw
= [J(g;
0;
E W[Lp;
IvlO]implies
P)IA(4Y
i.e. J c f ; x ; p) E W[Lp; lul'] for every p > 0. Moreover,
IlJM ";P)  f ( ~ ) I I W [ L ~ ; , u , " l = I J(f;
O;
P)
f(o)llp
+
IIJ(g;
0;
P)
 g@)Ilp,
and thus In order to establish (11.3.11) for d(x) = J ( f ; x; p) we use Lemma 11.3.1 and obtain by (11.3.8) for el < e2
1441 I [Rg'J(f;
O;
1 J8r
P)l(O)
=
 [RE'JM
O; P ) I ( O ) I I P
h''+')J(K;fi
0;
p)
dh
1
P
5 M m w ; .)pa"
j6rh1a &,
which tends to zero as el, e2 f O+ for each fixed p > 0. (ii) 5 (i). If 1 < p I 2, then (ii) in particular states that [lR~'fllp= 0(1) as E f 0+, andfE W[Lp; Ivla] by Theorem 11.3.3.
414
CHARACTERIZATION OF CERTAIN FUNCTION CLASSES
L e t p = 1.Ifweset (1 1.3.12)
it follows by Fubini's theorem that
Therefore, in view of the hypothesis and (5.1.2) we conclude that I?&(U)f
(v)  g W l I IIRF'f  gll1 = o(1)
(€+
O+),
and since lim,,o+ vE(v)= IuI" for all u E R by (11.3.10), this implies Iul"f"(v) = g"(v), i.e.f E W[LP; lola].In particular, Theorems 11.2.611.2.9 show that f has an ath strong Riesz derivative and Dr'f = g. Having discussed the case 0 < a < 2, it is clear how the extension to arbitrary a > 0 would proceed. We state Theorem 11.3.6. Let f lent:
E
Lp, 1 5 p I2, and a > 0. Thefollowing assertions are equiva
where j E N is chosen such that 0 < (11.3.14)
u
< 2j and
C,,,,= ( 1)j22jao[
Theorem 11.3.7. Let f E Lp, 1 I p 5 2, and equivalent: (i) f E WWP;1 ~ 1 ~ 1 , (ii) there exists g E Lp such that (0
0. The following assertions are
< 2j)
In this event, f has an ath strong Riesz derivative and D t ' f = g. For the proof of the last two theorems we refer to Problem 11.3.3. Theorems of BernstehTitchmarsh and H. Weyl We shall now give two applications of the preceding results on the operators R:"). The first one is concerned with a theorem of E. C. TITCHMARSH which extends the famous theorem of S. N. BERNSTEIN on the absolute convergence of Fourier series (cf. Problem 11.4.4) to Fourier integrals. 11.3.2
415
CHARACTERIZATION IN THE FRACTIONAL CASE
Theorem 11.3.8. Let f E Lp, 1 I p I 2. Then f
E
Lip* (LP; a), 0 < a I 2, implies
f E LQfor all q with p/(p  1 + ap) < q I p/(p  1). Proof. Let p, 0 < p < a, be arbitrary. By the inequality of HolderMinkowski
(€
Thus, if 1 < p I 2, Theorem 11.3.3 guarantees the existence of g lulBf"(u) = g"(u) a.e. Sincef E Lf,, for 1 I q I pr and
E
> 0).
Lp such that
for any N > 1 by Holder's inequality, we have f E Lqfor pqp'/(p'  q ) <  1, i.e., for all q with p / ( p  1 pp) < q I p / ( p  1). Since this is valid for all p with 0 < < a,the assertion follows. The proof forp = 1 proceeds similarly, and Theorem 11.3.8 is established.
+
Corollary 11.3.9. Let f E Lp, 1 5 p I 2. Then f f " E L'.
E
Lip* (Lp; a), a > l/p, implies
Next we consider the L'version of a theorem of H. WEYLconcerningthe connection between Lipschitz conditions and representations of continuous 2rperiodic functions as RiemannLiouville integrals (cf. Theorem 11.5.4, Problem 11.5.3).
Theorem 11.3.10. Let f E L'. If there exists p E BV such that f ( x ) = [R, dp](x) a.e., 0 < a < 1, thenf E Lip (L'; a). Conversely, i f f E Lip* (L'; a), 0 < a I 1, thenfor each p with 0 < p < a there exists g E L' such that f ( x ) = [R,g](x)a.e.
Proof. If f ( x ) = [R, dp](x) for some
p E BV and 0 < a < 1, then A; f ( x ) = (mh,,* dp)(x), and thus IlA,f [I1 I Ihla~~ml,,lllllpllsv by Lemma 11.2.2. Conversely, if f E Lip* (L1; a), then for any p, 0 < /3 < a, and 0 < el < e2
which tends to zero as el, e2 3 O + . Therefore there exists g E L1 such that lime+o+IIRLB'f gill = 0, and Theorem 11.3.5 implies that Iu16f"(u) = g"(u) for all u E R. Hence it follows by Lemma 11.2.2 that [A, f ]^(u) = [mh. * g]"(u), and an application of the uniqueness theorem gives Ahf ( x ) = (m,, * g)(x) a.e. Therefore by an integration over the interval [0, y ]
,
,
Letting h f co, Problem 11.2.2 shows that f ( x ) = [R,g](x) a.e., and the proof is complete.
416
CHARACTERIZATION OF CERTAIN FUNCTION CLASSES
Problems 1. Show that the kernels (Problem 3.1.14) of Jacksonde La Vallk Poussin and (3.1.33)
of GaussWeierstrass are of type (1 1.3.4). 2. Show that f~ V[Lp; [vl"] implies IIAhfllp = O( lhl") for 0 c a c r. 3. Prove Theorems 11.3.6, 11.3.7. (Hint: TREBELS [ll) 4. Give a proof of Theorem 6.3.14 via an application of Problem 11.3.2 and Theorem 11.3.7 which is thus independent of the lemma of E. M.Stein and Bessel potentials. 5. Let f E Lp, 1 I p I 2, and a > 0. Show that f belongs to V[Lp; {  i sgn u}lul'] if and only if
where j e M is chosen such that 0 c a < 2 j + 1 and m sinW + 1 (1 1.3.15)
j0y i T r du.
c C r , a , + 1= ( 1)'+129'+1=
6. Let f E Lp, 1 S p 5 2, and a > 0. Show that f belongs to W[Lp;{  i sgn u}lula] if and only if there exists g E LP such that (0 c a c 2 j 1)
+
7. Show that i f f € V[Lp; lola] for some a > 1, then f E W[LP; ( i ~ )for ~ ] every integer k, 1 5 k I [a]. 8. Prove the following theorem of E. C. TITCHMARSH: Let f E LP, 1 < p I 2. If ]If(. h)  f ( 0  h)IIp = ()(Au), 0 c a I 1, show that f* E: LQ for p / ( p  1 + ap) < 4 < P/(P  1). 9. Let f e LP, 1 I p 5 2, and 0 < a < 2. Show that + 0 + )
(.?
impliesf(x) = 0 a.e. (see also H. S. SHAPIRO [I, p. 421). 10. Show that for 0 c a < 2 the set offc X(R) with IIRIP'fllx(~= 0(1), e + 0+, becomes a normalized Banach subspace of X(R) under the norm (11.3.16)
IlfllxcR,
+ SUP IIRP!fllx(R) E>O
(0 < a c 2).
Show that in case X(R) = Lp, 1 4 p 5 2, this Banach space and the Banach space V[Lp; Ivl"] are equal with equivalent norms. (Hint: Prop. 10.5.7, Theorem 11.3.3)
11.4 The Operators Rp) on Xan Next we turn to counterparts of the preceding results for periodic functions. In this section we are concerned with characterizations of the classes W[Xan;J kla], V[&; lkl"] in terms of integrals of type (11.3.1). Here we prefer to proceed separately, whilst in the next section we transcribe the results of Sec. 11.2 to periodic functions by reduction. The treatment of this section will be much the same as for functions belonging to Lp, 1 I p I2. Indeed, letfE X a n and 0 < a c 2. Defining [RY)f](.x) as in (1 1.3.2) it
CHARACTERIZATION IN THE FRACTIONAL CASE
follows that RY'fc C,,. For each itself satisfying
e
417
> 0, Rr'fis a bounded linear operator of X,, into
(11.4.1) Here we suppose throughout that x,, satisfies the following conditions: (1 1.4.2) For p E A, p + PO, {X,,(X)} is an even, positive periodic approximate identity such that (i) x,, E Weanfor each p E A, (ii) 1  X W ) = O(lP  pola) (P +.Po), Jon
ublx;(u>l
=
O 0,P +.Po). Proof. Assertion (i) follows by Theorem 1.5.8. To prove (ii) we proceed as in the proof of Lemma 11.3.1 and apply the Taylor formula (11.3.6) to g(h) = I,,(n$f;x). As x,, has a uniform second derivative for each p E A, we may interchange the order of integration and differentiation (with respect to h) to obtain
IImx
'
0)
a2

I A W x) =
& lon [ a % f ( x+ h) + a % f ( x h)lx;(u) du
for each p E A, x E R. Therefore
= O ( s h (h
 I){/'
uUlx;(u)ldu
= O(halp

po14a).
Lemma 11.4.2. If{x,,(x)}satisfies (11.4.2), then f E Lip* (Xan; a), 0 < a < 2, implies that
Proof. Let p E A be fixed. Since x,, E W&,, we have (ik)2x;(k) = [X;]"(k) by Prop. 4.1.8, and therefore {x;(k)} E I1. As in the proof of Lemma 11.3.2 we may apply Prop. 4.1.6 to deduce m kh xph(k) sin2 T f n ( k )eikx. I , , ( E % f ;x ) =  4
2
k=m
Since {(ik),x;(k)} = {[x3'(k)} E I1 by hypothesis, we again have
(11.4.3)
for each x E R. Since moreover
418
CHARACTERIZATION OF CERTAIN FUNCTION CLASSES
it follows by Lemma 11.4.1 that
Theorem 11.4.3. Let 0 < assertions are equivalent:
a
< 2, k,(x)} satisfy (11.4.2), and f E Xaz. The following
0) f E V[Xan; lkI"1, (P
+ 
Po),
(€to+). Proof. In view of (1 1.4.2), the proof of (i) 0 (ii) is given by Problem 6.1.10. To prove the implication (iii) (ii) we proceed as in the proof of Theorem 11.3.3. Indeed, by (1 1.4.3) and the hypothesis
f ~ ~asp x z n+~po = uniformly for p E A. Conversely, let (ii) be satisfied. Then ~ ~ R ~ ~ )  p o l O(1) by Lemma 11.4.2, which already proves (iii) if we set Ip  pol = E, provided we can show that f belongs to Lip+ (Xzn;a). But this is given by the following
Lemma 11.4.4. Ler f E Xan and 0 < a < 2. Then f E V[Xzn; lkl"] implies f E Lip* (Xzn;a).
The proof is left to Problem 11.4.3. Concerning characterizations of the classes W[Xan; lkla] we have
Theorem 11.4.5. Let 0 < a c 2 and f E Xan.The following assertions are equivalent:
(0 f E WtXan; lkl"1, (ii) there exists g E Xan such that lim,,,, IIRF'f  gllxln = 0. The proof may be given as for Theorem 11.3.5, in fact, the argument is simpler. So far we have seen that in case 0 < a < 2 the whole program of Sec. 11.3 may be carried over to periodic functions. The same is true for the general case a > 0. Theorem 11.4.6. Let f (i) f E W a n ; lklaI,
E
Xanand a > 0. The following assertions are equiualent:
wherej E N is chosen such that 0
0. Thefollowing assertions are equivalent: 0 ) feW[Xa,; lklaI, (ii) there exists g E Xansuch that (0 c a < 2j)
For the proofs of the last two theorems and for related results we refer to the
Problems below. Problems 1. Show that the periodic kernel {&(x, f ) } of Weierstrass (cf. Problem 1.3.10) satisfies (11.4.2) with po = 0 and p = 4;. Rewrite (11.4.2) such that po = 00, and show that the kernel {jn(x)}of Jackson (cf. Problem 1.3.9) is an example in this case. 2. Let 0 < a < 1 and / r R~ be fixed. If mh,a,nh,crare defined by (11.2,5), (11.2.6), respectively, show that m;,.(x) =  mh,a(x Zrj) and n:.(x) = 4%%  m n h . a ( X 2vj) belong to Lin, and IImh*,aIILJ, I Ihlallml,allL1*IIn;.aIILJ, 5 Ihla//nl,allL1. Furthermore, [rn:,.]^(k) = (elhk l)lkla, [n;,a]A(k)= {  i sgn k} (elhk  l)lkl" for k # 0, and [m:,aI"(O) = [n~,al^(0) = 0. Show that [m?..I = n;,.. 3. Prove Lemma 11.4.4. 4. Prove the following theorem of S. N. BERNSTEIN: f E Lip (Can;a) for a > (1/2) implies f A E 1'. State and prove extensions analogous to Theorem 11.3.8. (Hint: HausdorffYoung inequality) 5. Prove Theorems 11.4.6, 11.4.7. 6. Show that f~ V[Xan; lkl"] implies ~ ~ A ~ f ~=~ O( x 2lhl") n for 0 < a < r. 7. Let f E Xan and a > 0. Show that f belongs to V[Xan; {  i sgn k}lkl"] if and only if
+
4s
+
+ +
where]€ N is chosen such that 0 c a < 2j 1 and Ca,aj+lis given by (11.3.15). 8. Let f E Xzn and a > 0. Show that f belongs to W[Xan; {  i sgn k}(kl"]if and only if there exists g E Xzn such that (0 < a < 2 j 1)
9. Show that i f f € V[Xan; lkl"]for some a > 1 , thenfE W[X,,; (ik)'] for every integer I, 1 I 1 I [all. 10. Let f E Xzn and a > 0. Show that f belongs to V[X2,; lkl"] if and only if where r E P is such that CL = r + y, 0 < y < 2. (For the definition of f ( l ) see Def. 11.5.10). (Hint: BUTZERGORLICH [l])
11.5 Integral Representations, Fractional Derivatives of Periodic Functions
In this section we briefly discuss suitable extensions of the results of Sec. 11.2 to periodic functions. Before we begin with this program, let us return to functions
420
f~ Lp, 1 Ip
CHARACTERIZATION OF CERTAIN FUNCTION CLASSES
I2, for a moment, For instance in the integral case, we have shown that
f E WIL1; (iu)'] if and only iff belongs to Wily the latter class being defined by
differentiabilityconditions upon$ It was this aspect which we tried to extend to the fractional case in Sec. 11.2. But equivalently, we may say (Theorem 5.1.16) that f E W[L'; (ivy] if and only if there exists g E L' such that (1 1.5.1)
s(X>
= m:J
dul
J:
du, * '
.I:,'
g(Ul.1dur
(~rg)(x),
where each of the iterated integrals represents a function of L'. In other words, we may also arrive at an integral representation of$ In the fractional case we have the following counterpart: Theorem 11.5.1. LetfE L' and 0 < a c 1. Thenf belongs to W[L'; lola] ifand only if there exists g E L1 such thatf(x) = [Rag]@)almost everywhere. Proof. Letfe W[L'; Iul"] and g E L' be such that lul.f"(u) 11.2.2 (11.52)
(elhu
 l)f^(u)
=
 1)lul "g"(u)
(elhu
=
= g^(u). Then by
[mh,.
Lemma
* g]^(u)
for all u # 0, and thus by the uniqueness theorem for each h E R (11.5.31 f ( x
+ h)  f ( x ) = h h , c r * g)(X) = [R,gl(x + h)  [Ragl(x)
a.e.
Integration with respect to x over [0, y ] gives
Letting h f GO we have (cf. Problem 11.2.2) that the righthand side tends to zero, and therefore&) = [R,g](x) a.e. Conversely,if there is g E L1 such thatf(x) = [R,g](x) a.e., we may invert the single steps of the above arguments to deduce (11.5.3) and (11.5.2), i.e.fE WIL1; Iula]. Regarding the extension to arbitrary values of a we first note Theorem 11.5.2. Let f E L' and r E N. Thenf belongs to WIL1; Iul'] ifand onIy i f f E La and there exists g E L' such thart (11.5.5)
[ H ' f ] ( x ) = Jx OD
dul
S"
dug.. .
g(ur),du, a.e.,
OD
where each of the iterated integrals exists as a function of L'. Proof. LetfE W[L'; lulr] and g E L' be such that lul'f"(u) = g"(u). Since r 2 1, this impliesf^ E La and thusfE La by the theorem of Plancherel. Therefore H f ( s f * ) and all powers HkA k E N,belong to La byTheorem8.1.7. Moreover, Hakf = ( 1)Yand H 2 k + l f = (1)"f" by Prop. 8.2.10. Let r be even, say r = 2k. Then (iu)ayA(u) = [( l)kg]^(u) by hypothesis and therefore (cf. (11.5.1)) f ( x ) = ( 1)"Jrg)(x) a.e.,
t We recall that H denotes the operation of taking Hilbert transforms (cf. Def. 8.0.1).
42 1
CHARACTERIZATION IN THE FRACTIONAL CASE
which proves (1 1.5.5) for even values of r. If r is odd, say r = 2k + 1, then by Theorem 6.3.14 there exists g , E L1 such that Iu]fA(u) = gT(u). Therefore IulakgT(u)= g"(u), and thus the result for the even case may be applied to deduce gl(x) = ( l)k(Jakg)(x)a.e. But, in view of Problem 8.3.3, IuI f"(u) = gy(u) implies the representation f  ( x ) = sf gl(u) du a.e. and therefore f "(x) = ( l)k(Jrg)(x) a.e., which completes the proof of (11.5.5). Sincefe L' n La, we may apply Prop. 8.3.1 so that the converse follows in view of (10.6.7).
Theorem 11.5.3. Let f E L' and a > 1 be such that following assertions are equiualent:
+ Is, 0
a. It follows that H ' f * p ) f  a belongs to X 2 , for each f e X,,. Indeed, if e.g. X,, = Can,thenfand thus H'fbelong to LS,, for every 1 < s < 00. On the other hand, vr"belongs to Canif r  a > 1, and to L;, 1 5 q < 2/(2  r + a), if r  a I1. Therefore H'f* vr a E Canby Prop. 0.4.1.
+
425
CHARACTERIZATION IN THE FRACTIONAL CASE
Theorem 11.5.7. Let f E X2, and ci > 0. Suppose that r E N is such that a < r. Then f E W [ X , , ; Ikl"]ifandonlyiff~L%,forsorneqwithl < q c coandH'f*rpr,be1ongs to w;c2,. Proof. If YE W[X,,; lkl"] and g E X, is such that Ikly"(k) = g"(k), we have just shown that H'fexists and belongs to Lz,. Therefore by Prop. 9.1.8 and the convolution theorem (ik)"H'f* rp,,]"(k) = (iky{i sgn k)'f"(k)lkla' = g"(k), and the assertion follows by Prop. 4.1.9, since H'f* pr"belongs to X2,. Concerning the converse, we have by Prop. 4.1.8 and 9.1.8 that there exists g E X,, such that g"(k) = (jkY[H'f*~ral"(k) = (iky{i sgn k)'f"(k)lklu' = Ikl"f"(k), i.e. .f belongs to W[X2,; lkla]. For the particular case X,, = Cznwe have
Corollary 11.5.8. Let f E CPn and a > 0, r E N being such that f E W[C2,; lkl"] with Ikl"f"(k) = g"(k), g E C ,, if and only iffor all x
a
< r. Then
(11.5.17)
+
This is to be seen in contrast with the representation f ( x ) = f"(0) (rp" * g)(x) obtained in Theorem 11.5.4. The same methods of proof may be employed to derive the following characterization of the classes V[X,,; JkJ"].
Theorem 11.5.9. Let f E X,, and a > 0, r E N being such that a < r. Then f E V[X2,; lk1"] ifand only i f f E l;, for some q with 1 < q < 00 and H ' f * rpra belongs to V;c& The following definition is now motivated. Definition 11.5.10. Let f E X,, and a > 0. If there is g E X,, such that Ikl"f"(k) = g*(k), then g is called the ath Riesz derivative o f f and denoted by f ('I. Thus we have proceeded quite differently to the case of functions belonging to L', 1 I p I 2. The present approach to the definition of a Riesz derivative of order a has the advantage that it is easier to deal with, at least in the periodic case. Note that we do not necessarily need the notation DP)f since the ath Riesz derivative may be interpreted in the strong sense as well as in the pointwise sense, in the latter case the definition would be given by (1 1.5.17). So far we have only discussed fractional derivatives of periodic functions based upon the Riesz integral. Also Weyl's definition of a derivative of fractional order may be treated similarly. Let a > 0, r E N, andfE X,,, We recall Prop. 4.1.8 which in particular states that f E W& implies (ik)'f"(k) = u(r)]"(k).Therefore, if there exists g E X2, such that (ik)"f "(k) = gA(k)t,then g may be regarded as a fractional derivative off 'f (ik)' is defined by (ikY = lkl" exp {i(na/2)sgn k), the principal branch.
426
CHARACTERIZATION OF CERTAIN FUNCTION CLASSES
known as Weyl's derivative of order a, in notation f ( = ) .In view of Problem 11.1.2 we may ask for its connection with the RiemannLiouville integral. Let us consider (11.5.18) Since (cf. (11.5.12), (11.5.15)) (11.5.19)
va(x) = ya(x) cos
na
+ $u(x)sin na 2'
it follows that is defined and belongs to L:, for every a > 0. In fact, the series (11.5.18) is the Fourier series of va. Therefore, if the Weyl derivative f@)off E Xzn exists for some a > 0, then f (x) = f"(0) + (va *f("))(x)since the Fourier coefficients of both sides coincide. On the other hand, we may proceed for 0 < a < 1 as in the Riesz case and deduce (cf. Problem 11.1.2) that for x > 0 1 where the series is absolutely and uniformly convergent for 0 I x I 27r. Moreover,
Here we have used the fact that JFf (")(u) du = 0 which follows since (ik)"f^(k) = Lf(")]"(k)by definition. Thus, if we assume that f E X,, is such that f "(0) = 0, then the existence of the ath Weyl derivative f(@off for 0 < a < 1 implies that f is almost everywhere equal to the RiemannLiouville integral (a =  co) of order a off(") (and vice versa). If a is a natural number, say a = r, then we recall Theorem 4.1.10 and (10.6.6) which show that (ik)'f"(k) = g"(k), g E Xan,implies
where pr is an algebraic polynomial of degree (r  1) at most. For further information concerning the Weyl derivative and RiemannLiouville integral of order a we refer to Sec. 11.6.
427
CHARACTERIZATION IN THE FRACTIONAL CASE
Problems
1. (i) Let f
E L1 and 0 < a < 1. Then f belongs to VIL1; Iola] if and only if there exists p E BV such that f ( x ) = [R, dp](x) almost everywhere. (ii) Let f E L1 and a 2 1. Then f belongs to VIL1; lul"] if and only if f E La and there exists p E BV such that
a = [a]
+ 8, 0 < 8 < 1 : R6 d p belongs to L' and
2. (i) Let f E Lp, 1 c p I2,and 0 < a < (l/p).Then f belongs to W[Lp; Iola] if and only if there exists g E Lp such that f ( x ) = [R,g](x) almost everywhere. (ii) Let f E Lp, 1 < p 5 2, and a > 0. Then f belongs to W[Lp; Iula] if and only if assertion (iv) of Theorem 11.0.1(for 1 < p 5 2) is valid. (Hint: Problem 11.2.2 and Theorem 6.3.14,the integrals exist in the sense of Theorem 5.2.21) 3. Prove the following theorem of H. WEYL:Let f E Can and 0 < a < 1. If there exists g E L& such that g(u) du = 0 and
En
(11.5.21)
f (x)
=
Ix
1
(x
u)"'g(u) du,
m
En
then f belongs to Lip (Can; a). Conversely, if f ( u ) du = 0 and f E Lip (Can;a), then for each 8 with 0 < 8 < a there exists g E Cansuch that (1 1.5.21)holds with a replaced by 8. 4. Let f E Xznand a > 0, r E N being such that 0 < a < r. If the ath Weyl derivative o f f exists (i.e., there exists g E Xan such that (ik)"f^(k) = gA(k)), then f belongs to Lip, (Xan;a). Conversely, iff E Lip, (Xan;a), then thepth Weyl derivative off exists for every 0 < 8 < a. 5. (i) Let CL > 0, r E N being such that 0 < a < r. Show that pa of (11.5.12)belongs to Lip, (L:n; a). (Hint: For r = 1 see Problem 11.4.2, for arbitrary r use the decomposition of rpa = rpa,, * rpU/,* . . * rp",, as an rtimes convolution) (ii) Discuss the extension of M,* of (11.5.9) to a 2 1 by the rtimes convolution of M:/,, where r E N is such that 0 < a < r. (iii) LetfE V[Xan; lk1"]forsomea > 0.Using~uof(11.5.12),showthatf~ W[Xan; Ikle] for every 0 < 8 < a. 6. Let a > 0 and 8 > 0, rl, ra being such that 0 < a < rl, 0 < rl /3 < ra. Then f E Lip,, (Xan;a) implies 936 * f E Lip,, (XSn;a /3). (Hint: ZYGMUND [7II,p. 136n 7. Let t, be a trigonometric polynomial of degree n at most, rn(x) = 2;= n ck exp {ikx). Show that the ath Weyl and ath Riesz derivative are given by (a > 0)
+
+
2 n (ik)'ck n
tF)(x) =
n
elkx,
kn
tF)(x) =
L\ Ikl'ck k= n
elkX,
respectively. Prove the following inequalities (of Bernstein type) IItP)IIxan 5 M1(a)naIItnIlxanr
Iltt(la)Ilxan Md~)naIItnIIxan* (Hint: cf. Theorem 2.3.1and BUTZERGORLICH [2,p. 3551, BUTZERSCHERER [I, p. 1071) 8. Let a in (11.1.2)be co and 0 c a < 1. Then f E WIL1;(io)"] if and only if there exists g E L1 such that f ( x ) = [L,g](x) a.e. (cf. Problem 11.2.7).
428
CHARACTERIZATION OF CERTAIN FUNCTION CLASSES
11.6 Notes and Remarks The notion of fractional differentiation is almost as old as the calculus itself. The origins of the subject are attributed to G. W. LEIBNIZ(see H. T. DAVIS[l], LEIBNIZ[l]) who mentioned it in a letter to G. F. A. DE L'HOSPITAL as early as 1695. N. H. ABELappears to have made use of the notion in 1823 in solving the tautochrone problem. But the first systematic study is due to H. J. HOLMGREN [l], J. LIOUVILLE [l], and B. RIEMANN [l]. More recent papers of importance are those of J. HADAMARD [l], H. WEYL[I], E. L. POST 111, G. H. HARDY and L. E. LITTLEWOOD [2], and M. RIESZ[2]. (For a rather complete bibliography of 173 items see the doctoral dissertation of M. GAER[l].) Chapter 11 was written in collaboration with WALTER TREBELS. Sec. 11.1. The authors have largely followed M. RIESZ[2, p. 1&26] in this section. The material is also treated in books on distribution theory, e.g. GELFANDSHILOV [lI, p. 1151, SCHWARTZ [l]; for the Hadamard approach see also ZEMANIAN [l, p. 15 ff]. For fractional derivatives of functions analytic in a domain of the complex plane see e.g. GAERRUBEL [l] and OSLER[l], who in particular gives applications to infinite series. For generalizations and applications connected with the theory of ordinary and partial differential equations see WEINSTEIN [ 1, 21. Sec. 11.2 and 11.5. The actual goal of Sec. 11.2 is to associate fractional derivatives with the class W[Lp; Ivl"] (a > 0) by trying to imitate the classical approach to the RiemannLiouville derivative for periodic functions, whereby for 0 c a c 1 one first integrates fractionally of order (1  a) and then takes the first (ordinary) derivative (compare e.g. ZYGMUND [7II, 133 ff]). This is obtained by replacing the RiemannLiouville by the Riesz integral and the operation dldx by (dldx)H, keeping in mind that f * Hrnhsl. = H L f * I t l h , l  a ] ( H being the Hilbert operator, all limits being understood in the strong sense, see Def. 11.2.5). The functions mh.ar nh.a were first introduced by OKIKIOLU [l]; for their present modified form, particularly the connection between them given by Lemma 11.2.2(iii), see BUTZERTREBEU[l, 21. The development leading to Def. 11.2.5 is due to TREBELS, see also NESSELTREBELS [ 11; for preliminaries in this direction see BUTZERTREBELS [2]. Whereas we have defined the Riesz derivative on X(W) as the strong limit of a generalized differencequotient, a more immediate approach would be via the transformed state, i.e., iff€ Lp, 1 Ip I2, it is the (unique) g E Lp (if it exists) given by Ivlaf*(u) = g"(v). The equivalence of both definitions is then given by Theorems 11.2.61 1.2.9. For Theorems 11.2.611.2.12 see also COOPER [2], BUTZERTREBELS [2]. Whereas the definition via the transformed state has the advantage that a fractional derivative is defined directly for all a > 0, Def. 11.2.8 actually reveals the character of D!"!fas a derivative (in the usual sense). Nevertheless for periodic functions we have chosen Def. 11.5.10 (see also BUTZERSCHERER [l, p. 101 ff]). The equivalence to a definition via a difference quotient is then given by Prop. 11.5.6 (for 0 c a c 1). In view of Theorem 11.5.7 the latter definition also works for arbitrary a > 0. Indeed, iff E W[X,,; lkl"] and r E N is chosen such that a c r, then H ' f * q+a = f * H ' ~ I ,  , = H'(f * V 7  a ) . Thus the ath Riesz derivative (in the sense of Def. 11.5.10) is obtained by first integrating f fractionally of order (r  a):f* T ,  ~ ,followed by the operation ((d/dx)H)I. The integral representations for functions on W given by Theorems 11.5.11 1.5.3 may be new. In order to obtain their periodic versions we applied the general reduction method of CALDER~NZYGMUND [2] (see Sec. 9.4) to the kernels lxlal and {sgn X } I X I ~  ~ ; see also ACHIESER [2, p. 1131, ZYGMUND [71, p. 691. Though we understood the integral representations mainly as inverse operations to fractional differentiation, we actually defined fractional integration. Thus for functions f E Xanwith f"(0) = 0, the convolution f * tpU (f*ria) is the Riesz (RiemannLiouvilleWeyl) fractional integral of order a > 0; see also TIMAN [2, p. 1181. The integral representations for periodic functions are wellknown and often used to define specific classes of functions. Thus, by Wa@), a > 0,s real, SZ.NAGY[3, 71,
CHARACTERIZATION IN THE FRACTIONAL CASE
NIKOLSKII [2], S T E ~ K [I I N] denote the class off is f = @ a , o * g, where
E
Can(with f"(0)
= 0) whose
429 representation
and g"(0) = 0, g E L&. Note that 0,. d x ) = y,(x), @ ,, .(x) = ~ , ( x ) ,@., l ( x ) = t,h,(x) in our terminology (cf. (1 1.5.121, (1 1.5.15),(1 1.5.18)). The particular classes Wa(0),Wa(a)are of special interest (e.g. the problem of best constants posed by FAVARD [l], see LORENTZ [3, p. 117 ff]). In view of (11.5.19) the RiemannLiouvilleWeyl integral o f f € XZn, f"(0) = 0, is a linear combination of the Riesz and conjugate Riesz integral off and vice versa. Correspondingly, for the fractional derivatives in XZn,
For similar relations for derivatives on X(R) we refer to BUTZERTREBELS [2] and the literature cited there; see also KOBER[3]; compare furthermore ALJANCIC[2], DZJADYK [l], EFIMOV[2, 31, MALOZEMOV [l], OGIEVECKL~ [l], TELJAKOVSKI~ [l, 21. Integrals of the type (1 1.3.1)were apparently first considered by MARCHAUD [ l ] (1927) for continuous functions defined on a finite interval. Theorem 11.4.3 for Cznspace is due to ZAMANSKY [31 for a = 1 and to SUNOUCHI [3] for 0 < a < 1. For Xznspace and 0 < a I 1 as well as for a somewhat different generalization for a > 0 (cf. Problem 11.4.10) see BUTZERGORLICH [l]; for the present form of Theorem 11.4.6 see SUNOUCHI [S]. The method of proof here is, in all cases, a generalization of one due to SALEMZYGMUND [ 11, modified by ZAMANSKY [31. In the case of functions defined on the line group these results, particularly Theorem 11.3.3, are given in BUTZERTREBELS [2]. See also Sec. 13.2.4. The form of the operation Rp'ffor e + O+ suggests that it is connected with the Riesz integral R, f of (1 1.1.16). This fractional integral of order a, also known as Riesz potential of order a, was defined for 0 Ia < 1 (fsufficiently smooth). If one would ask for a Riesz potential of negative order, one would have to consider Sec. 11.311.4.
for a > 0 and E O + . Since this expression in general diverges on a set of positive measure, we consider its 'Hadamard finite part' (see e.g. BUREAU[l, 2D for 0 < a < 2, namely
Thus the limit of Rp'ffor e + O+ may be regarded as an extension of the Riesz integral to [l], D u negative powers, or as a Riesz potential of negative order (compare BALAKRISHNAN PLESSIS[l], STEIN[l]); expressed otherwise, the limit of Rp) for e + O + is the inverse operation to R,. We have e.g. actually shown for 0 < a < 1 (see Theorem 11.3.5, 11.5.1): if g E L1 and Rag E L1, then slim,,o+ Rp)(R,g) = g; conversely, if f E L1 and slim,,o+ Rpyexists, then Ra(slim,+o+ RF)f) = f. HEYWOOD [l] established an analogous pointwise result for R,"f in case a > 0, 1 < p < l/a. The extension of Riesz potentials to arbitrary negative orders may also be achieved by using Hadamard's finite parts. But we will not regularize the divergent integral (11.6.1) using a Taylor expansion o f f (sufficiently smooth; see WHEEDEN [1,2]) but a (central) difference off of sufficiently high order. The resulting regularization may also be justified by the fact that the operator RF?, defined by
430
CHARACTERIZATION OF CERTAIN FUNCTION CLASSES
can be considered in the setting of general results due to WESTPHAL [l, 21; see also BERENSWESTPHAL[l, 21, BERENS [3]. Here the space Lp, 1 Ip I2, is replaced by a Banach space X, the group of translations by a oneparameter group of uniformly bounded operators G(5) of class (C,) on X. If the infinitesimal generator A of G(5) is defined by Af = slimc*o{'[G(lJ  Ilffor allfbelonging to the domain D(A) (cf. (13.4.8)), Miss WESTPHAL developed a theory of fractional powers of the generator A. More explicitly, for a > 0, f E D((  Aa)a'2) if and only if (1 1.6.2)
exists, Ca,a,being defined by (11.3.14). Comparing this general result with ours justifies calling slim,,, + RghJ a fractional Riesz derivative. In particular, for the translation group on Lp, 1 Ip s 2, for which A = d/dx, we have shown (Theorem 11.3.7) that the limit (1 1.6.2) exists if and only iff€ W[Lp; Iul'l, which thus characterizes D(( d2/dx2)aJa). The operator ( d2/dx2)u12was introduced by BOCHNER [41 (see FELLER [l]), and our results assert (see footnote to Def. 11.2.8) that (a > 0).
Concerning Theorem 11.3.8, the original result of BERNSTEIN [2] (see Problem 11.4.4), also generalized by SzAsz [l], may be found in ZYGMUND[71, p. 2401; for that of TITCHMARSH [3] (Problem 11.3.8) see TITCHMARSH [6, p. 1151; for stronger results, also in the setting of intermediate spaces, see HERZ[l], PEETRE 131, TAIBLESON [l]. Concerning Theorem 11.3.10, the result (1917) of WEYL[l] (see Problem 11.5.3) does [2, p. 731, not seem to have been considered in the literature apart from BUTZERTREBELS HERZ[l]. For the use of the integrals (11.3.2) in connection with strong asymptotic expansions of singular integrals compare Problems 3.4.6, 8.2.8, 9.2.7, and the literature cited there.
Part V Saturation Theory
This Part is devoted to a study of saturation theory for convolution integrals. The method to be employed is the Fourier transform method already familiar to us from Chapters 7, 10, 11. Chapter 12 deals with the more classical aspects of the method, thus treating the saturation problem in X,, and Lp, 1 I p I 2. Chapter 13 gives the extension to all X(R)spaces by duality arguments. Furthermore, the comparison of the error of approximation corresponding to two different processes is studied together with a brief account of saturation theory for strong approximation processes on arbitrary Banach spaces.
This Page Intentionally Left Blank
12 Saturation for Singular Integrals on X2Tand Lp, 1 5 p 5 2
12.0 Introduction The problem of determining the optimal order of approximation of a function f by a sequence of polynomials allows two different interpretations : either one varies the sequence of polynomials that approximates an f satisfying given properties, or, one keeps the approximation process fixed and varies the properties of the function f to obtain the optimal order. In the former case one obtains a result on the best order of approximation En(Xz,;f) for all f satisfying the given properties; but, in general, no information is available concerning the sequence of polynomials for which this optimal approximation is attained. In the latter case the result is that the approximation by the given process will be optimal for all functions belonging to a certain class, the socalled saturation class. The first interpretation is essentially due to P. L. CHEBYCHEFF (1857); the second was posed by J. FAVARD in 1947. Saturation theory is the study concerned with the critical or optimal order of approximation by a given approximation process. In this respect, let us consider the Fejtr means un(f; x) of the Fourier series off E Czz.Although these converge uniformly to f as n t a~(Cor. 1.2.4), they cannot tend too rapidly to f no matter how smoothf may be. Indeed, IIon(f;0) f(o)I[ czn= o(n') necessarily impliesf(x) = const. Thus for all nonconstant functionsf E Cznthe o,Cf; x) approximatefwith an order of a t most O(nl). On the other hand, this critical order is actually attained by particular functions in Cz,.Indeed, for fo(x) = exp {ix}
One says that the FejCr means are saturated in Cznwith order O(nI). The problem now is to characterize the class of elements f E C,, for which this optimal order of approximation off by .,,(Ax), namely O(nl),is in fact attained. For this purpose, we recall Theorem 2.4.7: in the case 0 c a < 1, IIon(f;
28F.A.
0)
f(o)IIc,,
=
O(n3
efE Lip (Czn; a).
434
SATURATION THEORY
But, as we have seen in Sec. 2.4, the methods of proof employed there are not sufficiently powerful to cover the critical case a = 1. In fact, we shall show (cf. Problem 12.2.8) that It may be expected that this particular instance has an analog when Can is replaced by an arbitrary Banach space and un(f; x) by a strong approximation process on X satisfying suitable conditions. Let X be a complex Banach space with norm 11 1 X , and let p be a positive parameter varying over some set A c R and tending to p o (cf. Def. 1.1.1). 0
Dewtion 12.0.1. A family {T,}, p E A,of bounded linear operators T, of X into X is called a (uniformly bounded) strong approximation process on X if for each f E X (12.0.1)
I l T D f l l X 5 M IlfIIx,
the constant M being independent of p (12.0.2)
E
A and f
E
X, and
lim llTPf  fllx = 0.
P+Po
The strong approximation process {T,} on X is said to be commutative i f T,,(T,,f 1 = TP1(TPIfIfor each P1, Pa E A andf E x. In view of the uniform boundedness principle, condition (12.0.2) alone implies the existence of a constant M such that 1 T,f IIx I M 1 f (Ix uniformly for all f E X and all p close to po. Condition (12.0.1) in addition assumes that M is independent of p E A.
Definition 12.0.2. Let {T,}, p E A, be a strong approximation process on the Banach space X. W e shall say that the process {T,} possesses the saturation property i f there exists a positive function ~ ( pon) A tending monotonelyt to zero as p po such that every f E X for which f
(12.0.3)
IITPf
 f IIX
= O(T(P))
is an invariant element of {T,}, i.e. T,f = ffor all p (12.0.4)
F[X; TPl = { f E x I IITPf  f
E
IIX =
(P
f
Po)
A,and i f the set
O(dP)), P +Po}
contains at least one noninvariant element. In this event, the approximation process {T,} is said to have optimal approximation order O(q~(p))or to be saturated in X with order O(dp)), and F[X; To]is called its Favard or saturation class. It should be emphasized that the saturation problem actually consists of two different questions : firstly, the question of whether saturation holds, i.e., the establishment of the existence of the saturation property of a given strong approximation process {T,}; secondly, the characterization of the Favard class F[X; T P by ] structural properties upon f.
t
This is not essential; pi(p) may be any function on
12.1.4, 12.2.1).
A tending to zero as p t po (cf. Problems
SATURATION FOR SINGULAR INTEGRALS
435
The solution of the latter problem will be referred to as an equivalence theorem for the Favard class. (As is to be expected by the example of FejCr means, the methods of proof of an equivalence theorem in the saturation case may be quite different from those employed in the nonoptimal cases.) The complete solution of both problems, thus of the saturation problem, is called a saturation theorem for the given approximation process. As in the nonoptimal case of Chapter 2, the direct part of a saturation theorem is the direct one of an equivalence theorem for the saturation class. The inverse part of a saturation theorem is defined correspondingly. The first two sections of this chapter are devoted to a study of saturation on the circle group and the next two sections to the line group. Whereas Sec. 12.1 is concerned with the existence and inverse problem of saturation theory for periodic convolution integrals, Sec. 12.2 deals with the direct problem and the associated Favard classes. The submethods presented and the applications thereto can usually be interchanged. Important submethods are those based on positive kernels, uniformly bounded multipliers, and functional equations. Sec. 12.3 treats the theory in Lp, 1 Ip I2, while Sec. 12.4 is reserved to a study of various convolution integrals as illustrations to the theory presented. Although the general underlying method of proof of our results in X,, as well as Lpspaceis the Fourier transform method, the results for both space types are presented in such a fashion that they are complementary with respect to the set of submethods of proof (each applicable to an important subclass of problems) as well as examples. Sec. 12.5 is concerned with saturation of higher order for both space types.
12.1 Saturation for Periodic Singular Integrals, Inverse Theorems
Let f E X,, and I,( f;x) be a periodic singular integral as given by (1.1.3) with kernel {x,(x)}. Our first aim is to give a sufficient condition such that the saturation property holds for ZP(Ax ) :
(12.1.1) Given a kernel {x,(x)} satisjying (1.1.5), let there exist a function #(k) on Z with #(O) = 0, #(k) # 0for k # 0, and a positive function cp(p) on A tending monotonely to zero as p + po such that
(k E Z). The latter limitrelation holds trivially for k = 0 since x;(O) = 1, p E A, by (1.1.1). Furthermore, (I 2. I. 1) implies limp x;(k) = 1 for every k E Z, whereupon (I. 1.5) and Problem 1.3.3 guarantee that lim, l[Zo(A0 )  f(0)IIx2, = 0 for every f E XS,. +
Proposition 12.1.1. Let f
E X,,
and {x,(x)} satisfy (12.1.1). If there is g E X, such that
(12.1.2) then f belongs to WIX,,;
#(k)].
436
SATURATION THEORY
Proof. We apply the (finite) Fourier transform method as in Sec. 10.1. Indeed, [Zp(f; o)]"(k) = x;(k) f"(k) by the convolution theorem, and thus by (4.1.2)
which in view of (12.1.1) and (12.1.2) givesf E W[X,,; #(k)]. Proposition 12.1.2. Iff E X,, and &,(x)} satisfies (12.1. I), then IlZP(A
0)

f(O)llXP,
= O(Cp(P))
(P+
Po)
implies that f is constant. Moreover, the constant functions are the only invariant elements of the singular integral Z p ( f ; x). Proof. Since (12.1.2) holds with g(x) = 0 in Xan,we conclude #(k)f"(k) = 0, k E Z, and since #(k) # 0 for k # 0, this impliesf"(k) = 0 for k # 0. Therefore f = const by the uniqueness theorem. Since x;(O) = 1, p E A, it is obvious that the constant functions are invariant elements of ZP(A x). Conversely, if f E Xzn is invariant, then h;(k)  11f"(k) = 0,k E Z, by the convolution theorem. By (12.1.1) it follows that #(k)f "(k) = 0,k E Z,and thusf = const as above. If the kernel of the singular integral I p ( f ;x) satisfies (12,1.1), then it is rather obvious that there exist noninvariant elementsf€ X,, such that actually (12.1.3)
IIIAf;
0)
 f(o)IIxa,
= O(V(P))
(P
+.Po).
For instance, for the function fo(x) = exp {ix} we have Zp(fo; x) = x;(l)fo(x), and hence
Therefore we may state
Theorem 12.1.3. Let f E X,,. If the kernel {x,(x)} of the singular integral I p ( f ; x) surisjes (12.1.1), then ZD(f; x) is saturated in X,, with order O(p(p)). Hence (12.1.1) is a sufficient condition such that the saturation property holds for the singular integral ZP(& x). On the other hand, there are also certain converse statements. For the sake of simplicity, let us assume that the kernel {xP(x)}is even and positive; then the coefficients $(k) are real numbers, bounded by one (cf. Sec. 1.5.2). Obviously, the assertions of Theorem 12.1.3 remain valid in this case if instead of (12.1.1) we only assume that (with y and $ given in (12.1.1), compare also (1.7.2)) (12.1.4)
exists (as a finite number). Concerning the converse, suppose that &,(A x) is saturated in X,, with order O(dp)), constants being the only invariant elements. Furthermore, assume that the saturation class contains all trigonometric polynomials. The left side of (12.1.4) defines a certain function f l k ) on Z with @O) = 0 and #(k) 5 0 for k # 0. In fact, #(k) is a
437
SATURATION FOR SINGULAR INTEGRALS
finite number for each k E+. For, suppose that $(m) = = exp {imx} (cf. (1.3.14))
03
for some
ni E Z.
Since for
fm(X)
this would imply that fm does not belong to the saturation class of I,( f ; x), a contradiction to the assumption. Furthermore, $(k) # 0 for k # 0. For, if there exists ko # 0 such that W o )= 0, then it follows (as above) that IIZp(fo;  fo(o)Ilxz, = a(&)) for fo(x) = exp{ikox>,which is a contradiction to the assumption that the constants are the only invariant elements. Therefore, under the present assumptions, condition (12.1.4) with y and $ given in (12.1.t) is not only sufficient but also necessary for the saturation property of IP(kx) to hold; compare also Sec. 12.5.2, 12.6, and Problems 12.1.4, 12.1.5, 12.2.14. 0)
Having checked that the saturation property holds for every singular integral x ) with kernel satisfying (12.1.1), we now have to treat the associated problem of characterizing the Favard class. In this respect, the following theorem gives a solution of the inverse part using the Fourier transform method as employed in Sec. 10.2.
Z,,(fi
Theorem 12.1.4. Zff fE WX,,; $(k)l.
E
X,
and
h,(x)}surisfies condition (12.1.1), then (12.1.3) implies
Proof. Firstly, let X,, be C,,. By hypothesis there is a constant M > 0 such that Il(q~(p))~[Z,(fi  f(o)]II I M as p + po. By the weak* compactness theorem for 0)
L,", there exists g E L,", and a sequence { p j } with lim,
pr = po such that
since exp {  ikx} E Li,, k E h. On the other hand, the lefthand side equals (12.1.5) Thus $(k)f"(k) = g"(k), k E H. For the cases X, = L;, 1 < p < to, the proof is much the same using the weak* compactness theorem for L;., Finally, let X,, = Li,. If we set
then {pp(x)}is a family of 2rrperiodic, absolutely continuous functions, and assumption (12.1.3) means that there is a constant M > 0 such that ) ) ~ , I ) B VI~ ~M as p f po. We now infer by the theorem of HellyBray that there exists p E BV,, and a sequence { p j } with lim,, pr = po such that
since exp {ikx} E Can,k E H. For the lefthand side we again have (12.1.5) and thus $(k)f^(k) = p"(k), k E Z, which completes the proof. Thus the Favard class F[X,,; ZJ of the singular integral Z,(fi x ) with kernel satisfying (12.1.1) is contained in V[X,,; #(k)]. To prove the converse implication (i.e.,
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438
the direct part of a saturation theorem) which would complete Theorem 12.1.4 to an equivalence theorem, one would need further information upon the kernel. This is postponed to the next section, whilst we here proceed by examining condition (12.1.1) for certain examples. Let us first consider the singular integral of FejCr as given by (1.2.25). Here the parameter p is discrete, namely n, with A = N and po = co. Since for the kernel {F,(x)} we have F;(k) = 1  (Ikl/(n 1)) for Ikl In and F,^(k) = 0 otherwise (cf. (1.2.43)), it follows that
+
(k E Z). Thus {Fn(x)}satisfies (12.1.1) with d n ) = n  l and #(k) =  Ikl.
Corollary 12.1.5. The singular integral of Feipr is saturated in X,, with order O(nl), and its Favard class is contained in V[X,,;  lkl]. But there are other examples of singular integrals for which (12.1.1) is not so easily evaluated. For instance, for the kernel {k,(x)} of the singular integral of FejkrKorovkin (cf. Sec. 1.6.1) we have
0, Ikl > n. Certainly, one could show by a Taylor series expansion that this kernel satisfies (12.1.1) with cp(n) = n  a and #(k) = (7r2/2)ka. On the other hand, if we would exploit the fact that the kernel is positive, we may reduce the verification of (12.1.1) to the particular case k = 1 (and k = 2). Indeed,
Proposition 12.1.6. If the kernel {xp(x)}with parameter p E A is even and positive and if q(p) is a positive function on A tending monotonely to zero as p + po so that
then the singular integral Io(f; x) is saturated in X,, with order O(&)), and its Favard class (12.1.6)(ii) is satisfied if (i) and the condition
is contained in ~ [ X Z , ;ck2]. In particular,
(1.5.27), i.e. (12.1.7) hold.
Proof. We first observe that (12.1.6)(i) implies limp+oox~(l) = 1, which by Prop. 1.3.10 IIIo( f ;  f ( o ) I l x 2 , = 0 to be valid for is a necessary and sufficient condition for lim, every f E Xan. 0)
Now for h(x) = ka(l  cos x )  (1  cos k x ) , k E E, we have (e.g. by the mean value theorem) that 0 Ih(x) Ik4x4/4!.Since the kernel is positive, hypothesis (12.1.6)(ii) implies that IAh; 0) = o(cp(p)), p +po. Furthermore, I,,(h; 0) = ka(l  x;(l))  (1  x;(k)), and thus X;(k)
 1 = ka Xz(1)  1
dP)
9.m
I d h ; 0) +.
dP)
439
SATURATION FOR SINGULAR INTEGRALS
Hence {xO(x)}satisfies (12.1.1) with $(k) = c ka by (12.1.61, and the conclusion follows by Theorems 12.1.3 and 12.1.4. Finally, in virtue of (1.5.6) we have 7r4
2n
n
u4x,(u) du I (1  x;(l)) 8
and thus (12,1.6)(ii) follows by (i) and (12.1.7). Returning to the singular integral of FejerKorovkin we have in view of kC(1) = cos (n/(n + 2)) and (1.6.6) that the kernel {kn(x)}satisfies (12.1.6)(i) and (12.1.7) with v(n)= n  2 and c = n2/2. Therefore Corollary 12.1.7. The singirlar integral of FejirKorovkin is saturated in Xzn with order O(n2), and its Favard class is contained in V[XZ,; (7rz/2)kz].
Problems 1. Show that Def. 12.0.2 is equivalent to the following: The strong approximation process {T,}, p E A, on the Banach space X is said to be saturated in X if there exists a positive function y(p) on A tending monotonely to zero as p + po such that for each f E X which is not invariant under the given process
'
IlTPffllx CldP), where C1> 0 depends only upon f, and if there exists at least one noninvariant element fo E X such that I I T P f O  f o l l x < CZdP), where Cz > 0 is another constant depending only upon f o . 2. Prove Theorem 12.1.4 by considering the quantity u,((&p))  [ 1,cf; 0 )  fb)] ; x). (Hint: Use Cor. 6.1.3; see also the proof of Theorem 12.3.6) 3. Show that Prop. 12.1.1 remains valid if (12.1.2) is replaced by
lim inf,,,,
II(dp))l
V,(L
0)
 f(o)l 
do)llxan
= 0.
4. Let the kernel {x,(x)} of the singular integral Z,( f ; x) be even and positive. (i) Show that Z,(L x) is saturated in Xanif and only if (12.1.4) holds, where $(k) > 0, k # 0, may be infinite for certain values of k but there exists at least one m E N such that $(m) < co. (Hint: As to the sufficiency, compare the proof of Theorem 12.1.3. As to the necessity, let ZP(L x) be saturated with order O(&p)), and constants being the only invariant elements. It follows that $(k) # 0 for k # 0. Furthermore, there exists at least one rn E N such that $(m) < 00; for otherwise, the saturation class does not contain a nonconstant element. Note that in any case $ ( k ) f A ( k ) is a bounded function on Z for every f E F[Xzn;Z,]) (ii) Show that Zp(f; x) is saturated in Xzn if and only if there exists m E N such that for all k # 0 lim SUP [1  x%)l/[l  x3m)l 1 $(k) > 0, ,+PO
where $(k) may be infinite for certain values of k. In this case, the saturation order of Z P ( L x) is O(1  x;(m)), p Po.. Extend to arbitrary kernels {x,(x)}. (Hint: For this Problem see also DE VORE[2]) 5. Let {x,(x)}, p E A, be a kernel satisfying (1.1.5). Suppose that there exists m E N such that x;;'(k) = I for Ikl Im, and that condition (12.1.1) is satisfied for a function 4 with $(k) # 0 for Ikl > m. Show that the set of invariant elementsf€ Xznof the singular integral fO(L x ) is given by T,. Furthermore, show that Z,(L x) is saturated in Xzxwith order O(V(p)). (Hint: See.also T U R E C K I ~[3]) f
SATURATION THEORY
440
6. Let {xB(x))satisfy (12.1.1).Show that the kernel {(xp*xB)(x)}satisfies (12.1.1) with y(p) and 2#(k). Thus the strong approximation process Z,”(f; x ) given by (1.1.23) is saturated in X,, with order O(dp)), and has Favard class F[Xzn;Z,”]contained in V[Xzn;2$(k)l. 7. Give further applications of Prop. 12.1.6 such as to the singular integrals Jncf; x ) of Jackson, N,*(f; x ) of Jacksonde La VallCe Poussin, etc. 8. Let {To},p E A, be a strong approximation process on the Banach space X and d p ) a positive function on A tending monotonely to zero as p f po. Show that the set X(Tp),cp(o)
=
ife
1 1ITo.f  f l l x = O(Cp(p)), p
+
PO}
becomes a normalized Banach subspace of X under the norm (12.1.8) l I f I I ( ~ ~ ) . c p ( o )= IIfIIx + ~~%[~P)I~IIZ‘B~fIIxX(IW)(Tp),B(P), In case X = X,,, X = X(R) we shall employ the notation Xan,(Tp),o(p), respectively. (Hint: See also BUTZERBERENS [l, p. 157 ff], BERENS [3, p. 23 ff])
12.2 Favard Classes
In view of Theorem 12.1.3, singular integrals with kernels satisfying (12.1.1) possess the saturation property. By Theorem 12.1.4 the corresponding Favard class is contained in V[X,,; $(k)]. It is the aim of this section to investigate whether VCX,,; #)] precisely makes up the Favard class and to characterize this class by more concrete conditions upon the original functionsf. The still missing direct part of a saturation theorem does not permit such a unified treatment as was given by Theorem 12.1.4 for the inverse part. Thus we shall consider the direct problem by different methods according to the diverse properties of the singular integrals in question. Nevertheless, the particular singular integrals to follow need not necessarily be treated by the method given, and the various methods may often be interchanged. 12.2.1 Positive Kernels
Our first method of proving direct theorems is closely connected with the moments of the corresponding kernels. Here we consider the particular case of even and positive kernels for which $(k) = c k2.
Theorem 12.2.1. Let f E X,, and h,(x)} be an even and positive kernel satisfying (12.1.1) with +(k) = c k2. Then the singular integral I,Cf, x) is saturated in Xznwith order O(dp)), and the Favard class F[X,,; I,] is characterized by any one of the following statements:
(9 f
E
w,,; c kal,
(ii) there exist g E ,L; i f X,, = C2,, p E BV,, i f X,, = L;, 1 < p < a,and constants a,, a , such that g(u2) du2
f ( x ) = ao
(iii)
\If(.
+ h) +
+
JI,
[a1
+
Ji:
(*(n) g(u2)
f(o
 h)  2f(o)IIx2,
=
Wa)
]] dul
g E Lg,
if X,,
=
Lg,,
(a.e.1,
d% (h f 0).
SATURATION FOR SINGULAR INTEGRALS
44 1
The proof is given by a cyclic argument which follows by Theorems 12.1.3, 12.1.4, 10.2.2, 1.5.8, and 4.3.13. Note that there may be added any further characterization of the class V[X,,; c k2]as given in Sec. 10.2. Let us mention that in view of Theorem 1.5.8, Prop. 12.1.6, and Problem 12.2.1 only the behaviour of h;(l)  11 and, e.g., of the fourth moment of the kernel determines the saturation property of even, positive singular integrals. Although from a general point of view the above theorem describes a rather particular situation, it may be applied to such important singular integrals as those of FejtrKorovkin, de La Vallte Poussin (cf. Sec. 12.2.3), Jackson, and Riemann. But we shall leave all details to the Problems of this section. The singular integral of Rogosinski may also be handled by the same arguments though the corresponding kernel is not positive (cf. Theorem 2.4.9 and Problem 12.2.3). Here we shall apply Theorem 12.2.1 to the (special) singular integral of Weierstrass as given by (cf. Problem 1.3.10) (12.2.1)
 u ) W , 0 du
Wt(f;x ) =
with kernel (12.2.2) t being a positive parameter tending to zero. Since
(12.2.3)
the kernel {8,(x, t ) } satisfies (12.1.1) with v(t)= t and $(k) = k2. Thus Theorems 12.1.3 and 12.1.4 apply, and since the kernel is even and positive, so does Theorem 12.2.1, giving a complete saturation theorem for Wt(f;x). Together with Problem 2.5.12 on nonoptimal approximation, this would complete the discussion on the approximation behaviour of the integral Wt(f; x). As we have seen in Chapter 7, the integral W t ( f ;x ) of Weierstrass is the unique solution of Fourier’s problem of the ring as posed in (7.1.4). At this point we examine the dependence of this solution upon its initial valuefand study the rate at which Wt(f; x ) approximatesfasf+O+. Proposition 12.2.2. Let f E Xan. (a) If there is g E X,, such that lim i n t o + [It l[ Wt(f;  f(o)]  g(o)11x2, = 0, then f is equal (a.e.) to a function in ACI, with second derivative in Xzn. In particular, i f (1 W t ( f ;0)  f(o)IIxan = o(t), t +O + , then f is constant (a.e.). (0 < I 1). 2a) (b) 11 W d f ;  f(o)IIxa, = o(ta) O fE Lip* This result reveals that if the solution Wt(f;x ) tends too rapidly (in X2,norm) towards the initial temperature distribution f, namely with order o(t) for t t 0 + , then f must be constant (a.e.). Thus W d f ;x ) approximates f with order of at most O(r); moreover, in case 0 < ct 5 1, the order of approximation is given by O(ra), t +. 0 + , if and only if the initial temperature distribution is a Lip* (Xan;2a)function.
.
0)
I %
Uniformly Bounded Multipliers Several important singular integrals are not covered by the results of the preceding subsection, so, for example, FejCr’s integral. To obtain a general solution of the 12.2.2
SATURATION THEORY
442
characterization problem for the Favard class F[X,,; Z,] we shall assume that, apart from (12.1 .I), h,(x)} satisfies the following condition: (12.2.4) Given a kernel {x,(x)} satisfying (1,1.5), let there exist #(k) with $(O) = 0, $(k) # 0 for k # 0, and T(p) > 0 tending monotonely to zero as p f po such that the functions A, defined by
k=O be multipliers, uniformly bounded with respect to p E A, of type
(LE, Grin) if X m
=
Can,
(BVan, BVan) if x,, = Liz, (Lg,, Lg,) if Xan= Lg,
1
0, then necessarily condition (12.2.4) holds with #(k) = lkl".
Proof. Let Xzn = Czn. We define an operator S of V[C,,; lkl"] into L& through Sf = g where g E L,", is such that Ikl"f"(k) = g"(k). Then gE L,", belongs to the range of S, i.e. g E S(V[C2,; lkl"]) if and only if g"(0) = 0. Indeed, let g E L& be such that g"(0) = 0 and vu be given by ( I 1.5.12). Then f, = q p g belongs to Cznand f,^(k) = Ikl "g"(k) for k # 0, f,^(0)= 0. Therefore Ikl"f,^(k)= g"(k) for k E H, and thus g E S(V[Czn; lk1"l). The opposite direction follows trivially. Hence, if we set L,",. O = { g E L,; I g"(0) = 0}, then S maps V[Czn; lkl"] onro L,",.O. Let g E LZn be arbitrary. Setting g(x) = g"(0) + gl(x), we have g l E L,",.O. Therefore there exists f,, E Can such that Ikluf;(k) = g;(k). Consequently, the identity 4?(k)gA(k)=
KdP))1{44hl;  f&)I 0)
+ g"(O)lW
SATURATION THEORY
444
holds for every k E Z,where A,(k) is defined as in (12.2.4) with f i k ) = lkl". In other words, we have for every g E LZn that A,(k)g"(k) is the Fourier transform of a function in L&, namely of [(q~(p))~{I,(&;x )  fpl(x)} + g"(O)], whichis actually anelement of Cm. Thus AD(k) defines a multiplier U, of type (L%, L&) for each p E A, and since fel E V[Ca,; Ikl"1 implies llIp~gl; fpl(o)IICln = O(q~(p))by hypothesis, there exists a constant Mg such that 1 U,gllL;n 4 Mg. The uniform boundedness principle then shows that the multipliers are uniformly bounded for p E A, giving the proof in Can. Analogously, the cases Xan = L%T are treated. 0)
The critical point concerning the application of Theorem 12.2.3 is the verification of condition (12.2.4).? Firstly, we observe that, in view of Problem 6.5.1, A, is a multiplier of type (BV,,, BV,,) if and only if it is of type (L,",, L,",), and since by Problem 6.5.2 a multiplier of type (LL, L,",) is also of type (LB, LE,), it suffices to prove that A, is a multiplier of type (L&, La",), bounded uniformly with respect to p E A. Moreover (cf. Prop. 6.5.3), a necessary and sufficientcondition upon a function A, on Z to be a multiplier of type (L&, L&) is that A, be the FourierStieltjes transform of a function V, E BVan. To verify that A, is the FourierStieltjes transform of some v p E BVan with I Y, I BV~, = 0(1), p E A, we may use any of the criteria given in Chapter 6, thus e.g. Theorem 6.1.2 or Cor. 6.3.9 on uniformly quasiconvex functions on Z. But for many singular integrals h,(k) is representable as A(k/p)and then the following property turns out to be the most convenient sufficient condition for (12.2.4) to be valid: (12.2.6)
Given a kernel k p ( x ) } ,p > 0, p o = co, satisfving (1.1.5), let there exist +(k) with +(O) = 0, $(k) # 0 for k # 0, ~ ( p >) 0 tending monotonely to zero as p + co, and h E NL1 such that the representation
holdsfor every k E Z, p > 0. Theorem 12.2.5. Let f E X,, and k,(x)} satisfy (12.2.6). Then IpCr;x ) is saturated in X,, with order O(q~(p)),and F[X,,; Z,] is characterized by any of the four statements of Theorem 12.2.3. Proof. Since h belongs to NL' and h" is continuous, lim, (x;(k)  l)/&p) = +(k)h"(O) = +(k) for each k E Z, and thus (12.1.1) is satisfied. Moreover, (12.2.6) implies (12.2.4). Indeed, by (3.1.29) and (5.1.58) the function h,* as defined by (cf. t
(6.3.17)) (12.2.7)
h,*(x)= p z / z ; ;
2
k=  m
h(p(x + 2kr))
belongs to Lin such that IIh,*)IL;. I llhllL1 and [h,*]"(k)= h"(k/p) = A,(k), k E Z, p > 0. Thus (12.2.4) follows by Problems 6.5.1, 6.5.2 (cf. also Prop. 6.5.3). Theorem 12.2.3 then completes the proof. In the particular instance of the Hilbert space Lgn condition (12.2.4) is satisfied (cf. Prop. 6.5.2) if and only if there exists a constant M such that Ih,(k)l 5 M for all k E Z, p E A (cf. Problem 12.2.4). This is easily verified in the applications.
SATURATION FOR SINGULAR INTEGRALS
445
Note that (12.2.6) in particular implies that the operators Upoccurring in the proof of Theorem 12.2.3 (cf. (12.2.5)) are convolution operators (cf. (6.5.2)). This enables us to prove
Theorem 12.2.6. Let f equivalent:
E X,,
and (X,(x)} satisfy (12.2.6). The following assertions are
(i) There exists g E X,, such that
(ii) f E W[X,,; W)l, (iii) there exists g E X,, such that for each p > 0
where the kernel {h,*(x)}is given by (12.2.7).
Proof. Since (12.2.6) implies (12.1.1), the proof of the implication (i) => (ii) follows by Prop. 12.1.1. Let g E X,, with $(k)f"(k) = g"(k). If h,* is defined by (12.2.7), then by the convolution theorem
and thus for each p > 0 (12.2.8) Moreover, since h E NL1,{ph(px)}is an approximate identity on R in t..e sense c Def. 3.1.4 which implies that {h,*(x)}is a periodic approximate identity (cf. Sec. 3.1.2). Therefore by Theorem 1.1.5, limo+ Il(g*h,*)  gllx,, = 0 for every g E X2=,which establishes (i). Regarding the verification of condition (12.2.6), the notion of quasiconvexity of functions on the real line (cf. Def. 6.3.5) and in particular Cor. 6.3.12 turn out to be most useful. As an example, let us consider the typical means of the Fourier series of YE Xznas introduced in Sec. 6.4.2. and given by the singular integral
with kernel (12.2.10) This kernel satisfies (12.1.1) with ~ ( n = ) (n
(12.2.11)
(K
+
> 0).
and #(k) =  lkl" since
(k E Z).
446
SATURATION THEORY
Furthermore, (12.2.6) is satisfied, for if we set r,?K(k)  1 a kK'k#O, lkl =  ( n + 1) I I
{
"(iiTi)
1,o I1x1 T(X)
I1
=
k = 0, IxI", 1 5 (XI, 1, ) at x1 = 1 where 7' has a then and 9' are locally absolutely continuous on ( 0 , ~except jump: f ( l + )  ~ ' ( 1  ) =  K . Furthermore, JOm
u Iq"(u)l du =
K(K
+ l)x"

dx =
K
+ 1,
i.e. is an even and quasiconvex function of Co.Thus by Theorem 6.3.11 as the Fourier transform of an even function h E L'. Therefore
is representable
Proposition 12.2.7. The typical means R,,, x ( f i x ) of the Fourier series o f f E Xanare saturated in Xzn with order O(nrC).The Favard class F[Xan; R,,, "1 is characterized by any of the four statements of Theorem 12.2.3 with #(k) =  lkl" as well as by the following:
where j E N is chosen so that 0 < K < 2j, (vi) there exists g E LZ,, for Xan = Can,p E BVZ, for Xan = Lin, and g E Lg, for X2n = LZ,, 1 < p < co, such that g"(0) = 0, p"(0) = 0, and f is representable as f ( x ) = f ^(O)
+
(%*g)(x)
{
((p,*dp)(X) he.), (%*g)(x) respectively, where pKis given by (1 1.5.12), (vii) f E LI, for some q with 1 < q < 00 and H'f*p,  belongs to V;(ln, where r E N is such that K < r and H denotes the operation of taking Hilbert transforms. For the proof we cite Theorems 12.2.3, 12.2.5, 11.4.6, 11.5.5, and 11.5.9. If K = 1, R,,, J f i x ) reduces to Fejtr's singular integral a,,(& x ) which is thus saturated in Xan with order O(nl). Its Favard class F[Xa,; a,,]is described by V[Xz,;  lkl]. Moreover, F[Xzn; a,] may also be characterized, apart from the seven assertions of Prop. 12.2.7 for K = 1, in terms of assertions concerning the class V[Xzn;  lkl] given by Sec. 10.3 (cf. Problem 12.2.8). The typical means are the particular case h = 1 of the Riesz means R,,, ", ,(A x ) of the Fourier series off E Xz,, for which we refer to Sec. 12.5. It is left to the reader to give further examples to Theorem 12.2.5. Generally, any of the conditions of Chapter 6 which is sufficient for the representation as the Fourier transform of an L1function may be used to examine (12.2.6), and thus to provide exact characterizations of Favard classes. Moreover, it is sufficient that (12.2.6) holds with hA replaced by the FourierStieltjes transform of a function Y E NBV. Finally, note that of course also those singular integrals for which po is finite are covered by Theorem 6.3.11 (cf. Problem 12.2.5).
Functional Equations Thus far condition (12.2.6) has been an essential tool for characterizing the Favard class of periodic singular integrals. It is actually a condition upon the kernel (xp(x)} of the singular integral which is expressed by the equation (12.2.8) for the subclass of functions f E W[X2,; $@)I. On the other hand, the difference [!,(A x)  f ( x ) ]
12.2.3
447
SATURATION FOR SINGULAR INTEGRALS
may also be determined via certain functional equations which are valid for all
f E X,, and arise from the specific structure of the process in question. As an example we consider the general singular integral of Weierstrass of Sec. 6.4.1. It is defined by
(12.2.12) with kernel m
(12.2.13)
wt,&)
=
2
k=m
The kernel {wt,K(x)}satisfies (12.1.1) with
etlklr eikx
(K
> 0).
v(t) = t and +(k) =  lklKsince
(12.2.14)
(k E Z).
Thus Theorems 12.1.3 and 12.1.4 apply. For the direct part of the saturation theorem, let f E V[C2,; such that  IkI"f^(k) = g^(k). We have for k E Z
 lkl"] and
g E L,",
Pt
The change of the order of integration in the last equality is justified by Fubini's theorem since 1 W7,K ( g ;o)llm I M , 1 gll uniformly for all T > 0, and thus
Therefore, by the uniqueness theorem and the continuity of all expressions involved,
=
for all t > 0 and x E R. Iff E V[LE,;  lkl"], 1 < p c co,then for each t 0 (12.2.15) holds almost everywhere. Finally, if f E [L:,;  lkl"] and  lklKf"(k) = p"(k), p E BV,,, then for each t > 0 Ct
(12.2.16) Thus f E V[X,,;  lkl"] implies 1 Wt,K( f ; 0)  f(0)llXzn = O(t), and we may state Proposition 12.2.8. The general singular integral Wt,K ( f i x) of Weierstrass is saturated in X2, with order O ( t ) , t + 0+, and the Favard class F[X,,; Wt,K ] is characterized by any one of the seven statements of Prop. 12.2.7. Moreover, f E W[X2,;  lkl"] ifand only fi there exists g E X,, such that (12.2.17)
SATURATION THEORY
448
For the proof of the latter assertion we note that iff holds, and therefore
E
W[Xan;  lkl"] then (12.2.15)
in view of the HolderMinkowski inequality. For further characterizations in the particular cases K = 1 and K = 2 we refer to Sec. 12.2.1 and Problem 12.2.13, respectively. The above proof of the implication V[X,,;  lkl"] c F[X,,; Wt,K ] actually makes use of the socalled semigroup property of the general singular integral of Weierstrass. This means that the operators Wt, satisfy the functional equation (in the parameter t ) (12.2.18)
wtl.
K{
wta,
E f
1 = wti
+ ti. K
(ti, t a >
f
0).
This is easily established since [ Wt, f ](x) =
(12.2.19) for allfE Xanand
m
2
k=m
e'lklKfn(k) elks
z > 0, and therefore m
[wtl,K{wt,,Xf>l
=
2 etl'k'"{etz'k'yA(k)> efkx =
[Wt,+tZ,Kfl(x)a
k=  m
For details we refer to Sec. 13.4.2 where we shall give a short account of approximationand saturationtheoretical questions for semigroups of operators on a Banach space. Apart from the semigroup property, other types of functional equations may be useful to establish characterizations of Favard classes. As an example we consider the singular integral of de La VallCe Poussin defined by (cf. Sec. 2.5.2) (12.2.20)
j'
V'cf; x) =  f(x  U)V,(U)dU 27r '
with kernel (12.2.21)
u,(x) = (2c0s;)2n
=
$ k = n
(n!)a eikx (n  (kl)!(n Ikl)!
+
Condition (12.1.1) is satisfied with ~ ( n = ) nl and I,@) =  k 2 since (12.2.22) and thus Theorems 12.1.3 and 12.1.4 apply. But for the characterization of the Favard class, Theorem 12.2.5 is not applicable since v;(k) does not permit a representation such as v;(k) = A(k/n).On the other hand, the kernel is even and positive, and since +(k) = ka, Theorem 12.2.1 may be used. But here we prefer to proceed as follows. We need the equation (cf. (2.5.19)) (12.2.23)
k2UC(k)= na[u;(k)
 Gl(k)]
(n € N, k E rn)
SATURATION FOR SINGULAR INTEGRALS
which follows by direct computation. Let f g E Lg,. Then for m > ti and k E Z
E
449
V[C2,; k2] and k2fA(k) = g"(k),
and therefore by the uniqueness theorem V n ( L x)
 Vm(f;
"
1
XI= j = n + l J5 vj(g;XI.
Letting m f 00 we have in view of (2.5.15) that for all n E N and x E R (12.2.24) the righthand side being convergent in Can since
Iff E V[Lg,; ka], 1 < p < a,then for each n (12.2.24) holds almost everywhere, whereas f o r f e V[L:,;  k a ] we obtain for each n E N Vn(f;X)  f ( x ) =
(12.2.26)
2"
jntlJ
1
5 Vj(dp;X)
a.e.7
where p E BV,, is such that k2fA(k) = p"(k). In any case, f 1 V n ( A0)  f(o)IIxan= O(l/n),and therefore (cf. Cor, 2.5.11)
E V[Xzn;
k2]implies
Proposition 12.2.9. The integral V n ( f ;x) of de La Valle'e Poussin is saturated in X,, with order O(n'), n + 00, and the Favard class F[X,,; V,] is characterized by any of the three statements of Theorem 12.2.1 (with c = 1). Moreover, f E W[X,,;  k2] if and only i f there exists g E X2, such that
For the proof of (12.2.27) we observe iff + j  , , then if we set cn =
x?=
E W[X,,;
k2],then (12.2.24) holds, and
which tends to zero as n +. co by (2.5.15). Now (12.2.27) follows by (6.2.14). Whereas for the general singular integral of Weierstrass the functional equation is 29F.A.
SATURATION THEORY
450
given by the semigroup property (12.2.18), for the singular integral of de La Vallke Poussin the equation
(12.2.28) [Vn(f; o)l”(x) = nawn(f; 4  Vndf; 41 is essential. (12.2.28) holds for everyfE Xan and every x E R, n E N. Problems 1. (i) Let f~ Xan and the kernel k,(x)} of the integral Z,V, x) be even and positive. Complete the proof of the following list of equivalences (cf. Prop. 12.1.6 and Theorem 12.2.1):
(p
2.
3. 4.
5.
6.
0)
7. 8.
f
Po),
(6) 1,Cf; x ) is saturated in Xan with order 0(1  $(1)), and the Favard class is characterized as in Theorem 12.2.1. (Hint : cf. GORLICHSTARK [1,2]) (ii) Show that the singular integrals Jn(f; x ) of Jackson, N,*V, x) of Jacksonde La Vallte Poussin, and KnV,x) of FejkrKorovkin are saturated in Xzn with order O ( ~ Z  ~n )f , co, the Favard class being characterized as V[Xzn;c ka]with c =  3/2,  3/2, 7ra/2, respectively. Show that the integral means A d f ; x) o f f E Xzn are saturated in Xan with order O(ha), h + 0 + , with Favard class characterized as V[Xan; c ka],c =  1/6. Show that the same is true for the rth integral means A l ( f ; x) with c = r/6. Thus an rtimes repeated application of the smoothing operation Ah effects neither the order nor the class of saturation. Show that the singular integral &(f; x) of Rogosinski is saturated in Xan with order O ( ~ Z  ~with ) , Favard class characterized as V[Xan; ( va/8)ka]. (Hint: Theorem 2.4.9) Given a kernel {xp(x)}, let there exist #(k) with #(O) = 0, #(k) # 0 for k # 0, and y@) > 0 tending monotonely to zero as p f po such that IX,(k)l I M for all k E E, p E A, X,(k) being defined as in (12.2.4). Show that f E V[L&; #(k)] implies IlZ,(f; 0 )  f(o)Ila = O(y@)) (cf. also Prop. 6.5.2). Given a kernel {xp(x)}, p E A, po c a, satisfying (1.1.5), let there exist #(k) with $(O) = 0, #(k) # 0 for k # 0, d p ) > 0 tending monotonely to zero as p + po, and h E NL1 such that x;(k)  1 = d p ) #(k)h”(lp  Polk) holds for each k EZ,p E A. Show that the singular integral ZP(f; x ) is saturated in Xan with order O(y(p)), p f po, and that the Favard class F[Xan; I,] is characterized as V[Xan; #&)I. (Hint: BERENSGORLICH [l]) Let {xp(x)} satisfy (12.1.1) and (12.2.4). Show that the iterated integral Z,”(f; x) is saturated in Xan with order O(y(p)) and that the Favard class F[Xan;Z,”]is characsatisfies (12.2.6), so does the iterated terized as V[Xan; 2#(k)l. Furthermore, if {xxp(x)} kernel {(xP*xg)(x)}.(Hint: Use Problem 12.1.6 and the estimate (cf. Problem 1.1.4) IIIp”(f;  f(0)IlXl. 5 (IIXPIIl + 1) IlZP(f; 0 )  f(0)IIXln) Establish the equivalence of the statements (i) and (iv) of Theorem 12.2.3. (i) Use the identities ( U , , ~ C ~x); denote the (C, 2) means of the Fourier series of f E Xan, cf. Problem 1.4.4) = (n + N d f ;X)  un,a(f; x)), [G(f;o)l’(x) un,a(f; X )  an  i.4A XI = 2[Ccf; o)l’(x)/n(n 2)
+
SATURATION FOR SINGULAR INTEGRALS
451
to deduce that Ilu,(f; 0 )  f(0)IIXan = O ( n  l ) if and only if Ilahl)(f; o)IIxax = 0(1), n + 00. (Hint: See also ZAMANSKY [2], ZYGMUND 171, p. 1231, LORENTZ [3, p. 1001) (ii) Show that the singular integral u,V, x ) of FejCr is saturated in Xan with order O ( n  l ) , the Favard class F[Xan;u,] being characterized by any of the seven statements of Prop. 12.2.7 for K = 1. Furthermore, show that the following statements are equivalent for f to belong to F[Xzn;a,]: (a) There exist g E L& if Xan = Can, p E BVan if Xan = Lin, g E Lgn if Xan = LE,, 1 < p < m, and a constant a such that
}
g(u) du
f"(4= a + /:n+p(u)
(a.e.1,
g(u) du = OClhl) (b) I I f " ( 0 + h)  f"(.)IIx,, (h 01, (c) ]If(. + h)  f ( O ) I I X p , = o(lhl), provided Xan = Lgn, 1 < p < 00. (iii) Show that the (C, a) means u,,,(f; x ) of the Fourier series of f€ Xan are saturated in Xznwith order O(n'), the Favard class being characterized as V[Xan; a lkl]. (Hint: cf. BUTZERGORLICH [2, p. 3881) 9. Let f E W[Xzn;ka] and kaf"(k) = g"(k), g E Xan. Show that for the integral means f
Ah(f; X )
and therefore
10. Let f E Xznand {x,,(x)} satisfy (12.1.1). If YE W[Xan;#(k)] implies the approximation IIMf;0)  f ( 0 ) IIx,, = O ( d d ) , then in fact
where g E X,, is such that #(k)f"(k) = g*(k). (Hint: Use the theorem of Banach[2]) Steinhaus with T,, as the dense subset, cf. BUTZERGORLICH 11. Show that for f E W[Can;ka] the following limit relations of Voronovskaja type are valid (cf. (1.5.29), (1.6.7)):
Show that the converse statement is also valid. State and prove analogous results in LE,space. State and prove analogous results for the singular integrals J,(f; x ) of Jackson, N,*(f; x) of Jacksonde La VallCe Poussin, V,V, x ) of de La Vallte Poussin, etc. (Hint: BUTZERG~RLICH [21)
452
SATURATION THEORY
12. Show that forfE W[Can; lkl] the following limit relations of Voronovskaja type are valid (cf. Problem 9.2.5) (i) lim I l M o )  df; 011  ( f “ Y ( o ) I I C a n = 0, n m
Show that the converse statement is also valid. State and prove analogous results in LE,space. 13. (i) Show that the singular integral P,V, x ) of AbelPoisson is saturated in Xzn with orderO(1  r ) , r + 1 ,withFavardclassFIXan;Pr]characterizedasVIX,,; lkl]. Give further characterizations of F[X2,; Pr](cf. Problem 12.2.8). (ii) In the light of (i) and Cor. 2.5.5 examine the dependence of the solution of Dirichlet’s problem for the unit disc (cf. Sec. 7.1.2) upon its boundary valuef, and study the rate at which Pr(f;x) approximates f as r + 1  . (iii) Show that the solution of Dirichlet’s problem (7.1.17) for a given boundary ,I also solves Neumann’s problem (7.1.25) for a boundary value value fo E X go E Xan if and only if Ikl f ; ( k ) = g;(k). Conversely, the solution of Neumann’s problem for go solves Dirichlet’s problem for fo, where fo is determined by ql*go (except for an additive constant), q1 being defined through (11.5.12) (cf. also the 11, p. 119 where also the conproof of Prop. 12.2.4). (Hint: BUTZERBERENS nection of simultaneous solutions of Dirichlet’s and Neumann’s problem with the class V[X,; lkl] is studied) 14. Let S,,(f; x), unV,x ) be the singular integrals of Dirichlet and Fejtr of f E Xan, respectively. Show that the kernel of the integral
a,
(1 2.2.29) satisfies (12.1.1) and (12.2.6) so that I,,(f; x) is saturated in Xan with order O(n’), the Favard class F[X,; I,] being characterized as V[Xan; lkl]. Does the kernel form an approximate identity? 15. Treat the saturation problem for the singular integral O,,(fi x ) defined in Prablem 1.6.16. Apply to the integrals of Fejtr, FejkrKorovkin, and Jackson. 16. Let the kernel {x,,(x)} of the singular integral Zo(f; x ) satisfy (12.1.1) and (12.2.4). Show that the Banach spaces V[X2,; I+%@)],endowed with norm (10.4.7), and X2n,~lp1,(D(o), endowed with norm (12.1.8), are equal with equivalent norms. (Hint: Compare with Theorems 12.1.4, 12.2.3, see also Prop. 0.8.5)
+
12.3 Saturation in LP, 1 I p i 2
12.3.1 Saturation Property Let f E X(R) and Fejtr’s type)
x E NL’. In
Sec. 3.1 we have seen that the singular integral (of
(12.3.1) defines a family of bounded linear transformations J ( p ) of X(R) into itself having uniformly bounded norms, i.e. (12.3.2)
IIJcf; ”; P)llXrn,
llxll1Ilf Ilxm
(P
’01,
SATURATION FOR SINGULAR INTEGRALS
453
and which approximate the identity, i.e. (12.3.3)
lim
0 m
IIJW
0;
P )  f(.)llxc~
=
0.
In the study of the corresponding saturation problem? we shall restrict ourselves in this chapter to the spaces X(R) = Lp, 1 I p I 2. Then saturation theory of periodic singular integrals suggests that the following conditions are of interest: (12.3.4)
Gioen x E NL', let there exist constants c # O# and a > 0 such that
(12.3.5) Given x E NL', let there exist constants c # 0, a > 0, and a function v E NBV such that for all u # 0
Since v is normalized and v" is continuous by Prop. 5.3.2, condition (12.3.5) implies (12.3.4). But the converse assertion is not necessarily true (cf. Problem 12.3.1).
Proposition 12.3.1. Let such that
x satisfy (12.3.4). Iffor f E Lp, 1 I p I 2, there exists g E Lp
(12.3.6) then f belongs to W[Lp;c IuI"]. Proof. Letp and thus
=
1. In virtue of Theorem 5.1.3 we have [ J ( f ;0 ; p)]"(v) = x"(u/p)f"(u),
for all u E R. Hence by (5.1.2)
uniformly for all u E R. But (12.3.4) in particular implies (1 2.3.8)
and therefore c lul"f"(t7) = g"(u) for all v E R. For 1 < p I 2 Theorem 5.2.12 yields (12.3.7) for almost all u E R which implies
t We shall confine ourselves to singular integrals of Fejkr's type. For results concerning the more general integral (3.1.3) we refer to the Problems. $ Let us emphasize the importance of the existence of a nonzero constant c. At the end of SeC. 12.5 we shall construct an example of a function x E NL1 for which lim,+o ( ~ " ( u )  l)/lul" = 0 for every a > 0 and which indeed does not possess the saturation property.
45 4
SATURATION THEORY
by Titchmarsh's inequality, In view of (12.3.8) and Fatou's lemma it follows that IIc  g"(o)llp, = 0, and thus the assertion.
Io~" (o)
Proposition 12.3.2. Let x satisfy (12.3.4). Iffor f
IIJ(f;
";PI

f(0)
IIP
E
Lp, 1 5 p 5 2,
(P
=Np9
f
001,
then f(x) = 0 a.e. Moreover, the nullfunction is the only invariant element of the singular integral (12.3.1) in LP.
Proof, Since (12.3.6) holds with g(x) = 0 a.e., we conclude IvlyA(v)= 0, and thus f"(v) = 0 (a.e.). Therefore f ( x ) = 0 a.e. Obviously, the nullfunction is an invariant element of J ( f i x; p). Conversely, i f f € LP is such that llJ(S; a ; p)  f(o)llP = 0 for  1)f "(0) = 0 for every p > 0, every p > 0, then by the convolution theorem kA(u/p) and hence as abovef(x) = 0 a.e. Theorem 12.3.3. I f f E Lp, 1 Ip 5 2, and x satisfies (12.3.5), then the singular integral J c f ; x; p ) is saturated in Lp with order O(pp3.
Proof. In view of Prop. 12.3.2 we only have to show that the order of approximation
llJ(A
(12.3.9)
O
;P)

f ( 0 )
I p = O(p  9
(P
,a))
is attained for at least one functionf E Lp different from the nullfunction. This is given by the following (see also Problem 12.3.8) ~~
12.3.4. Let @ E C& andx satisfy (12.3.5). Then for 1 Ip 5 2
Proof. Since @ E Cz0 it follows by Problem 11.2.13 that @ has strong Riesz derivatives of every order. In particular, Dp)@ E LP for every 1 I p s 2, and IuIu@"(u) = [DP'@l"(u). Moreover, [D!"'@]"E L' and therefore
I
c " " [W@I(x  4 f f v ( p 4 4%
by Prop. 5.1.10, 5.3.3. Since Y is normalized, this implies
pap(@; x;p)
1 W?'@l(x
c "  @(x)]  c[Dp)@](x)= = 4t/2?r
 u)  [Dj"'@I(~))dV(PU),
and (12.3.10) follows by Problem 3.1.16. We here recall Prop. 1.7.4 on positive bounded linear polynomial operators. This suggests that for even and positive functions x the exponent a of (12.3.4) may be restricted to 0 < a s 2. Indeed (cf. also Problem 12.3.2) Proposition 12.3.5. If x is an even and positive function satigying (12.3.4), then the exponent u is restricted to 0 < u s 2.
455
SATURATION FOR SINGULAR INTEGRALS
Proof. Since xA is an even and real function bounded by 1, it follows that c < 0 under (12.3.4). Suppose a > 2. Then for R > 0 (12.3.11)
I
lulrI
XAM
=
hJ:/om
(1  cos uu)x(u) du
But lim u0
lo It
1  cos uu Ua
x(u) du
=
IOR
u2x(u) du > 0,
so the righthand side of inequality (12.3.11) tends to infinity as tr f 0, a contradiction to (12.3.4). Thus 0 < a 5 2. 12.3.2 Characterizations of Favard Classes: p = 1
The next problem is to characterize the Favard class of the singular integral J ( f ; x; p). To this end we proceed separately for p = 1 and 1 < p I 2. Theorem 12.3.6. Let f
L'. (i) If x satisfies (12.3.4), then (12.3.9) implies the existence of p E BV such that c lulafA(u) = p"(u)for aN u E R. (ii) I f i n addition x satisfies (12.3.5), then f E VIL1;c lula] implies (12.3.9). E
Proof. (i) By the convolution theorem and the hypotheses (12.3.12)
uniformly for large p, and therefore c Iul.fA(u) defines a bounded continuous function on R. In order to apply CramCr's representation theorem, let us consider the arithmetic means (cf. the proof of Theorem 10.5.4) (x E R, R > 0).
By (12.3.12) and Lebesgue's dominated convergence theorem it follows that
Hence by Fatou's lemma and (5.1.18)
SATURATION THEORY
456
Since the latter expression is bounded independently of R > 0, (i) follows by an application of Theorem 6.1.5. (ii) By Prop. 5.3.1 and Theorems 5.1 $3,5.3.5
r
1
1"
nm
the convolution of p and v being understood in the sense of Prop. 0.5.2. Thus by the uniqueness theorem for FourierStieltjes transforms we have for each p > 0
It follows that
uniformly for all p > 0, which completes the proof. Thus the saturation class for singular integrals (12.3.1) with kernel (12.3.5) may be characterized as the class VIL1; c Ivla]. Moreover,
x
satisfying
Theorem 12.3.7. Let f E L' andx satisfy (12.3.5). Then the following assertions are equiualent for the singular integral J ( f ; x ; p ) : ( 9 IIJM 0 ; P )  f ( 0 ) I l l = O ( p  9 (ii) f E VIL1; c Iula],
( p + a),
11
(iv) f E WIL1; c Iula] , (v) there exists p E BV such that (12.3.13) holds, (vi) there exists p E BV such that for every s E Co m
!zJ
S(U)f"[J(f;
u ; P)
 f (41
= Jmm s(u) C ( U h
(vii) for a # [a] (a = [a])the [a]th ([[a 11th) strong Riesz deriuatiue o f f exists arid (D'O'f = f)
(E
where j
E N
is chosen such that 0
0, so that the representation (12.3.5) with c = 1 follows by Theorem 6.1.5, since Iol"[xA(u)  11 is bounded on R by (12.3.16). In view of Theorem 6.5.6 and Problem 6.5.5 we may formulate the result of Theorem 12.3.6(ii) and Prop. 12.3.8 as follows:
459
SATURATION FOR SINGULAR INTEGRALS
Corollary 12.3.9. Let x E NL1. Then VIL1; c lola] i f and only i f h as defined by
h(u) =
(12.3.19) is a multlplier of type (BV,
p, I
C
{fe L1 I
IIJV,
0 ;
p)
 f(o)II1
= O(p")l
U Z O 17
u = o
BV).
12.3.3 Characterizations of Favard Classes: 1 < p I2 Next we turn to characterizations of Favard classes in Lpspace, 1 < p I2. The results follow straightforwardly, but instead of carrying over the method of proof of e.g. Theorem 12.3.6 to the present situation which obviously is possible, we shall indicate another way of proof, intimately connected with that of Theorem 6.1.5 and applicable to the case p = 1 as well. Theorem 12.3.10. Let f E LP, 1 c p 5 2. (i) I f x satisfies (12.3.4), then (12.3.20)
I J(f;
";PI f(O)llP = O ( p  3
(P
+a)
implies the existence of g E Lp such that c Iu["fn(u) = g"(u) a.e. (ii) Ifin addition x satisfies (12.3.5), then f E V[Lp; c IuI"] implies (12.3.20).
Proof. (i) First we observe that the hypothesis implies by the convolution theorem and Titchmarsh's inequality that
uniformly for large p. Hence (12.3.4) and Fatou's lemma yield c lu[af"(u) E Lp'.l. Let @ belong to the class F as defined by (6.1.10). Then by the Parseval formula (5.2.27) and Holder's inequality
uniformly for large p. Furthermore, limo+mpa'[xA(u/p)  llf"(u) = c IulyA(u)a.e. by (12.3.8), and since the latter function belongs to Lp', in view of (12.3.21) we may apply Prop. 0.1.11 to obtain
t If p = 2, the assertion follows from this point on by Plancherel's theorem, see also Problem 12.3.10.
SATURATION THEORY
460
for every @ E F, observing that @ E F implies 0,@" E Lq for every 1 5 q I 03. Together with (12.3.22) this yields
Thus the integral on the left defines a bounded linear functional on F considered as a subspace of LP'. Since F is dense in LP' (cf. Problem 6.13, we may extend this functional to a bounded linear one on all of LP' (cf. Prop. 0.7.1). Therefore by the Riesz representation theorem there exists g E L P such that

m
c ~u~"fA(u)@"(u)du =
Jmm

g(u)@(u) du = m
m
J
m
g " ( u ) w ) du,
the latter equality being valid by (5.2.27). Thus for every @ E F
and since [F]" is dense in Lp as well, part (i) follows by Prop. 0.8.1, 0.8.11. (ii) Let g E Lp be such that c Iul*f^(u) = g"(u) a.e. Then by (5.2.27) and (5.3.5) for every @ E F
m m
Since ( g * dv(p0)) belongs to L P by Prop. 0.5.5 and since F is dense in LP', it follows that for each p > 0
Hence by (0.5.9)
uniformly for all p > 0, and the theorem is completely established. Thus in Lpspace, 1 < p I 2, the saturation class for singular integrals (12.3.1) with kernels x satisfying (12.3.5) may be characterized as the class V[LP; c Iul"] just as for p = 1. Moreover, by the preceding arguments we have immediately Theorem 12.3.11. Let f E Lp, 1 < p 5 2, andx sarisfy (12.3.5). Then the following assertions are equivalent:
(9 I I m
0;
p)
 f(o)II,
(ii) f E W[Lp; c Ivla],
=
w"1
( p + a),
46 1
SATURATION FOR SINGULAR INTEGRALS
(iv) there exists g E LP such that (12.3.23) holds, (v) there exists g E Lp such that limo+m Ilp"[J(f; (vi) f has an ath strong Riesz derivative, (vii) there exists g E Lp such that
0;
p )  f (011  g(o)/lp = 0,
wherej E N is chosen such rhat 0 < tl < 2j, and Ca,2, is a constant given by (1 1.3.14); in this case, f has an tlth strong Riesz derivative.and DP'f = g, (viii) there exists g E L P such rhat for 0 < a < 1 : Ra,,.g E Lp andf(x) = [RdlP(RLIIP,g)](x) a.e., a 2 1 : RP,,,,g,RB/p(Rp,P#g) belong to Lp and [H[*'fl(x)=
Ix 1"' ~ U I
m
m
d ~ 2. ..
J::'
[ R ~ / p ( R ~ / p * g ) l ducal ( ~ l a J a.e.,
ivhere tl = [a] + /3, 0 5 B < 1, and Rog = g . For statements (vi)(viii) werefertoTheorems 11.2.611.2.9, 11.3.7, and Problem 11.5.2.
Problems 1. Give an example of a function x E NL' which satisfies (12.3.4) but not (12.3.5). (Hint: SUNOUCHI [8] gives an example due to P. MALLIAVIN) 2. Let f E L' and x E NL' beeven and positive. Show that if IIJ(f; 0 ; p )  f ( o ) 1 l 1 = O(p") for some a > 2, then f ( x ) = 0 almost everywhere. (Hint: See also BUTZERKONIG [l]) 3. Let x E NL' be even and positive. Show that (12.3.4) holds for a = 2 if and only if the 2nd moment of x exists. In this case we have c = m(x; 2)/2. (Hint: Prop. 5.1.19) 4. Let f E Lp, 1 5 p 5 2, and y, be an even and positive function satisfying (12.3.4) with OL = 2. Show that the corresponding singular integral J ( f ; x ; p ) is saturated in Lp with order O ( P  ~ ) and that the Favard class F[LP; J ( p ) ] for e.g. p = 1 is characterized by any one of the following statements: (i) There exists p E BV such that cvzf"(u) = p"(v) for all u E R, (ii) there exists p E BV such that f ( x ) = st dul dp(uz) a.e., (h 0). (iii) 11 f(0 Iz) f(0  11)  2f(o)II1 = O(h2) (Hint: Theorems 12.3.6(i), 10.5.4, and 5.3.15) 5. (a) As an application of the preceding Problem show (1 5 p I2): (i) The singular integral of GaussWeierstrass is saturated in Lp with order O(r), t + O + . (ii) The singular integral of Jacksonde La Vallke Poussin is saturated in L P with order O ( P  ~ )p, + co. (iii) The moving averages are saturated in L P with order O(hz), h 3 O+ (iv) The Bessel potentials are saturated in Lp with order O(p"), p 3 a. In each case, the saturation class is characterized as V[LP; cu2] with c =  1;  312;  1/6; a/2, respectively. (b) Show that for a twice continuously differentiable f E W[LP; Iu12], 1 5 p 5 2, the following limit relations of Voronovskaja type are valid (cf. (3.3.9), Problem 3.4.5) :
+ +
f
.
(iii)
SATURATION THEORY
462
x satisfy (12.3.5). Show that the following statements are equivalent: (i) There exists g E Lp such that Ilpa[J(f; 0 ; p)  f (a)]  cg(O)II, = 0, (ii) there exists g E L P such that lulaf"(u) = g"(u) (a.e.), (iii) there exists g E L P such that
6. Let f E LP, 1 I p 5 2, and
In any event, the ath strong Riesz derivative off exists and DPY = g. As already mentioned, the results of this section may also be established for general singular integrals Z(f; x; p) as given by (3.1.3). Instead of conditions (12.3.4), (12.3.5) we have (p is a positive parameter tending to infinity): ; let there exist a function $(u) defined (12.3.24) Given an approximate identity { ~ ( xp)}, ) and continuous on R with isolated zeros, and a positive function ~ ( p tending monotonely to zero as p + co such that
(12.3.25)
Given an approximate identity { ~ ( xp)}, ; let there exist 4, tp as in (12.3.24) and vp E 0V such that
(12.3.26)
foralluERandIlvpll~"~Mforallp>O. Given an approximate identity { ~ ( xp)}, ; let there exist 4, tp as in (12.3.24) and an approximate identity {h(x;p)} such that for all u E R
7. Let f E L', 1 5 p I 2, and { ~ ( xp)} ; satisfy (12.3.24). Show that the following conclusions are valid : (i) If there exists g e LP such that limp+ ll(p(p))l[Z(f; 0 ; p)  f(o)]  g(o)]II, = 0, then f E W[Lp; I,&)]. (ii) If IlZ(f; 0 ; p)  f(o)llp = O(p(p)), then f E W P ;+(u)I. ; in addition satisfies (12.3.25), then conversely f E V[Lp; $(u)] implies (iii) If { ~ ( xp)} I l U 0 ; p)  f(0)llP = O(p(p)). 8. (i) Letf E Lp, 1 I p I2, and {x(x; p)} satisfy (12.3.25). Show that f * dvp E W[LP; $(v)] and $(u)lf* dv,]"(u) = (cP(p))l[Z(f; 0 ; p)  f(o)]^(u) for every f E Lp. Thus, if { x ( x ; p)} in addition satisfies (12.3.26), show that W[Lp; $(u)] is dense in Lp. (ii) Let f E Lp, 1 5 p 5 2, and { ~ ( xp)} ; satisfy (12.3.26). Show that f s W[Lp;$(u)] if andonlyifthereexistsgE Lpso that limo+mIl(&))'[Z(f;o;p) f(o)] g(o)llp = 0. 9. Let f e L' and {x(x; p)) satisfy (12.3.26). Show that the following assertions are equivalent :

L1
(iv) f E W L ' ; 4(4l , (v) there exists p E BV such that for each p > 0
463
SATURATION FOR SINGULAR INTEGRALS
(vi) there exists p E BV such that for every s E Co
State and prove analogous results for Lpspace, 1 < p 5 2 (cf. Theorem 12.3.1 l(i)(v)). 10. Given X E NL', let there exist constants c # 0, a > 0 such that IX(u)l IM for all u E R, h being defined as in (12.3.19). Show thatfE V[L2; c lola] (cf. also Theorem 6.5.5) implies IWf; 0 ; P )  f ( O ) I l a = O(v(p)). 11. Instead of using CramBr's representation theorem prove Theorem 12.3.6 with the aid of Theorem 6.2.1 or 6.2.3. (Hint: See also BUTZERKONIG[I]) 12. Letftz Lp, 1 I p I 2, and x satisfy (12.3.4). Suppose that h(u) as defined by (12.3.19) is a multiplier of type (Lp, Lp). Show that the corresponding singular integral J ( f ; x ; p) is saturated in Lp with order O ( P  ~ and ) saturation class F[Lp;J(p)] characterized as V[Lp; c Iula]. (Hint: Theorems 6.5.56.5.7) What can be said about the necessity of the multiplier condition ? 13. Let the kernel x of the singular integral J(f; x; p) satisfy (12.3.5). Show that for 1 I p I 2 the Banach spaces V[Lp; 101~1,endowed withnorm (10.5.11), and Lpu(p),,La, endowed with norm (12.1.8), are equal with equivalent norms. (Hint: Compare with Theorems 12.3.6, 12.3.10)
12.4 Applications t o Various Singular Integrals
The object here is to present applications of the general theorems obtained in the preceding section. In all of these, condition (12.3.4) is readily verified. To prove (12.3.5) we shall mainly make use of the actually stronger conditiont
(12.4.1) Given x E NL1, let there exist constants c # 0, 01 > 0,and afunction h E NL1 such that for all v # 0
Thus, in this section we are concerned with different ways to verify (12.4.1) for a number of examples. Once this is done, the results here follow as immediate consequences of those of Sec. 12.3.
Singular Integral of Fej& A first application can be made to the singular integral of FejCr as defined by
12.4.1
(12.4.2)
(cf. (3.1.14), Problem 5.1.2). It is of type (3.1.8) with kernel F, (12.4.3)
F(x) =
y
FA@) =
IuI 2 1.
t In the particular instance of the Hilbert space La we may apply Problem 12.3.10, the hypothesis of which is easily verified in the applications.
SATURATION THEORY
464 Since (12.4.4)
it is obvious that the FejCr kernel satisfies (12.3.4) with a = 1 and c =  1. Regarding (12.4.1) we have by (12.4.4) that [FA(v) l]/lvJ is locally absolutely continuous and belongs to L2, together with its derivative. Hence (12.4.1) follows by Prop. 6.3.10. It is left to Problem 12.4.1 to verify that the corresponding function h is even and given by (x > 0).
(12.4.5)
Thus, the integral of FejCr is saturated in Lp, 1 I p < 2, with order O(p'), and its Favard class F[Lp; o(p)] is characterized as V[Lp;  101]. More explicitly, we have Proposition 12.4.1. Let f E Lp, 1 Ip I2. (a) There exists g E LP such that (12.4.6) lim IIPb(S; P )  f O ;
P
(013
 g(0)IIp
=
0
(0
ifand only iff E W[Lp; lull, i.e., ifand only iff "(x) isequa1a.e. t o a function of AC,,, with derivative in LP. In particular, if (12.4.7) IlQ; 0 ; P )  f ( o ) I I p thenf(x) = 0 a.e. The following statements are equivalent:
0) I I a (iv) .(fI\
0;
P)
= 4pl)
(P
+
a),
f@)llYJ = w9
+ h) 
f(o)\Ip
= O(Jh)),prouided1 < p
< 2.
For the proof of part (a) we refer to Problems 12.3.6 and 8.3.3, whereas part (b) is a consequence of Theorems 8.3.6, 10.5.6, 12.3.6, and 12.3.10. For further characterizations of the Favard class F[LP; o(p)] we refer to Theorems 12.3.7 and 12.3.11. 12.4.2 Generalized Singular Integral of Picard The next application is to the generalized singular integral of Picard, given by
where cK(x)is defined by its Fourier transform (12.4.9)
c;(u)
=
(1
+
)u)K)l
(K
> 0).
In view of Problem 6.4.5, c, belongs to NL' for every K > 0 and, in fact, it is a positive function for 0 c K I2. In particular, c2(x) = d q exp { IxI}, and C,(f; x; p) reduces to the (special) singular integral of Picard as considered in Problem 3.2.5.
465
SATURATION FOR SINGULAR INTEGRALS
In view of the identity (12.4.10)
CC(/I)

 lotK
1
= c,^(v)
it follows that the Picard kernel satisfies (12.3.4) with a = K and c =  1. Obviously, (12.4.10) already furnishes the representation (12.4.1). Proposition 12.4.2. The generalized singular integral C,(f; x ; p) of Picard is saturated in Lp, 1 5 p 5 2, with order O(p,). The Fawrd class F[Lp; CK(p)]is characterized by any of the nine [eight]statements of Theorem 12.3.7 [12.3.1I], respectively. In particular, we have (cJstatement (v) [(iv)]): f E F[Lp; C,(p)] if and only i f there exists p E BV if p = 1 a n d g E L P i f l < p 5 2suchthat (12.4.11)
Note that in view of (12.4.10) equation (12.4.1) reduces to a 'recurrence formula' for the generalized singular integral of Picard. Hence the direct part of the saturation theorem follows by the uniform boundedness of the operators CK(p)as defined by the original singular integral. 12.4.3 General Singular Integral of Weierstrass The general singular integral of Weierstrass is defined by m
(12.4.12)
dU,
the kernel wK being given by its Fourier transform (12.4.13)
w;(u) = exp { IvlK}.
(K
> 0)
In view of Sec. 6.4.1, w, belongs to NL1 for every K > 0 and, in fact, it is a positive function for 0 < K 5 2. Therefore, (12.4.12) defines a singular integral of type (3.1.8) with parameter p = t   l j K , t  + O + . For K = 1 and K = 2 the functions w, admit explicit representations given by (3.1.39) and (3.1.33); then the integral (12.4.12) is associated with the names of CauchyPoisson and GaussWeierstrass, respectively. Now, w, satisfies (12.3.4) with a = K and c =  1 since (12.4.14)
To establish (12.4.1) we could apply Theorem 6.3.11 (cf. Problem 12.4.2). But as in the periodic case of Sec. 12.2.3 we shall prefer to use the identity (12.4.15)
This implies that
~OF.A.
466
SATURATION THEORY
where the change of the order of integration is justified by Fubini's theorem since
Thus w, satisfies (12.4.1) with h(x) = ji T  ' / ~ W , ( T  ' ' ~ X ) dT.
Proposition 12.4.3. The general singular integral W,df; x ; t ) of Weierstrass is saturated in Lp, 1 2 p 4 2, with order O(t), t f O+ The Fauard class F[Lp; WK(t)]is characterized by any of the nine [eight] statements of Theorem 12.3.7 [12.3.11], respectiuely. In particular (cf. statement (v) [(iv)]): f E F[Lp; Wx(t)]ifand only i f there exists p E BV if p = landgELPifl 0 (12.4.19)
~ " ( u ) 1 = /3
then (12.3.5) is satisJed for
fd"'
H"(u)uBl du,
a = /3.
Proof. Since H A E C,, the integral in (12.4.19) exists as a Riemann integral (at least improper) for every /3 > 0. Changing variables we obtain for each v # 0
say. Here R;;(u), obviously being the Fourier transform of some Rn E L', denotes the approximate Riemann sum of the integral. It follows that (12.4.20)
uniformly for all n E N. Theorem 6.2.3 then gives the existence of some v E BV such that (12.3.5) holds with Q = j?. Turning to applications, we consider the integral of BochnerRiesz as defined by
SATURATION THEORY
468
the kernel bl being given by its Fourier transform (A > 0).
(12.4.22)
By Problem 6.4.9, b,, belongs to NL1for every X > 0 and, in fact, admits the explicit representation ((112) + A)'J (12.4.23) bh(x) = 2"(1 + A)c>lxl(1,2)+A(lXl), where J,(x) is the Bessel function of order y. It follows that (12.4.24)
and thus b,, satisfies (12.3.4) with a = 2 and c =  A . To establish (12.3.5) we have in view of the identity b?(o)  1 = 2
(12.4.25)
I0'"'
[  Ab? l(u)]u du
that b,, satisfies (12.4.19) with ,B = 2 and H ( x ) = Xb,,(x). applies for X > 1. In fact, for all h > 0
Proposition 12.4.6. Let f (a) The approximation
E
LP,
Therefore Prop. 12.4.5
1 c p I 2.
1 BA(f; '; 
f ( O )
/IP = O ( p  1'
(P
+
00)
impliesf ( x ) = 0 a.e. (b) The following assertions are equivalent:
(i) IlBA(f; '; p)  f(.) 1 P = O(P 2) ( P + 001, (ii) there exists g E LP such that IIp2[Bl(f;0 ; p)  f ( o ) ]  g(o)llp = 0, (iii) there exists g E LP such that Xoaf"(v) = gA(u) a.e., (iv) f is equal a.e. to a function in AC:,, with first and second derivative in Lp, (v) there exists g E L P such that f ( x ) = (l/A)
1% 1'' du,
m
(vi)
Ilf(0 + h)  2f(.) +
f ( 0
 h)IIP
m
=
g(uz) duz a.e.
(h += 0).
O(h7
Proof. In view of (12.4.24), part (a) follows for any X > 0 by Prop. 12.3.2. Concerning part (b), for X > 1 the equivalence of (i), (ii), and (iii) follows by Theorem 12.3.11 since (12.3.5) holds. Thus let 0 c X I1. Then, according to (12.4.24), (i) implies (iii) by Theorem 12.3.10. If (iii) holds, we use the identity (12.4.26)
1
B A ( ~X ;; P)  f ( x ) = B~+i(f; X ; P)  f ( x )  hpl B,+(g;X; P)
469
SATURATION FOR SINGULAR INTEGRALS
which follows by comparing the Fourier transforms of both sides. Moreover, (iii) implies (A + 1)u2fA(c~)= [ X  l ( X + I)g]^(o) a x . ; thus (iii) is valid for h + 1, too. SinceX + 1 > l/BA+I(h";p)  f ( o ) l l p = O(p'), and sillce IIBA(g; p)Ilp 2 Ilgllp, (i) follows by (12.4.26). Regarding (ii) we again use (12.4.26). Indeed, if (iii) holds, then O ;
1
 X {BA(g;
p)
 g(x)>.
+
Since X 1 > 1, the first term on the right tends to zero in LPnorm as p+m, whereas limp.+m(IBA(g;0 ; p)  g(o)IIp = 0 by Theorem 3.1.6. Finally, the equivalence of (iii)(vi) follows by Theorems 5.2.21 and 10.5.4. This completes the proof. Naturally, there exists an analogous result in L1space. Thus the singular integral BA(f;x; p) of BochnerRiesz is saturated in LP, 1 Ip I2, with order O ( P  ~ ) and , f belongstotheFavardclass F[Lp; &(p)]if and only if .(fI/ + h)  2f(0)+ f ( 0  h)llp = O(hZ), h + 0. 12.4.5
Riesz Means Finally we consider the Riesz means
of the Fourier inversion integrals (5.1.6) and (5.2.19) o f f € Lp, 1 5 p 5 2. In view of the Parseval formulae (5.1.5) and (5.2.18), R,,,(S; x; p) may be rewritten as a singular integral of FejCr's type (12.4.28) By Problem 6.4.8, r K , Abelongs to NL' for each form given by
K
> 0,X > 0 and has Fourier trans
(I 2.4.29)
In fact, re, is a positive function for 0 < K I 1 and h 2 I , The integral BA(f;x; p) of BochnerRiesz is the case K = 2 of the Riesz means R,,,(S; x; p). Since (12.4.30) the kernel r,,,,, satisfies (12.3.4) with (Y = K and c = A. To establish (12.4.1) we may apply Theorem 6.3.11. For, if we set
SATURATION THEORY
470
then for h 2 1, K > 0, vK,A and 7L,Aare locally absolutely continuous on (0, co) except that v:,Ahas a jump for h = 1 at x1 = 1: &,1(1 +)  7;. 1(1) =  K . Furthermore, u l$,A(~)I dv < 03 for every h 2 1 and K > 0. Thus for these values of K and A, vK,A is an even and quasiconvex function of class Co,and so by Theorem 6.3.11 it is representable as the Fourier transform of an even function of class L1. Moreover, for all K > 0, h > 0
Proposition 12.4.7. The Riesz means RK,A(f ; x ; p ) are saturated in Lp, 1 Ip I 2, with order O ( P  ~ ) and , the Fauard class F[Lp; RK,A(p)]is characterized as V[Lp;  A lul"]. Proof. Since (12.3.4) holds for each K > 0, h > 0, Prop. 12.3.2 and Theorems 12.3.6(i) and 12.3.1qi) apply. Let h 2 1. Then condition (12.4.1) was shown to hold for each K > 0, and the proof follows by Theorems 12.3.6(ii) and 12.3.10(ii). If 0 < h < 1 and p E BV is such that  A lul.fn(v) = p"(v) for all u E R, we use the identity
and complete the proof as in the case of the integral of BochnerRiesz. For further characterizations of the Favard class of the Riesz means we refer to Theorems 12.3.7 and 12.3.1 1. Problem 1. Show that the following singular integrals satisfy (12.4.1) with
(i) FejQr:a = 1, c = 1,
(ii) CauchyPoisson: a = 1, c =
 1, 1
1
h(x) = log  9
4%
(iii) GaussWeierstrass: a = 2, c = h(x) =
+ x= X=
 1,
z/z(ex914
 X I m xia
eu9 du}.
Here h is an even function, and the representations hold for x > 0. (Hint: cf. BUTZER[7]) 2. Show by an application of Theorem 6.3.1 1 that the kernels of the general singular integral of Weierstrass and the Bessel potentials satisfy (12.4.1). 3. Let f E LP, 1 5 p 5 2. Show that iff is equal a.e. to a function 0 E ACIo, with W E Lp, then the following limit relations of Voronovskaja type are valid :
.
(9 limprm I I / U ( o )  a(f;
0;
p)I 
@'(o)Iln
= 0,
SATURATION FOR SINGULAR INTEGRALS
47 1
12.5 Saturation of Higher Order 12.5.1 Singular Integrals on the Real Line
So far we have seen that the most fundamental condition upon the kernel x such that the corresponding singular integral J c f ; x ; p) may possess saturation phenomena is given by (12.5.1) for some constants a > 0, c1 E @ (cf. (12.3.4)). If xA is a function of IuIa, say ~ " ( v )= r](luIa), then (12.5.1) reduces to
and this means that r ] has a nonzero righthand derivative at t = 0. This suggests to consider higher order Taylor expansions of at t = 0, or, more generally, Peano expansions at t = 0. We are therefore led to discuss condition (12.5.2)
Given x E NL', let there exist constants a > 0 and cl, c2, c, # 0, r E N, such that
. . ., c, E C with
r1
 'C cj I U p
x^(u)  1
f1
lim
= c,.
blar
v0
In connection with (12.3.5) and (12.4.1) we expect that also the following will be needed : (12.5.3)
Given x E NL1, let there exist constants a > 0 and cl, c2, and afunction h, E NL' such that
c
.. ., c, E @, C,
# 0,
u = 0.
The interest in these conditions is also motivated by the following considerations. If the kernel x satisfies the condition (12.5.4)
for some hl E NL', then f o r t € W[LP; cl[uIu],1 representation
and therefore by Theorem 3.1.6 lim Ilp'[J(f; n m
(12.5.6)
0;
p)

f(0)I
u=o we have by Problem 12.3.6 the
Ip I2,
 c1D5a)fIIP= 0.
412
SATURATION THEORY
On the other hand, by the right side of (12.5.5) there may be defined for all g E X(R) a new singular integral Jl(g; x; p) of Fejk's type, namely
which is generated by the kernel hl. In these terms, the asymptotic approximation (12.5.6) for the original integral J(f; x; p) may be interpreted as ordinary convergence of the integral Jl(L x ; p). Moreover, we may investigate the degree of convergence of g by the integralJl(g; x; p) and in particular ask for its saturation, thus for second order safurnfion of the original integral J(f; x; p). In studying the difference [h;'(v)  11, conditions of type (12.3.4) and (12.4.1) may be posed, thus that there exist constants fl > 0 and y # 0 such that h;'(v)  1 lim (12.5.8) = y, V+O
lvlE
and that furthermore there exists ha E NL' such that (12.5.9)
I
1,
v = 0.
Going back to (12.5.2) and (12.5.3) for r = 2, one expects that one must verify (12.5.8) and (12.5.9) with y = cz/cl and fl = a, i.e., that the exponent fl of (12.5.9) for hl must be equal to the exponent a of (12.5.4) for the original x. This will do to indicate the connection between the saturation problem for the integrals J(f; x ; p) and Jl(f; x ; p).
Theorem 12.5.1. Let f E LP, 1 Ip 5 2, and x satisfy (12.5.2). (i) If the a(r  1)th strong Riesz derivative o f f exists and
thenf(x) = 0 a.e.
(ii)
If the a(r  1)rh strong Riesz derivative o f f exists and
then f E V[Lp; c, Iular]. (iii) If, in addition, x sarisjies (12.5.3),thenconverselyf E V[Lp;cr IuIar] implies(l2.5.11).
Proof. According to Problem 11.2.9 the existence of the a(r  1)th strong Riesz derivative o f f implies that the flth strong Riesz derivative of f exists for all 0 < fl 5 a(r  1) and [D6E'f]"(v)= IuIBf"(v). Furthermore, setting
we have
(12.5.13)
[V,,J(f;
0;
p)]"(u) =
[x"(:)
1 
'5' cj 1;
Iuj]fA(v).
j= 1
Thus, if (12.5.10) holds, II[p"'V,J( f; 0 ; p)]^(o)llp. = o(l), p + 00. Therefore (12.5.2) implies c,lvl"'f^(v)= 0, and (i) follows. Likewise, the proofs of (ii) and (iii) follow along the lines of the proofs of Theorems 12.3.6 and 12.3.10. We only recall the result
SATURATION FOR SINGULAR INTEGRALS
473
of Problem 11.2.9 that f E V [ L p ;crluIar],1 Ip I2, implies the existence of the pth strong Riesz derivative off for every 0 < /3 < ar. As a first example let us consider the generalized singular integral C,cf; x ; p) of Picard. In view of (12.4.10) we have hl(x) = c,(x). Moreover, c;(v)  1 (12.5.14)
r1
 f 2 I
( 1)’Ivp =c m ,
( l)r(Vyr
which already establishes (12.5.3) with a the notation of (12.5.12)
=
K , cj
= (
l)’, and hr(x) = c,(x). Therefore in
Proposition 12.5.2. Let f E L’, 1 5 p 5 2, and K > 0. (a) The following assertions are equivalent: (i) The K(r  11th strong Riesz derivative o f f exists and IIV,C,(f; 0 ; p)llp = O(p”3, (ii) the K(r  1)th strong Riesz derivative o f f exists and p E BV i f p = 1 or g E L P i f 1 < p 5 2 such that
wherej E N is chosen such that 0 < Kr c 2jand C,,, 2f is a constant given by (11.3.14). (b) Furthermore, the following statements are equivalent: (v) There exists the K(r  1)th strong Riesz derivative o f f and g E L p such that 0, /lpKrVxrCAfi 0 ; p )  g(o)IIp (vi) the Krth strong Riesz derivative o f f exists, (vii) there exists g E L p such that ( l)’luln‘.fn(u)= gn(v). All of the assertions (i)(vii) are equivalent i f p # 1. The proof follows by Theorem 12.5.1 in view of (12.5.14) (cf. Sec. 12.4.2). Whereas for this integral it was very easy to establish condition (12.5.3) since it reduces to the recursive formula (12.5.14), for most of the other examples the following generalization of Prop. 12.4.5 works.
Proposition 12.5.3. If f o r an even function that f o r some p > 0, r E N (12.5.15)
~ “ ( u ) 1

x E NL’
r1
2 c,luIs’
I =1
= p/’“’(lo18 0
there exists an even H E L’ such
 U ~ ) ” H ” ( U ) U B  ~du,
then there exists vr E NBV such that condition (12.5.3) is satisjied for a = p, hr(u) being replaced by v;(v).
Proof. The proof follows as for Prop. 12.4.5. Indeed, changing variables we have by (12.5.15) that for all u # 0
474
SATURATION THEORY
where R, belongs to L1 and integral. It follows that
K(u)denotes
the approximate Riemann sum of the
uniformly for all n E N. Hence Theorem 6.2.3 gives the existence of some v, E NBV such that (12.5.3) holds with a = /I, hf'(u)being replaced by vr(u). As an application, let us consider the singular integral W ( f ;x ; t ) of GaussWeierstrass, the particular case K = 2 of (12.4.12). Since
we see that (12.5.15) is satisfied withp = 2, cj = ( l)'/j!:and H ( x ) = {(  l)'/(r  l)!}wz(x). Concerning (12.5.12) for W ( f ; x ; t ) we have a = 2, p = t  l ' a , t  t O+. Observing that j ] , thus by Theorems 5.1.16, Dla')fexists if and only iff belongs to W[Lp;( l ) ' ( i ~ ) ~ and 5.2.21 ifandonlyif thereexists @~ACg;lwith@(~)~Lp,O s k I2j,such thatf(x) = @(x) a.e., the expression (12.5.12) for the GaussWeierstrass integral is given by r1 r f
2 7Wn(x).
VarWcf;x ; t ) = Wcf; x ; t )  f ( x ) 
(12.5.17)
j1
*
Hence Proposition 12.5.4. For the integral W ( f ;x ; t ) of Gauss Weierstrass off E LP, 1 Ip I 2, we have: (a) ZffEWILP;lvlaralandI I V a r W ( f ; o ; t ) l l p = o ( r r ) , t  + O + , t h e n f ( x ) = O a . e . (b) The following assertions are equivalent: (i) fbelongsto W[LP;[vlzr21and IIVarWCf;O;t)llp= O(tr),t+O+, (ii) f belongs to V[Lp;Ivlar], (iii) f belongs to W[Lp; lvla'  ' 1 and there existsp E BV i f p = 1 andg E Lp i f 1 < p I2 such that
(iv) there exists p E BV i f p = 1 and g E
Lp
if 1 < p
I2
(v) there exists a 2rrh Peuno difference off such that (vi) I I W f ( 0 ) I l P = O(har)
such that
I[OTf(0)Ilp = O(har),h + 0, (h + 0).
The proof is left to Problem 12.5.1. 12.5.2 Periodic Singular Integrals There is a completely analogous theory of saturation of higher order for periodic singular integrals Z,V, x ) as defined by (1.1.3). But if we proceed as above and try to generalize (12.1.1), for example, in the way we generalized (12.3.4) to (12.5.2), then on account of the general dependence of the kernel upon the parameter p the coefficients in the respective expansions would be general functions pj and tjj, j = 42, ,r. Although this general approach is possible, the consequence would be
...
SATURATION FOR SINGULAR INTEGRALS
475
a very intricate calculus, difficult to interpret. Therefore we restrict the discussion to the following case which covers many of the examples considered so far: (12.5.18)
Let p be a positive parameter tending to injinity and let the (periodic)kernel h,*(x)} ofZ,*(f; x ) be generated by x E NL1 (4 (3.1.28)):
Then [x,*]"(k)= x^(k/p) by (5.1.58). In view of Sec. 12.3, 12.4 it now seems reasonable to restrict (12.1.1) to (12.3.8), i.e., let there exist a > 0 and c1 E @, c1 # 0, such that for each k E Z (12.5.19) For the same reasons let there exist h, E NL' such that (12.5.20) Now we might expand [xp*]".But this would correspond to the expansion of x'' (cf. (12.5.2) and (12.5.3)), and the periodic theory would be reduced to that of the real line, i.e., to Sec. 12.5.1; then one might formulate a general theorem, however this we shall leave to Problem 12.5.2. But let us continue with the Riesz means Rn, ,(A x ) of the Fourier series off€ Xan (cf. Problem 6.4.8)
with kernel (12.5.22)
(K,
h > 0).
Here, condition (12.1.1) is satisfied with q(n) = (n + 1)" and $(k) =  A lklK since (12.5.23) Moreover, the kernel {r,,,K, ,,(x)}is generated via(12.5.18) by rK,,,(x)(cf. (12.4.29)), thus m
rn,x,h(x) =
6j =2 m
+
(n + l)rK,,,Kn l)(x
+ 2j~)).
In view of Sec. 12.4.5, r K S satisfies h (12.4.1), and therefore the representation (12.5.20) holds for {r,,,K , &)}. Hence the Riesz means R,,, K , h ( f ; x ) are saturated in X,, with order O(n.), and the Favard class F[X,,; R,,,K , is characterized as V[X,,;  A lkl"].
476
SATURATION THEORY
Concerning higher order saturation we note that in view of the Taylor expansion ( r E N , x > 0,A > 0) (12.5.24)
(1
x
r1
)= ~
2 (1)j
()xl
10
r;
A(~)
+ (l)'(?)r
Jox(x  .>''(I
 ~)"'d7:
of (12.4.29) satisfies
(12.5.25) rGA(u)  1 
r1
2 (l)'()
1=1
= (
IuIK'
I)'(
?)rK
jo'"'
UK)T 1r:
Thus the representation (12.5.15) holds for r K , Awith fl = K , c, H(x) = ( 1)'r (12.5.26)
(3 ,,
=
 ,(U)UK (1)'
 1 du.
h (i)' and
rK,  r(x). Therefore with (cf. Problem 12.5.2) r1

Rn, K . A ( f i x )
VKrRn,K . A(A x )

2
(
A ( 1)' j ) ( n
j=l
+ l)KY(Kj)(x)
Proposition 12.5.5. For the Riesz means Rn,K , ,,(A x ) of the Fourier series o f f E X,, we haue: (a) r f f E W[X,,; lkl"('  ')I and llVJ3m,
K,
A(fio)IIx,,
= o(n)('),then f is constant.
(b) The following assertions are equivalent:
(9 f
(ii) f
E W[Xan; Ikl"" E V[X,,; lkl"'].
"1
and IIVmRn, K , A(A O)Ilx,,,
=
W"3,
(c) The following assertions are equiualent: (i) f E W[X,,; IklK(' l)] and there exists g E X,, such that lim Il(n + 1 Y V A n . K , A ( f i
n+ m
0)
 g(O)ll;C,, = 0,
(ii) f E W[X,,; lkl"']. In case h = 1, 1 E N, a new phenomenon occurs for the Riesz means Rn,K , A( f ; x). Then (1  x)' is an algebraic polynomial of degree I, and therefore the expansion (12.5.24) breaks ofat r = I + 1. Indeed, (1 x)' =
(12.5.27)
'% (  l ) j ( JI x j
+ (l)'X[,
1=0
and hence (cf. (12.4.29), (12.5.3), (12.5.25)) (12.5.28)
rcl(v)  1 
for all v E R, where (12.5.29)
11
2 (l)j(!)J
J= 1
IvIK1= (1)' IulK'h;,(u)
SATURATION FOR SINGULAR INTEGRALS
477
Now it is easy to see that the function defined by (12.5.29) satisfies the conditions of Theorem 6.3.11 for every K > 0 and I E N, and therefore it is actually the Fourier transform of a function hK,I E NL1. Since r; '(k/(n 1)) = r c K , I(k),the periodic kernel {rn,K , has the representation
+
for every k E Z, where (h:, K, Jx)} denotes the periodic approximate identity produced by h K .I via (12.5.18), thus
(12.5.32)
2T
1"
" t,(x  u)h: K , &) du = t,(x).
Therefore
Proposition 12.5.6. Let f E Xan and K > 0, 1 E N. If the d t h Riesz derivative o f f exists denotes the best trigonometric approximation o f f ( K l ) , then and if En(X2,;f (12.5.33) Il(l)'(n
+ 1)rl VKIRn,
K. i(f;
0)
.f(K1)~)ll~2,,
=
O(En(X,n;f("'))) (n +. a).
Proof. In view of (12.5.30) and Def. 11.5.10 we have (12.5.34)
( 1)Yn
+ 1)rl VKIRn,
K.
i(f;
x) =
U"') * G,
K.
i)(x).
Therefore, if t,*(f("')) denotes the trigonometric polynomial of best approximation of degree n forf(Kl),it follows by (12.5.32) and (3.1.29)(i) that
I (
1)Yn
+ IY'VxIRn.
I(.L 5 Il(f(")  t,"K'') K.
5
0)
(IIhK,
lllL1
f(K1)(0)II~z.l
+ l))' (fI/
* hn*.
+ Ilt,*(f(Ki))  f (K"llx~~
K, ~ l l x l n
 tn*(f(Kf))IIX1n'
This proves (12.5.33). Now, the righthand side of (12.5.34) may be extended to all of X2,. This defines a new singular integral 1: K , '(S; x) o f f € X Z n , namely
with parameter n E N, n +. m (note that K > 0, I E N are certain fixed numbers). Since {h:, K , l(x)}is a (periodic) approximate identity, we have lim
n
m
llln*. df;  f(o)IIxzn K,
0)
=
0
478
SATURATION THEORY
for everyfE Xan.Moreover, it follows by (12.5.32) (as in the proof of (12.5.33)) that for everyfE X,, 11In*,K, I(f; 0 )  f ( 0 ) 1I Xan = O ( & o ( ~ n ;f)) (n +. CQ). Since by Corollary 2.2.4, E,,(X,,;$) tends to zero arbitrarily rapidly for n + oc) by taking f sufficiently smooth, it follows that the integral I:, K, I ( f ; x) does not possess the saturation property. This is a consequence of the fact that [h:, K , ,]^(k) = 1 for lkl 5 n 1, which implies that the approximate identity {h:, K, I(x)>does not satisfy (12.1.1). Indeed, if k E Z is arbitrary, then [h:,K,l]"(k)= 1 for all n 1 lkl  1 which implies $(k) = 0 for k E Z regardless of p(n) # 0. Finally, we observe that hK,I of (12.5.29) furnishes an example of a function which belongs to NL1but does not satisfy (12.3.4). Indeed, since hc, = 1 for all Iu( I 1, we have
+
for every u > 0. Problems 1. Prove Prop. 12.5.4. (Hint: See also BUTZER [ll]) 2. (i) Let x satisfy (12.5.3) and {xP+(x)} be generated via (12.5.18) by x. Setting VaJ,*(f; X ) = Z,*(f;
X)
 f(x) 
r1
2
cjpajf(oj)(x),
j=l
show that the assertions of Prop. 12.5.5 hold for I,*cf; x). (ii) Give applications to the integrals Wt(Ax ) of Weierstrass and P,(f; x) of AbelPoisson. (Hint: See also BUTZERSUNOUCHI [l]) 3. Show that if condition (12.5.2) holds for some r E N, then (12.5.2) holds for every m E N with 1 < m Ir. What can be said concerning condition (12.5.3)? 4. (i) Show that for the Fej6r means of the Fourier series o f f € W[Can;lkl] (compare Problem 12.2.12) (n + a). Iln[f(o)  0Af; .)I  (f)'(O)Ilcan = O(En(Czn; (f")')) (ii) Show that for the typical means R n , z ( f ;0 ) of the Fourier series of a twice continuously differentiablef (n f a). Il(n + V{R*,a(A 0 )  f(0N r(o)IIc2, = O(En(C2,;f")) State and prove analogous results for the singular integral N:(f; x ) of Jacksonde La Vallte Poussin. (Hint : See also BERENSGORLICH [l])

12.6 Notes and Remarks
The concept of saturation was first introduced by JEAN FAVARD for summation methods of Fourier series in a lecture in 1947 (cf. [2]). However as early as 1941 ALEXITS [l] had already shown that for the Fej6r means Ilun(f;0 )  f(o)Ilc2, = O ( n  l ) if and only if f " E Lip (Czn: 1). The proof of the important fact that Ila,(f; 0 )  f(o)IIc2, = o ( n  l ) implies f ( x ) = const, which in fact establishes the existence of the saturation property, (see also his paper [3]) in a letter to HILLEin July 1940 was communicated by ZYGMUND (cf. HILLE[2, p. 3521). As a matter of fact, HILLE[l] in 1936 had already considered such
SATURATION FOR SINGULAR INTEGRALS
419
oresults for particular singular integrals which are semigroup operators. The first important paper largely concerned with saturation is the doctoral dissertation (1949) of MARC ZAMANSKY [ l ] which pays particular attention to the singular integrals of FejCr and Jacksonde La Vallee Poussin. Although FAVARD does not seem to have been aware of the ALEXITS result, which was followed by a further paper [2] in 1952, he was the first to realize the importance of the concept as such. In 1956 BUTZER[2, 31 showed that semigroups of operators on Banach spaces possess the saturation behaviour. Applications were given to the singular integrals of AbelPoisson and GaussWeierstrass. These investigations were followed by a large number of papers concerned with saturation theorems for various particular methods of summation of Fourier series. It seems that FAVARD [41 was the first to recognize in 1957 that the knowledge of the behaviour of &(k)  1 for fixed k and p +po (cf. condition 12.1.1) suffices to determine saturation for general methods of summation of Fourier series. This paved the way for the general approach to the subject, and was developed in 1958/60 by BucHwALTERt [l], HARSILADSE [2], and especially by SUNOUCHIWATARI [l], SUNOUCHI[l I. Furthermore, in 1959/60, BUTZER[571 announced an integral transform method in order to solve saturation theorems, the setting being that of the theory of singular integrals. Depending upon the nature of the problem in question one would use Fourier transforms, or finite Fourier transforms, or Laplace transforms, or finite Legendre transforms, etc. The choice of the transform may be compared with the choice of a particular transform method in the solution of initial and boundary value problems for partial differential equations (cf. Chapter 7). The advantage of transform methodst as a whole lies in the fact that they possess a common methodological basis and allow a unified theory. See the short survey articles by BUTZER[12, 131. Sec. 12.1. The definition of saturation that has been chosen (cf. Def. 12.0.2) is the one used
in BUTZER[3]. It is equivalent to one given by ZAMANSKY [l] (cf. Problem 12.1.1) yet somewhat different to the one used, for example, in BUTZERBERENS [l, p. 881. Condition (12.1.1) which is sufficient for the saturation property to exist, may be compared with the definition of the infinitesimal generator (in the transformed state) of semigroup theory as applied to operators of convolution type (cf. (13.4.8)). The Dirichlet singular integral (which is not an approximate identity) does not satisfy (12.1.1). Indeed, it fails to have the saturation property since it furnishes approximations of arbitrarily high order (cf. (2.4.1)). Now on the other hand, the kernel of the integral In(f;x ) of (12.2.29) satisfies (12.1.1) and (12.2.6) so that In(f;x ) is saturated in X,, with order O(n'), the saturation class F[X,,; I,] being characterized as V[Xzn; (1/2) lkl]. For a n example of an approximate identity which does not possess the saturation property we refer t o the end of Sec. 12.5.2. As already remarked, Prop. 12.1.2 for the singular integral of FejBr (or AbelPoisson) may be found in ZYGMUND [3]. Theorem 12.1.4, a main result of this section, is due to SUNOUCHIWATARI [l] and HARSILADSE [2], see also ALEXITS[2] and FAVARD [4]. For weak* compactness arguments of this type we refer to SUNOUCHI [III] (see also Sec. 10.7). For Cor. 12.1.5 in Cznspace we recall ZAMANSKY [l]; for Lg,space (and Czn) see ALEXITS [2], FAVARD [4], and SUNOUCHIWATARI [l]. Prop. 12.1.6 may be found in BUTZERGORLICH[2, p. 3751. The important r6le of condition (12.1.7) was first noted by KOROVKIN [3]. The fact that the FejBrKorovkin integral is saturated in Xzn (Cor. 12.1.7) is shown in BUTZERGORLICH [2, pp. 371, 3771, see also KOROVKIN [3].
t BUCHWALTER wrote four noteworthy Comptes Rendus notes dealing with saturation theory, unfortunately written in a brief and somewhat too compact style. Since he did not publish longer articles with proofs and details, his papers perhaps did not receive the credit they actually deserved at the time. Often others rediscovered his results and published them in full. $ For Laplace transform methods in saturation theory, see BERENSBUTZER [l], BERENS[I]; for Legendre transform methods and surface spherical harmonics see BUTZER[12] WESTPHAL and BERENSBUTZERPAWELKE [l]; for Mellin transform methods see KOLBENESSEL [l, 21.
480
SATURATION THEORY
Sec. 12.2. Concerning Theorem 12.2.1, the assertion f E F[C2,; Z,,] if and only if f~ Lip*(Czn;2) is given in TURECKI~ [3], where also most of the relevant examples may be found. The Favard class of the (special) integral W,(f;x ) of Weierstrass (cf. Prop. 12.2.2) was first determined in L#,space, 1 < p < co, as an application of a general theorem on semigroup operators (cf. Sec. 13.4.2). For an interpretation of the saturation theoretical behaviour of solutions of partial differentialequations in terms of their initial or boundary behaviour see e,g. LEIS[l, 21, BUTZER[6, 9, 111, KANIEV [I]. The idea of solving the direct part of the saturation problem using the concept of uni[2], but is worked formly bounded multipliers (cf. Theorem 12.2.3) is due to BUCHWALTER out in SUNOUCHI [3,6]. Earlier results assumed the uniform convexity or quasiconvexity of the sequences {A,(k)) of (12.2.4); see BUCHWALTER [l] and SUNOUCHI [3]. In any case, the essential point is to have suitable sufficient conditions for (12.2.4) to be valid. Here (12.24, in connection with the criteria of BEURLING [l] and SZNAGY[3] (cf. Sec. 6.3.3), is widely applicable; see also BERENSGORLICH [I]. Concerning the saturation of the typical means, SZ.NAOY[3] showed for fractional K > 0 that f E V[Czn;  lkl"] implies I\&, x(f;0 )  f(o)IIcp, = O(n'9 while ZAMANSKY [31 and HAR~ILADSE [2] established the converse for K E N. For fractional K both directions [2] and SUNOUCHIWATARI [l]. were proved almost simultaneous!y by ALJAN~IC The idea of using functional equations in solving saturation problems may perhaps be traced back to the case of semigroup operators. The proof of Prop. 12.2.8 which rests upon the important identity (12.2.15) was carried out by NESSEL[23; the saturation class itself for Wt,.Cf; x ) may be found in BUTZERGORLICH [2, p. 3671. The recursive formula (12.2.28) was employed by BERENS 131; for a paper making use of functional equations for [l]. solving similar problems we cite BUTZERPAWELKE For Problems 12.2.l(ii)12.2.3 see ZAMANSKY [I], SZ.NAGY[3, p. 501, SUNOUCHIWATARI [l], T U R E C K [3],~ and BUTZERGORLICH [2, p. 3791. For further investigations concerned with the Ceshro (C, a), Holder (H, a), Riesz, Abel, Lambert, Euler, Borel, Riemann, Voronoi, and Norlund methods of summation of Fourier series we cite ALJANCIC [l, 21, BUCHWALTER [I, 21, HARSILADSE [l, 21, NATANSON [7], TURECKII [l51. For instance, if m E N, ALJANCIC[I, 31 showed that the Hdder ( H , m)method has saturation order O((1og"  'n)/n) and class V[Xan; lkl]. However, the methods of proof in the latter papers are often restricted to particular types of summation processes and cannot be generalized to a wider class. There are also a number of results on 'local saturation'. For instance, if f E Can and (a, 6) is an arbitrary subinterval of [  ~ , 7 r I , then u,,n(f;x )  f ( x ) = O(nl) uniformly over (a, b) if and only if (f")' is bounded over (a, 6). See SUNOUCHI [2, 71, MAMEDOV [8], SUZUKI [I], CHEN TUNPING[I], IKENOSUZUKI [I]. For results on saturation concerning [5]. strong approximation see FREUD A first result concerning pointwise saturation (for the Fejtr means) was already mentioned in the Notes and Remarks to Sec. 1.7. As precursory material one may regard the lemma of H. A. Schwarz (cf. Problem 10.1.3) as well as its following extension due to DELA VALL~E POUSSIN (cf. RTCHMARSH [6, p. 1531, BUTZER [lo], and the literature cited there): If f E C[a, 61 is such that limh,o I ~  ~ [ f ( xh) + f ( x  h)  2f(x)] = g(x) existsfinitelyfor I:' allx E (a, 6)withg E L1(a,b), then there are constants ao,a1such rhatf(x) = a0 t alx + g(ua) dua. This leads to the following extension for the FejCr means (cf. BERENS [4, 51): Let f E La,, be finite in some interval (a, b)and limnrmu,,(f ;x ) = f ( x )for all x E (a, 6). If the limit limna n [ U n ( f ; x )  f ( x ) l = g(x) existsfinitelyfor all x (a, 6) with g B L1(a,b), then there is a constant a. such that f " ( x ) = a.  I: g(u) du for almost all x E (a, 6). This result may be representative for pointwise saturation theorems obtained recently by several [I], AMEL'KOVIC [ 11, M ~ L B A C H authors. Let us, in addition, mention BAJ~ANSKIBOJANI~ [l, 21, LDRENTZ~CHUMAKER[l],MICCHELLI [I]. We remark that it is possible to deduce theorems on nonoptimal approximation from [4, 61, BUTZERGORLICH 121, BUTZERcorresponding saturation theorems. See SUNOUCHI
+
SATURATION FOR SINGULAR INTEGRALS
48 I
SCHERER [ll, FREUD [2,4], and GORLICHSTARK [2] and the literature cited there; compare also Sec. 13.3.3. The saturation problem has furthermore been considered for (i) algebraic approximation, [2], DE LEEUW[l], in particular, by the polynomials of Bernstein and Landau (LORENTZ SUNOUCHI [lII]), (ii) linear summation methods (determined by matrices) of series (which [3, 41, need not necessarily be trigonometric), see the original approach by FAVARD BUCHWALTER [l], ZELLER [2], also ZYGMUND [71, p. 1231, (iii) trigonometric interpolation [5], KALNI’BOLOCKAJA [l]). processes (RUNCK[ 11, SUNOUCHI Sec. 12.3. The basic results of this section are to be found in BUTZER[571, where the fundamentals on the Fourier transform method for functions defined on the line group are developed. The limiting relation (12.3.4) suffices to prove the inverse part, while the representation of the fundamental quotient of (12.3.5) as a FourierStieltjes transform enables one to prove the direct part of the saturation theorems. The importance of condition (12.3.5) for the method in question was first recognized by H. KONIG(see the remarks in BUTZER [7, p. 4011; compare also BUCHWALTER [3]). (It is of interest to note that condition (12.3.5) may also be expressed directly in terms of the kernel x itself, see KONIG [l]). The independent investigation [l] by SUNOUCHI does not contain condition (12.3.5) and is thus mainly concerned with inverse saturation theorems. For Lemma 12.3.4 see KOZIMASUNOUCHI [I], for Prop. 12.3.5 see BUTZERKONIG [l]. For the equivalences of (v) [(iv)] with (i) of Theorem 12.3.7 [12.3.11] see BUTZERNESSEL [2]. The fact that condition (12.3.5) is indeed a necessary one for the direct part to hold (cf. Prop. 12.3.8) was noted by BUTZERKONIG [l]. The proof of Theorem 12.3.10 is modelled after BUCHWALTER [3]. For the integral (12.3.1) does the ratio [xn(u)  l]/lu18 not only play an intrinsic r81e in the saturation case, say for p = a, but its behaviour for any 0 < p < a may just as well be related to nonoptimal approximations; see the funda[2], where X,,approximations of mental paper of SZ.NAGY[3] as well as TELJAKOVSK~ the corresponding (periodic) singular integral (3.1.30) of FejBr’s type are considered in case it represents a polynomial summation process of Fourier series (i.e. xn(v) has compact support). On the other hand, condition (12.3.5) alone, i.e., the behaviour of [x”(u)  l ] / l ~ [ ~ in the saturation case /? = a, covers all nonoptimal approximations O ( p  @ ) ;see Sec. 13.3.3. Regarding Problem 12.3.5, for saturation of the integrals of GaussWeierstrass and Jacksonde La VallBe Poussin see BUTZER[7] and SUNOUCHI [lI]. For extensions of the theory as given in Problems 12.3.712.3.9 we may, for example, refer to BUTZERNESSEL [2]. We may also mention a series of papers of MAMEDOV in which a systematic account on approximation by socalled rnsingular integrals (cf. also HARSILADSE [3]) defined through
is given. Similar extensions are discussed on the circle group. See e.g. [4, 5, 71 and in particular his monograph [9] (unfortunately written in Azerbaijani as are also many of his papers). Sec. 12.4. For the material on the generalized integral of Picard as well as on the general
integral of Weierstrass, see BERENSGORLICH [l], NESSEL [2]; for saturation of convolution integrals (cf. Sec. 3.6) in the sense of HIRSCHMANWIDDER [5] see DITZIAN[l, 21. Formula (12.4.19) of Proposition 12.4.5, which is a generalization of the semigroup case treated by NESSEL[2] (cf. (12.4.15)), was used by KOZIMASUNOUCHI [l] to prove direct parts of saturation theorems for various processes. Obviously, one may replace H E L1 by some r) E BV, thus HA(u) by r)”(u). In connection with Prop. 12.4.4 there is an extension by KURAN[l], stating that llfllP  IlP(f; 0 ; y)IIp = o(y) implies f ( x ) = 0 a.e. if f~ L’, 1 < p .c 00, and f keeps constant sign almost everywhere i f f € L’. (The results which are actually given for functions in several variables are proved by using the theory of subharmonic functions.) 31~.~.
482
SATURATION THEORY
For the integrals of BochnerRiesz and Riesz see BERENSGORLICH [l], NESSEL[2], KOZIMASUNOUCHI [l], BERENS [3]. The proof of Prop. 12.4.6 and 12.4.7 for 0 < h I1 is due to H. BERENS (unpublished). Saturation theory has already been treated in texts on approximation. Apart from the brief accounts in LORENTZ [3, pp. 981081 and ZYGMUND [71, pp. 1221241 we can refer to the stimulating presentation of H. S. SHAPIRO [l, 41; see also MAMEDOV [9]. Sec. 12.5. The approximation of the GaussWeierstrass integral W ( f ;x ; t ) by Taylor polynomials, particularly the equivalence of parts (i) and (ii) of Prop. 12.5.4(b), was first presented by BUTZER as a conjecture in a lecture at the Institut Henri Poincard, Paris, in Dec. 1958 (see BUTZER [l 11). Subsequently, BUTZERTILLMANN [l, 21 considered approximation and saturation of general semigroup operators {S(t)}, t 2 0, by Taylor polynomials (tk/k!)AkJA being the infinitesimal generator of the semigroup (see also Sec. 13.4.2; for details BUTZERBERENS [l, Sec. 2.21). Further papers in their order of appearance are BUCHWALTER [4], LI SJUNCZIN[l], BUTZER[6, 111, Leis [l], SUNOUCHI [6], BERENS [2], BERENSGORLICH [l]. In BUTZERSUNOUCHI [l] particular interest is given to asymptotic expansions of the solutions of Dirichlet’s problem (7.1.17) for the unit disc and Fourier’s problem (7.1.4) for the ring. Condition (12.5.15) for general kernels is given in KOZIMASUNOUCHI [l]; it generalizes relation (12.5.16) for the Weierstrass kernel noted in BUTZER1113. The curious phenomenon stated in Prop. 12.5.6 was observed by BERENSGORLICH 111, for the Fejkr means already by Lr SNNCZIN[l].
2;;;
13 Saturation on x(R)
13.0 Introduction
In the final chapter of this volume we wish to treat : (i) a number of questions, considered particularly in the last three chapters, from a unified source including the LPcase, 2 < p < co,(ii) the comparison of the error of approximation corresponding to two different processes, and (iii) saturation on arbitrary Banach spaces. In Chapter 12 we have seen that for the FejCr and CauchyPoisson kernel condition (12.3.5) is satisfied with the same exponent (a = 1). Thus the saturation classes of the associated convolution integrals in Lp, 1 Ip 5 2, are both characterized by the same class (apart from a constant factor), namely V[LP; c IuI 1, whereas those of GaussWeierstrass and Jacksonde La VallCe Poussin, for example, are characterized by V[LP; ma] (see Problem 12.3.5).This suggests that the order of approximation of the first pair of integrals should be comparable, similarly for the second pair. Indeed, we havefor0 < a I1 (p+oo,y+O+)
IIdf;";PI  f(0)IIP = o(P3 * I I W ";Y ) f(o)IIp
= O(Y").
(If 0 < CL < 1, the equivalence follows via the class Lip (Lp; a) by the results of Sec. 3.5.) This leads to a study of socalled comparisth theorems which relate the order of approximation of a pair of approximate identities directly (and not via equivalent function classes such as V[X; c l u l ] ) so as to cover optimal as well as nonoptimal approximation. Whereas in the saturation case the quotient h"(u) l)/lula played the key rde, for the comparison of a pair of convolution integrals we shall see that the quotients
will do the same, particularly their representations as FourierStieltjes transforms. Thus the notion of divisibility in the ring of FourierStieltjes transforms is significant.
484
SATURATION THEORY
For this purpose, following H. S. SHAPIRO [l], it is useful to rewrite the difference Jcf; x; p)  f ( x ) (even higher order differences, the operators RIP),the strong Riesz derivative, etc.) in a unified form as an integral (13.0.1) with suitable u E BV and parameter 7 > 0. Thus, let x E NL' andfe X(R). The difference J ( f ; x; p)  f ( x ) may be rewritten in the form (13.0.1) with 7 = l / p and a as defined by
6 being the Dirac measure of (5.3.3). Consequently, the error J ( f ; x; p)  f ( x ) of the approximation off by the integral J c f ; x; p) is expressible as
D,(f; x; 7 ) = (l/&)
(13.0.3)
jmmf(x 
TU)
da(u),
u being
defined by (13.0.2), and we are interested in the behaviour of this integral as 730+. To this end, it is no longer necessary to restrict attention to functions a of type (13.0.2). Thus, letting a E BV be such that JZm do(#) = 0, i.e. a E BV,, we call 1 D,(f; 0 ; ~ ) l l the x ~ adeuiation off in X(R). Correspondingly, ~
U
o
W
;f; ) 0 = SUP I &(f; OSZSt
0
; 7 )I X(W)
is referred to as the amodulust off in X(R). Noting that D,(f; 7 ) tends in X(R)norm to zero$ as 7 3 0+, a major aim will be to compare (for fixedf) the rate of decrease of the adeviation and omodulus for different choices of U. As a further example, if o is the binomial measure 6,(x) of Problem 5.3.1, then the amodulus is the ordinary modulus of continuity of order r off. In particular, supposing u to be the dipole measure S,, then (13.0.3) becomes { f ( x + 7 )  f ( x ) } . Thus the statement 11 Ddlcf; T)~~x(R)= O ( Tfor~ some a > 0 expresses thatfE Lip (X(R);cc). On the other hand, if a is defined (13.0.2), the corresponding statement expresses that IIJ(f; p)  ~(.)IIx(w, = O(p").Thus, as H. S.SHAPIROremarkS,both 'smoothness' and 'order of approximation' assertions can be expressed in the form (13.0.3) with suitable function a. From this point of view there is no essential difference between a direct and an inverse approximation theorem. Both types are merely comparison theorems relating the behaviour of DulCj.;x; 7 ) to that of D,;(f; x ; 7 ) for a pair of functions al, a2. 0 ;
0;
0 ;
t This terminology is justified since umoduli behave much the same as the ordinary modulus of smoothness. $ (13.0.3) is not only defined for (I E BVo but also for Y E NBV. However, thenfcX(U8) implies lim,,o+ Il&V, 0 ; 7 )  ~(.)I[X(R) = 0 (cf. Problem 13.1.1).
SATURATION ON
x(w)
485
Sec. 13.1 is concerned with the saturation problem for D,(f; s;T ) in X(R)space, in particular, in LP for 2 < p < co. In the latter case, dual arguments are employed in the proofs. Sec. 13.2 is devoted to applications. Whereas those for LPspace for 2 < p < 00 round off previous results begun in the last chapters, at the same time a number of theorems of Chapters 1012 are reestablished from a unified source. Sec. 13.3 is reserved to a compact account of comparison theory, including reductions to periodic functions. This yields yet further proofs of the fundamental theorems of Jackson and Bernstein on best approximation. Sec. 13.4 treats saturation of strong approximation processes on arbitrary Banach spaces. 13.1 Saturation of D , ( f ; x;
T)
in X(R), Dual Methods
The saturation problem for the integral (13.0.3) shall be investigated under the following assumption: (13.1.1) Given u E BV,, let there exbt a homogeneoust function #(v) of order a > 0 with #(O) = 0, #(v) # 0 for 0 # 0, and Y E NBV such that uy(u) = #(v)v"(u) for all u E R. The importance of this condition for the discussion in Lpspace, 1 I p I 2, is revealed by the results of the preceding chapter. In fact, we obtain by the methods of proof of Sec. 12.3
Proposition 13.1.1. Let f E Lp, 1 I p 5 2, and u satisfv (13.1.1). (a) If 1 D,(f; 0 ; T ) 1 = o ( P ) , T + 0 + , then D,(f; x; T ) = 0 a.e. for each T > 0. (b) The following assertions are equiiralent:
Proposition 13.1.2. Let f E Lp, 1 I p I 2, and u satisfy (13.1.1). The following assertions are equivalent: (i) There exisls g E Lp such that lim,,,, I l ~  " D d f ;0 ; 7 )  g(o)IIp = 0, (ii) f E W[Lp; #(v)l, (iii) there exists g E Lp such that for each T > 0
Srn
D,(f; x ; 7 ) = TU
4~  m g ( x
 TU) dv(u) a.e.
The proofs are left to Problem 13.1.2. This solves the saturation problem for D,(f; x ; T) in LPspace, 1 Ip I 2. Concerning extensions to all X(R)spaces, we introduce the following notations : Dewtion 13.1.3. The ut$oint integral of D,cf; x; T ) is given by D,.(f;
X; T)
j r nf ( x 
=
6
TU)
du*(u),
OD
where u*(u) = u(u).
t A function f l u ) defined on R is called (positive) homogeneous of order a > 0 if $(TO) = F$(u) for every 7 > 0 and u E R.
SATURATION THEORY
486
Definition 13.1.4, Let f E W[Lp; t,h(u)], 1 s p I 2. If g E Lp is such that t,h(u)f^(u) = g^(u), then g is called the $derivative o f f , in notation: D*J
To illustrate the latter definition let us consider some examples of functions $. In case $(u) = 1uIa, it was the result of Theorems 11.2.611.2.9 that D@fis equal to D f ' f , the ath strong Riesz derivative off. If #(u) = (iu)', r E N, then f E W[Lp; (iu)'] implies by Theorem 5.2.21 that f is equal a.e. to some 0 E ACi&I with E Lp, 1 I k I r, and D*f = W). For the interpretation in case +(u) = {  i sgn u>(iu)'we refer to Problem 8.3.3. Now, if u E BV,, then lim,,o+ (1 Du.cf; 0 ; T ) ~ \ X C W=, 0 for every f E X(R). Moreover, if u satisfies (13.1.1) and $* is defined by t,h*(u) = $(u), then (cf. Problems 13.1.3, 13.1.4) (13.1.2)
for every s E W[Lp; +*(u)], 1 5 p 5 2, and thus lim ~ ~ T  ~ D , 0. ;( sT );  (D**s)(o)lIp= 0.
(13.1.3)
140+
We may now treat the saturation problem for the integral (13.0.3) in Lpspace, 2 0.
(iii) there exists g E C such that
13.2 Applications to Approximation in Lp, 2 < p
0
(iv) there exists g E Lp such that lim IIP"[J(f;
Dm
0;
P)  f@)l
 cg(o)II,
=
0.
The proof is given by Theorem 13.1.5 and Prop. 13.1.6. The respective result in C is left to Problem 13.2.2. Now one might give applications to the various singular integrals discussed in Sec. 12.3, 12.4 in order to supply saturation theorems in all X(R)spaces. But this is left to the reader (cf. Sec. 13.2.5).
490
SATURATION THEORY
13.2.3 Strong Riesz Derivatives
In order to extend the results of Sec. 11.2 to LP, 2 < p < co,let us first rewrite Def. 11.2.5 in the setting of integrals of type (13.0.3). Let 0 < a < 1. Putting
?.W
=
S_xm mZ
IaW
du,
where mi', is given by (1 1.2.6), Lemma 11.2.2 shows that qe belongs to BVofor each 0 < a c 1. Moreover h  Y f * mL I~)(x)= h"D,,(f; x; h), thus the ath strong Riesz derivative off for 0 < a < 1 is the strong limit (if it exists) of h"D,,,(f; x; h) as h + 0. Now, satisfies (13.1.1) with +(v) = 101" since by Lemma 11.2.2
where m E NL' is given by (5.1.27). We have
Proposition 13.2.2. Let f E LP, 2 < p < 03, and a > 0. The ath strong Riesz derivative o f f exists ifand only ifthere exists g E LP such that for every s E W[Lp';lola] (13.2.1)
In this case,
Or)f = g.
Proof. For 0 < a < 1 the proof follows by the equivalence of assertions (ii) and (iv) of Theorem 13.1.5 as applied to v4.
Let a = 1. We recall (Theorem 8.1.12) that f E Lp, 2 < p < co, implies that the Hilbert transform f " also belongs to LP. The first strong Riesz derivative off has been defined as the limit in norm of h'Lf"(x + h)  f " ( x ) ] for h +0 (see (11.2.16)). In other words, h)  f " ( x ) = Ddl(f" ;x ; h), Dll'fexists if and only if the limit in norm of since f " ( x h' Dslcf";x ; h) exists. Setting [(u) = iu, in view of Theorem 13.1.5, (ii) 9 (iv), this is equivalent to the existence of g E Lp such that for every s E W[LP'; [(u)]
+
(13.2.2)
Since 1 ,< p' < 2, we have by Prop. 8.3.1 that s E W[Lp'; [(u)] if and only if s E W[Lp ; lul I. Moreover, [(Dts)"]"(v) = {isgn u}(iu)sh(u) = Iv[sA(v)= [Dj')sIA(u). Therefore by (8.2.1)
f "(x)( Dcs)(x) dx =
 Jmm f ( x ) (Dts)"( x ) dx
=
SF,
f(x)(D!l's)(x) dx,
which together with (13.2.2) proves (13.2.1) for a = 1. For arbitrary 01 > 1, the proof follows in view of the iterative process in defining DF'f. Indeed, if e.g. a = 1 + /I, 0 < 9, < 1, we have by the preceding two cases that DPYexists if and only if Dl'lfexists and there is g E Lp such that (13.2.3)
lmm ( D ~ 1 ~ f l ( x ) ( D ~ 8 'dx s ) ( x=)
SATURATION ON
for every s E W[LP'; lol'].
49 1
Moreover, DS'Yexists if and only if for every s E W[Lv'; l u l l
S_mmf(x)(Dl"s)(x)dx
(1 3.2.4)
x(R)
=
(D:")(x)S(x)
/mm
dx.
However, s E W[Lp'; Iula] if and only if Djulsexists and DL4)sE W[Lp'; IuI]. Thus (13.2.3) and (13.2.4) imply (13.2.1) for 1 < a < 2. The converse assertion as well as the proof for arbitrary a > 0 is left to Problem 13.2.3.
The Operators RJ") We start off by rewriting Rplfas a singular integral of FejCr's type. This shall enable
13.2.4
us to apply Prop. 13.2.1. Let 0 < a < 2. Then RFlfis defined on X(R) by (11.3.2). Setting 0, 1x1 I1 (1 3.2.5) k,(x) = , x ,  ( l + , ) , 1x1 > 1
{;a
e

=
&I)
E
f ( x  u)k,(u/e) du
a
c 2),
>0
we have k , E NL1. Moreover, for everyfe X(R) and
&(fix;
(0
0. Since ha is even, this implies A;(u) = satisfies (12.4.1)for all 0 < a < 2.
d%va(u). Thus we have shown that k,
Now we are ready to apply the results of Sec. 12.3 or Prop. 13.1.1, 13.1.2 to k,, which would again give Theorems 11.3.3, 11.3.5. For the extension of the latter to Lp, 2 < p < 00, we may apply Theorem 13.1.5 or Prop. 13.2.1 to deduce by (13.2.6)
Proposition 13.2.3. Let 0 < equivalent:
a
< 2. For f
E
Lp, 2 < p < 00, the following assertions are
(ii) there exists g E Lp such that for every s E W[Lp'; Ivl"]
(iii) there exists g E L P such that
Concerning the extension of Theorems 11.3.6, 11.3.7 to LP, 2 < p < 00, we note that for given a > 0 and 0 < a < 2j,j E N, we may by similar computations rewrite the integral (1 1.3.1) as a singular integral of FejCr's type. The corresponding kernel again satisfies (12.4.1) so that Theorem 13.1.5 furnishes the desired extension (cf. Problem 13.2.4). 13.2.5 Riesz and Fej& Means Let us finally collect the results so far obtained in order to give a saturation theorem for the Riesz and Fejer means in Lpspace, 2 c p < a.The singular integral of Riesz as given by (12.4.28) satisfies (12.4.1) so that we have by an application of Prop. 13.2.113.2.3
SATURATION ON
x(R)
493
Proposition 13.2.4. The Riesz means RK. ,,(A x ; p ) are saturated in Lp, 2 < p < 03, with order O(p"), p + 00, and the Favard class F[Lp; RK, i(p)] is characterized by any one of thefollowing statements: (i) There exists g E L p such that for every s E W[LP';Ivl"]
(ii) there exists g E L p such that limo+ Ilp"Lf(0)  RK,,,(A (iii) f has a Kth strong Riesz derivative, (iv) there exists g E L p such that
0 ;
p)]
 Ag(o)ll, = 0,
where j E N is chosen so that 0 < K < 2j, and C,, 2f is a constant given by (1 1.3.14). In atzy case, we have DF'f = g. For the particular instance K = 1 we note (cf. Problems 13.2.1, 13.2.5)
Proposition 13.2.5. The singular integral ~ ( fx ; p ) of Fejir is saturated in L p , 2 < p < co, with order O(p'), p + 03, atzd the Favard class F [ L p ; a(p)] is characterized as tize set of functionsffor which Ilf"(0 h)  f"(o)II, = O(lhl>,h+0. Thus, in view of Problem 8.2.5 the Favard class F[Lp; d p ) ] for 2 < p < co is given by
+
L i p ( L p ; 1). Problems 1 . Let f E Lp, 2 < p < 00, and x satisfy (12.3.5). Show that f is an invariant element of J(f; x ; p) if and only if f ( x ) = 0 a.e. In Cspace the same is true if and only if f ( x ) = const. 2. State and prove the analog of Prop. 13.2.1 for functions f E C. 3. (i) Let a = 1 p, 0 < p < 1. Show that if (13.2.1) is satisfied, then the ath strong Riesz derivative of fexists. (Hint: Compare the proof of Prop. 10.6.4, (ii) * (i), as well as (1 1.2.15)) (ii) Prove Prop. 13.2.2 for arbitrary values of a > 0. (iii) Let f E Lp, 2 < p < a,and suppose that for some a > 0 there exists g E L P such that (13.2.1) holds for every s E W [ L P ' ; Ivla]. Show that for each 0 < j3 < a there exists g, E L p such that JTm f(x)(Dis)s)(x)dx = J y m g,(x)s(x) dx for every s E W [ L P ' ; lvl,]. (iv) Let f~ C. The ath strong Riesz derivative o f f exists if and only if there exists g E C such that for every s E WIL1;lola]
+
S_mmm f
(x)( DP)s)(x)dx =
.Imm
g(x)s(x) dx.
4. Prove the equivalence of assertions (i) and (iv) of Prop. 13.2.4. 5. LetfE Lp, 2 < p < 03. Show that the first strong Riesz derivative off exists if and only if Ilf + h)  f " ( 0 ) I I P = O(lhl>,h + 0. "(0
13.3 Comparison Theorems 13.3.1
Global Divisibility
We start with a very simple comparison theorem based on the global divisibility of FourierStieltjes transforms.
SATURATION THEORY
494
Definition 13.3.1. u E BV divides ( E BV globally?, if there exists A E BV such that ["(u) = A"(u)u"(u) for all u E R. Theorem 13.3.2. If.
E
(13.3.1)
BV divides [ E BV globally, then for euery f
IIDeCf; ";T)llXOW 2
E X(R) (7
I I A I I B V I I D ~ c f ;O ; T)llX@)
> 0).
Proof. In view of Theorem 5.3.5 and Prop. 5.3.11, ("(u) = A"(u)u"(u) implies ((x) = (A * du)(x), i.e., for everyfe X(R) and 7 > 0 (cf. Problem 5.3.9) f(x
 7u) d&)
=
1 1/5f  a
f(x 
TU
 TY) du(u) dA(y) (a.e.).
f  m
Hence by the HolderMinkowski inequality
which proves (13.3.1).
Theorem 13.3.3. I f f , ul, 02, . . .,a, E BV are such that T ( u ) = 25= A ~ ( u ) u ~ ( ufor ) every u E R with some Al, . .,A, E BV, then for euery f E X(R)
.
I
(1 3.3.2)
11 oe(f;; 7)[IX@) O
5
2 11 A, 11 BV 1 D,,Cf; 1= 1
O
(7
> 0).
1) divides (,yp(u)
 1)
;7)I(X ( w )
This is an immediate consequence of Theorem 13.3.2.
(x;(u) 
Corollary 13.3.4. Let xl, X a E N L 1 and suppose that globally. Then for eueryfe X(R) (13.3.3)
IIJicf; 0 ; P)
 f ( o ) l l x ( ~5 I l ~ l l ~ ~ I I J a c f ; P)  f ( o ) l l ~ ~ ~ ) 0;
(P > 01,
where Jlcf; x; p) and J2(f; x ; p) are the singular integrals of Fejbr's type corresponding to the kernels x1 and x2, respectiuely. For the proof, set 5(x) = SZ a xl(u) du  6(x) and u(x) = S? a x a ( U ) du  6(x), where 6 is the Dirac measure (cf. (13.0.2)), and apply Theorem 13.3.2 with T = l/p.
Definition 13.3.5. We say that the kernel x1 E NL' is better than the kernel X a E NL' i f there exists a constant M such that for euery f E X(R) and p > 0 (1 3.3.4)
ItJ i ( f ; ;P)  f I1xm) 0
(0)
M
I Jdf; ;P )  f 6)I x@c)~ 0
Two kernels, each better than the other, are called equivalent.
Proposition 13.3.6. Let xl, xa E N L 1 be such that globally. Then x1 is better than x2.
t This will also be expressed by saying that u"
(xQ(u) 1) divides (,yT(u)  xG(u))
divides f' globally.
SATURATION ON
x(w)
495
Suppose there exists h E BV such that xT(v)  x;(v) = X"(u)&~(v)  1). Then x:(v)  I = (h"(v) S"(o))&;(v)  l), and the result follows from Cor. 13.3.4.
+
Proposition 13.3.7. The kernel of FejPr is equivalent to that of CauchyPoisson, i.e., there exist constants Ml, M 2such that (13.3.5) MlIIfYf; ";1lP)  f(0)IIxm 5
IIu(f;
";P )  f ( O ) l l X ( S 5 M2IIW; 0 ; IlP)
 f(O)IIX(W)
for every f E X(R) and p > 0.
Proof. For these kernels we have (cf. Problem 5.1.2)
Now h,(o) = (p"(u)  F"(u))/(F"(v)  1) is an even function, and by an elementary calculation it can be shown that h, satisfies the assumptions of Prop. 6.3.10 and thus is the Fourier transform of an L1function. Prop. 13.3.6 then gives thatp is better than F. On the other hand, writing ha(@= (F"(v)  p"(u))/(p"(v)  I), then again h, is an even function, which by the same reasons is the Fourier transform of an L'function. Thus F is better than p , and the proof is complete. For a selection of further comparison theorems between familiar singular integrals we refer to Problem 13.3.1. 13.3.2 Local Divisibility If a" does not divide .$" globally, we cannot in general expect an inequality of type (13.3.1). However, an inequality slightly weaker than (13.3.1) which nevertheless implies (13.3.1) in many cases of interest may be drawn under the weaker hypothesis that u divides t only locally. More precisely, Definition 13.3.8. a E BV divides E BV locally, vthere exists X E BV such that T ( u ) = h"(v)o"(v) for all v of a neighbourhood of v = 0. In this case we also say that a" divides (" locally. Instead of Theorem 13.3.2 we now have Theorem 13.3.9. Let u be a realvalued?function of BV which does not vanish identically. If we suppose that a dioides 5' E BV locally, then there is b, 0 < b < 1, such that for every f~ X(R) and 7 > 0 m
(13.3.6)
I &(f;
";T ) ~ ~ x ( R5) IIhll~vllDdf; O;
T)IIx(R)
IA
2
I=0
II00cf;O; Bb'T)IIx(R),
where A , B are constants depending only on ( and a.
t If u is complexvalued, then we have e.g. to suppose that there are two intervals symmetric to the origin where u" does not vanish.
SATURATION THEORY
496
Proof. Since u real implies ~ " ( u )= u"(u) (cf. Prop. 5.3. l), we observe that in view of Prop. 5.3.2 and 5.3.11 there exist constants uo > 0 and b, 0 < b < 1, such that u"(u) # 0 on [b2uo,uo]. Let K(u) be the continuous function which equals one for I Ib2uo,zero for IuI > buO,and is piecewise linear ('trapezoid function'). Then by Prop. 6.3.10 there exists k E NL' such that K(u) = k"(u) for all u E R. Let L(u) = K(u)  K(bu). Then t ( u ) = 1"(u), where 1 E L1 and I(x) = k(x)  (l/b)k(x/b).NOW,L(u) = 0 except for u in the intervals (b200,u,) and (uo,  bauo),and on these intervals u" does not van