An Introduction to Harmonic Analysis, Third edition (Cambridge Mathematical Library)

AN INTRODUCTION TO HARMONIC ANALYSIS Yitzhak Katznelson Third Corrected Edition Preface Harmonic analysis is the st...

Author: Yitzhak Katznelson

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AN INTRODUCTION TO

HARMONIC ANALYSIS Yitzhak Katznelson

Third Corrected Edition

Preface

Harmonic analysis is the study of objects (functions, measures, etc.), defined on topological groups. The group structure enters into the study by allowing the consideration of the translates of the object under study, that is, by placing the object in a translation-invariant space. The study consists of two steps. First: finding the "elementary components" of the object, that is, objects of the same or similar class, which exhibit the simplest behavior under translation and which "belong" to the object under study (harmonic or spectral analysis); and second: finding a way in which the object can be construed as a combination of its elementary components (harmonic or spectral synthesis). The vagueness of this description is due not only to the limitation of the author but also to the vastness of its scope. In trying to make it clearer, one can proceed in various ways† ; we have chosen here to sacrifice generality for the sake of concreteness. We start with the circle group T and deal with classical Fourier series in the first five chapters, turning then to the real line in Chapter VI and coming to locally compact abelian groups, only for a brief sketch, in Chapter VII. The philosophy behind the choice of this approach is that it makes it easier for students to grasp the main ideas and gives them a large class of concrete examples which are essential for the proper understanding of the theory in the general context of topological groups. The presentation of Fourier series and integrals differs from that in [1], [7], [8], and [28] in being, I believe, more explicitly aimed at the general (locally compact abelian) case. The last chapter is an introduction to the theory of commutative Banach algebras. It is biased, studying Banach algebras mainly as a tool in harmonic analysis. This book is an expanded version of a set of lecture notes written † Hence

the indefinite article in the title of the book.

iii

IV

A N I NTRODUCTION TO H ARMONIC A NALYSIS

for a course which I taught at Stanford University during the spring and summer quarters of 1965. The course was intended for graduate students who had already had two quarters of the basic "real-variable" course. The book is on the same level: the reader is assumed to be familiar with the basic notions and facts of Lebesgue integration, the most elementary facts concerning Borel measures, some basic facts about holomorphic functions of one complex variable, and some elements of functional analysis, namely: the notions of a Banach space, continuous linear functionals, and the three key theorems—"the closed graph", the Hahn-Banach, and the "uniform boundedhess" theorems. All the prerequisites can be found in [23] and (except, for the complex variable) in [22]. Assuming these prerequisites, the book, or most of it, can be covered in a one-year course. A slower moving course or one shorter than a year may exclude some of the starred sections (or subsections). Aiming for a one-year course forced the omission not only of the more general setup (non-abelian groups are not even mentioned), but also of many concrete topics such as Fourier analysis on Rn , n > l, and finer problems of harmonic analysis in T or R (some of which can be found in [13]). Also, some important material was cut into exercises, and we urge the reader to do as many of them as he can. The bibliography consists mainly of books, and it is through the bibliographies included in these books that the reader is to become familiar with the many research papers written on harmonic analysis. Only some, more recent, papers are included in our bibliography. In general we credit authors only seldom—most often for identification purposes. With the growing mobility of mathematicians, and the happy amount of oral communication, many results develop within the mathematical folklore and when they find their way into print it is not always easy to determine who deserves the credit. When I was writing Chapter Ill of this book, I was very pleased to produce the simple elegant proof of Theorem 1.6 there. I could swear I did it myself until I remembered two days later that six months earlier, "over a cup of coffee," Lennart Carleson indicated to me this same proof. The book is divided into chapters, sections, and subsections. The chapter numbers are denoted by roman numerals and the sections and subsections, as well as the exercises, by arabic numerals. In cross references within the same chapter, the chapter number is omitted; thus Theorem llI.1.6, which is the theorem in subsection 6 of Section 1 of Chapter Ill, is referred to as Theorem 1.6 within Chapter IlI, and

P REFACE

V

Theorem Ill.1.6 elsewhere. The exercises are gathered at the end of the sections, and exercise V.1.1 is the first exercise at the end of Section 1, Chapter V. Again, the chapter number is omitted when an exercise is referred to within the same chapter. The ends of proofs are marked by a triangle (J). The book was written while I was visiting the University of Paris and Stanford University and it owes its existence to the moral and technical help 1 was so generously given in both places. During the writing I have benefitted from the advice and criticism of many friends; 1 would like to thank them all here. Particular thanks are due to L. Carleson, K. DeLeeuw, J.-P. Kahane, O.C. McGehee, and W. Rudin. I would also like to thank the publisher for the friendly cooperation in the production of this book. Y ITZHAK K ATZNELSON Jerusalem April 1968

The 2002 edition

The second edition was essentially identical with the first, except for the correction of a few misprints. The current edition has some more misprints and “miswritings” corrected, and some material added: an additional section in the first chapter, a few exercises, and an additional appendix. The added material does not reflect the progress in the field in the past thirty or forty years. Almost all of it could, and should have been included in the first edition of the book. Stanford March 2002

Contents

I

Fourier Series on T 1 Fourier coefficients . . . . . . . . . . . . . . . . 2 Summability in norm and homogeneous banach spaces on T . . . . . . . . . . . . . . . . . . . . 3 Pointwise convergence of σn (f ). . . . . . . . . . 4 The order of magnitude of Fourier coefficients . . . . . . . . . . . . . . . . 5 Fourier series of square summable functions . . 6 Absolutely convergent fourier series . . . . . . . 7 Fourier coefficients of linear functionals . . . . 8 Additional comments and applications . . . . .

. . . .

1 2

. . . . . . . .

8 17

. . . . .

. . . . .

22 27 31 35 48

II The Convergence of Fourier Series 1 Convergence in norm . . . . . . . . . . . . . . . . . . . 2 Convergence and divergence at a point . . . . . . . . . ?3 Sets of divergence . . . . . . . . . . . . . . . . . . . . .

55 55 60 64

III The Conjugate Function 1 The conjugate function . . . . . . . . . . . . . . . . . . 2 The maximal function of Hardy and Littlewood . . . . 3 The Hardy spaces . . . . . . . . . . . . . . . . . . . . .

71 71 84 93

. . . . .

. . . . .

IV Interpolation of Linear Operators 105 1 Interpolation of norms and of linear operators . . . . . 105 2 The theorem of Hausdorff-Young . . . . . . . . . . . . 111 V Lacunary Series and Quasi-analytic Classes 117 1 Lacunary series . . . . . . . . . . . . . . . . . . . . . . 117 ?2 Quasi-analytic classes . . . . . . . . . . . . . . . . . . . 125

vii

VIII


VI Fourier Transforms on the Line 1 Fourier transforms for L1 (R) . . . . . . . . . . 2 Fourier-Stieltjes transforms. . . . . . . . . . . 3 Fourier transforms in Lp (R); 1 < p ≤ 2. . . . . 4 Tempered distributions and pseudo-measures 5 Almost-Periodic functions on the line . . . . 6 The weak-star spectrum of bounded functions 7 The Paley–Wiener theorems . . . . . . . . . . ?8 The Fourier-Carleman transform . . . . . . . . 9 Kronecker’s theorem . . . . . . . . . . . . . . VII Fourier Analysis on Locally Compact Abelian Groups 1 Locally compact abelian groups . . . . . 2 The Haar measure . . . . . . . . . . . . 3 Characters and the dual group . . . . . . 4 Fourier transforms . . . . . . . . . . . . 5 Almost-periodic functions and the Bohr compactification . . . . . . . . . . . . . .

. . . .

. . . .

. . . .

. . . . . . . . .

. . . . . . . . .

. . . . . . . . .

. . . . . . . . .

. . . . . . . . .

132 133 144 155 162 170 184 188 193 196

. . . .

. . . .

. . . .

. . . .

. . . .

201 201 202 203 205

. . . . . . . . 206

VIII Commutative Banach Algebras 1 Definition, examples, and elementary properties 2 Maximal ideals and multiplicative linear functionals . . . . . . . . . . . . . . . . . 3 The maximal-ideal space and the Gelfand representation . . . . . . . . . . . . . . 4 Homomorphisms of Banach algebras . . . . . . 5 Regular algebras . . . . . . . . . . . . . . . . . . 6 Wiener’s general Tauberian theorem . . . . . . . 7 Spectral synthesis in regular algebras . . . . . . 8 Functions that operate in regular Banach algebras . . . . . . . . . . . . . . . . . . 9 The algebra M (T) and functions that operate on Fourier-Stieltjes coefficients . . . . . . . . . . . 10 The use of tensor products . . . . . . . . . . . .

209 . . . . 209

. . . . 213 . . . . .

. . . . .

. . . . .

. . . . .

220 228 236 241 244

. . . . 250 . . . . 259 . . . . 264

A Vector-Valued Functions 1 Riemann integration . . . . . . . . . . . . . . . . . . . 2 Improper integrals . . . . . . . . . . . . . . . . . . . . . 3 More general integrals . . . . . . . . . . . . . . . . . .

272 272 273 273

CONTENTS

4

IX

Holomorphic vector-valued functions . . . . . . . . . . 273

B Probabilistic Methods 1 Random series . . . . . . . . . . . . . . . . . . . . . . . 2 Fourier coefficients of continuous functions . . . . . . P 3 Paley–Zygmund, when |an |2 = ∞ . . . . . . . . .

275 275 278 279

Bibliography

283

Index

285

Symbols

Kn , 12 P(r, t), 16 Vn (t), 15 Trimλ , 277 rn , 276 e ], 3 S[f f ∗ g, 5 f ∗ g , 179

HC(D), 210 A(T), 31 Bc , 14 C(T), 14 C n (T), 14 C m+η (T), 48 Dn , 13 En (ϕ), 48 M (T), 38 Pinv , 41 S[µ], 35 S[f ], 3 Sn (µ), 36 Sn (µ, t), 37 Sn (f ), 13 Lipα (T), 16 Ω(f, h), 25 H, 28 Hf , 40 δ , 38 δτ , 38 fˆ(n), 3 χX , 153 lipα (T), 16 L1 (T), 2 L∞ (T), 16 Lp (T), 15 µf , 40 ω(f, h), 25 σn (µ), 36 σn (µ, t), 37 σn (f ), 12 σn (f, t), 12

M

fτ , 4 D, 202 R, 1 T, 1 Z, 1 ˆ , 205 D

x

AN INTRODUCTION TO

HARMONIC ANALYSIS

Chapter I

Fourier Series on T

We denote by R the additive group of real numbers and by Z the subgroup consisting of the integers. The group T is defined as the quotient R/2πZ where, as indicated by the notation, 2πZ is the group of the integral multiples of 2π . There is an obvious identification between functions on T and 2π -periodic functions on R, which allows an implicit introduction of notions such as continuity, differentiability, etc. for functions on T. The Lebesgue measure on T, also, can be defined by means of the preceding identification: a function f is integrable on T if the corresponding 2π -periodic function, which we denote again by f , is integrable on [0, 2π) and we set Z

f (t)dt =

Z

2π

f (x)dx.

0

T

In other words, we consider the interval [0, 2π) as a model for T and the Lebesgue measure dt on T is the restriction of the Lebesgue measure of R to [0, 2π). The total mass of dt on T is equal to 2π and many of our formulas would be simpler if we normalized dt to have total mass 1, that is, if we replace it by dx/2π . Taking intervals on R as "models" for T is very convenient, however, and we choose to put dt = dx in order to avoid confusion. We "pay" by having to write the factor 1/2π in front of every integral. An all-important property of dt on T is its translation invariance, that is, for all t0 ∈ T and f defined on T, Z

f (t − t0 )dt =

Z

f (t)dt†

† Throughout this chapter, integrals with unspecified limits of integration are taken over T.

1

2 1

A N I NTRODUCTION TO H ARMONIC A NALYSIS FOURIER COEFFICIENTS

1.1 We denote by L1 (T) the space of all (equivalence† classes of) complex-valued, Lebesgue integrable functions on T. For f ∈ L1 (T) we put Z 1 kf kL1 = |f (t)|dt. 2π T

It is well known that L1 (T), with the norm so defined, is a Banach space.

D EFINITION : A trigonometric polynomial on T is an expression of the form (1.1)

P ∼

N X

an eint .

n=−N

The numbers n appearing in (1.1) are called the frequencies of P ; the largest integer n such that |an | + |a−n | 6= 0 is called the degree of P . The values assumed by the index n are integers so that each of the summands in (1.1) is a function on T. Since (1.1) is a finite sum, it represents a function, which we denote again by P , defined for each t ∈ T by (1.2)

P (t) =

N X

an eint .

n=−N

Let P be defined by (1.2). Knowing the function P we can compute the coefficients an by the formula Z 1 (1.3) P (t)e−int dt an = 2π which follows immediately from the fact that for integers j, ( Z 1 if j = 0, 1 ijt e dt = 2π 0 if j 6= 0. Thus we see that the function P determines the expression (1.1) and there seems to be no point in keeping the distinction between the expression (1.1) and the function P ; we shall consider trigonometric polynomials as both formal expressions and functions. †f

∼ g if f (t) = g(t) almost everywhere

I. F OURIER S ERIES ON T

3

1.2 D EFINITION: A trigonometric series on T is an expression of the form S∼

(1.4)

∞ X

an eint .

n=−∞

Again, n assumes integral values; however, the number of terms in (1.4) may be infinite and there is no assumption whatsoever about the size of the coefficients or about convergence. The conjugate‡ of the series (1.4) is, by definition, the series S˜ ∼

∞ X

−i sgn (n)an eint .

n=−∞

where sgn (n) = 0 if n = 0 and sgn (n) = n/|n| otherwise. 1.3 Let f ∈ L1 (T). Motivated by (1.3) we define the nth Fourier coefficient of f by Z 1 ˆ (1.5) f (n) = f (t)e−int dt. 2π

D EFINITION : The Fourier series S[f ] of a function f ∈ L1 (T) is the trigonometric series ∞ X S[f ] ∼ fˆ(n)eint . −∞

e ] and referred to The series conjugate to S[f ] will be denoted by S[f as the conjugate Fourier series of f . We shall say that a trigonometric series is a Fourier series if it is the Fourier series of some f ∈ L1 (T).

1.4

We turn to some elementary properties of Fourier coefficients.

Theorem. Let f, g ∈ L1 (T), then

(a) (f\ + g)(n) = fˆ(n) + gˆ(n). (b) For any complex number α d)(n) = αfˆ(n). (αf

(c) If f¯ is the complex conjugate§ of f then fˆ¯(n) = fˆ(−n). ‡ See

Chapter III for motivation of the terminology. by: f¯(t) = f (t)) for all t ∈ T.

§ Defined

4


(d) Denote fτ (t) = f (t − τ ), τ ∈ T; then fˆτ (n) = fˆ(n)e−inτ .

(e) |fˆ(n)| ≤

1 2π

R

|f (t)|dt = kf kL1

The proofs of (a) through (e) follow immediately from (1.5) and the details are left to the reader. 1.5 Corollary. Assume fj ∈ L1 (T), j = 0, 1, . . . , and kfj −f0 kL1 → 0. Then fˆ(n) → fˆ0 (n) uniformly. 1.6 Theorem. Let f ∈ L1 (T), assume fˆ(0) = 0, and define Z t f (τ )dτ. F (t) = 0

Then F is continuous, 2π -periodic, and 1 Fˆ (n) = fˆ(n), n 6= 0. in P ROOF : The continuity (and, in fact, the absolute continuity) of F is evident. The periodicity follows from Z t+2π f (τ )dτ = 2π fˆ(0) = 0, F (t + 2π) − F (t) =

(1.6)

t

and (1.6) is obtained through integration by parts: Z 2π Z −1 2π 0 1 −int 1 1 −int ˆ F (t)e dt = F (t) F (n) = e dt = fˆ. J 2π 0 2π 0 −in in 1.7 We now define the convolution operation in L1 (T). The reader will notice the use of the group structure of T and of the invariance of dt in the subsequent proofs. Theorem. Let f, g ∈ L1 (T). For almost all t, the function f (t − τ )g(τ ) is integrable (as a function of τ on T), and, if we write Z 1 (1.7) h(t) = f (t − τ )g(τ )dτ, 2π

then h ∈ L1 (T) and (1.8)

khkL1 ≤ kf kL1 kgkL1 .

Moreover (1.9)

ˆ h(n) = fˆ(n)ˆ g (n)

for all n.


5

P ROOF : The functions f (t − τ ) and g(τ ), considered as functions of the two variables (t, x), are clearly measurable, hence so is F (t, τ ) = f (t − τ )g(τ ).

For every τ , F (t, τ ) is just a constant multiple of fτ , hence integrable dt, and Z Z Z 1 1 1 |F (t, τ )|dt dτ = |g(τ )|·kf kL1 dτ = kf kL1 kgkL1 2π 2π 2π Hence, by the theorem of Fubini, f (t−τ )g(τ ) is integrable (over (0, 2π)) as a function of τ for almost all t, and ZZ Z Z Z 1 1 1 1 |F (t, τ )|dt dτ |h(t)|dt = F (t, τ )dτ dt ≤ 2π 2π 2π 4π 2 = kf kL1 kgkL1 which establishes (1.8). In order to prove (1.9) we write ZZ Z 1 1 −int ˆ f (t − τ )e−in(t−τ ) g(τ )e−inτ dt dτ h(n) = h(t)e dt = 2π 4π 2 Z Z 1 1 = f (t)e−int dt· g(τ )e−inτ dτ = fˆ(n)ˆ g (n). 2π 2π As above the change in the order of integration is justified by Fubini’s theorem. J 1.8 D EFINITION : The convolution f ∗ g of the (L1 (T) functions) f and g is the function h defined by (1.8). Using the star notation for the convolution, we can write (1.9):

(1.10)

fd ∗ g(n) = fˆ(n)ˆ g (n).

Theorem. The convolution operation in L1 (T) is commutative, associative, and distributive (with respect to the addition).

P ROOF : The change of variable ϑ = t − τ gives Z Z 1 1 f (t − τ )g(τ )dτ = g(t − ϑ)f (ϑ)dϑ, 2π 2π that is, f ∗ g = g ∗ f.

6


If f1 , f2 , f3 ∈ L1 (T), then ZZ 1 f1 (t − u − τ )f2 (u)f3 (τ )du dτ = [(f1 ∗ f2 )∗f3 ](t) = 4π 2 ZZ 1 f1 (t − ω)f2 (ω − τ )f3 (τ )dω dτ = [f1 ∗ (f2 ∗ f3 )](t). 4π 2 Finally, the distributive law f1 ∗ (f2 + f3 ) = f1 ∗ f2 + f1 ∗ f3

is evident from (1.7).

J

1.9 Lemma. Assume f ∈ L1 (T) and let ϕ(t) = eint for some integer n. Then (ϕ ∗ f )(t) = fˆ(n)eint .

P ROOF : 1 (ϕ ∗ f )(t) = 2π

Z

in(t−τ )

e

int

f (τ )dτ = e

PN

Corollary. If f ∈ L1 (T) and k(t) =

(1.11)

(k ∗ f )(t) =

1 2π

−N

N X

Z

1 2π

Z

f (τ )e−inτ dτ.

J

an eint , then

an fˆ(n)eint .

−N

EXERCISES FOR SECTION 1 1. Compute the Fourier coefficients of the following functions (defined by their values on [−π, π): (a)

(b)

f (t) =

∆(t) =

(√

2π

0

(

|t| < 1 2

1 2

≤ |t| ≤ π .

1 − |t|

|t| < 1

0

1 ≤ |t| ≤ π .

What relation do you see between f and ∆ ?

(c)

g(t) =

  1

−1 < t ≤ 0

−1

0 0. Let g be a continuous function on T such that kg − f kL1 < ε/2; thus kfτ − fτ0 kL1 ≤ kfτ − gτ kL1 + kgτ − gτ0 kL1 + kgτ0 − fτ0 kL1 = = k(f − g)τ kL1 + kgτ − gτ0 kL1 + k(g − f )τ0 kL1 ≤ ε + kgτ − gτ0 kL1 .

Hence limkfτ − fτ0 kL1 < ε and, ε being an arbitrary positive number, (H-2’) is established. 2.2 D EFINITION : A summability kernel is a sequence {kn } of continuous 2π -periodic functions satisfying: Z 1 (S-1) kn (t)dt = 1 2π

(S-2) (S-3)

1 2π

Z

|kn (t)|dt ≤ const

For all 0 < δ < π , lim

n→∞

2π−δ

Z

|kn (t)|dt = 0 δ

10


A positive summability kernel is one such that kn (t) ≥ 0 for all t and n. For positive kernels the assumption (S-2) is clearly redundant. We consider also families kr depending on a continuous parameter r instead of the discrete n. Thus the Poisson kernel P(r, t), which we shall define at the end of this section, is defined for 0 ≤ r < 1 and we replace in (S-3), as well as in the applications, the limit “limn→∞ ” by “limr→1 ”. The following lemma is stated in terms of vector-valued integrals. We refer to Appendix A for the definition and relevant properties. Lemma. Let B be a Banach space, ϕ a continuous B -valued function on T, and {kn } a summability kernel. Then: Z 1 kn (τ )ϕ(τ )dτ = ϕ(0). lim n→∞ 2π

P ROOF : By (S-l) we have, for 0 < δ < π , Z Z 1 1 kn (τ )ϕ(τ )dτ − ϕ(0) = kn (τ )(ϕ(τ ) − ϕ(0))dτ 2π 2π ! Z δ Z 2π−δ (2.2) 1 + kn (τ )(ϕ(τ ) − ϕ(0))dτ. = 2π δ −δ Now

1 Z δ

(2.3) kn (τ )(ϕ(τ ) − ϕ(0))dτ ≤ max kϕ(τ ) − ϕ(0)kB kkn kL1

2π −δ |τ |≤δ B

and

(2.4)

1 Z 2π−δ

kn (τ )(ϕ(τ ) − ϕ(0))dτ ≤

2π δ B Z 2π−δ

1 |kn (τ )|dτ. ≤ max ϕ(τ ) − ϕ(0) B 2π δ

By (S-2) and the continuity of ϕ(τ ) at τ = 0, given ε > 0 we can find δ > 0 so that (2.3) is bounded by ε, and keeping this δ , it results from (S-3) that (2.4) tends to zero as n → ∞ so that (2.2) is bounded by 2ε. J 2.3 For f ∈ L1 (T) we put ϕ(τ ) = fτ (t) = f (t − τ ). By (H-1’) and (H-2’), ϕ is a continuous L1 (T)-valued function on T and ϕ(0) = f . Applying lemma 2.2 we obtain


11

Theorem. Let f ∈ L1 (T) and {kn } be a summability kernel; then Z 1 (2.5) f = lim kn (τ )fτ dτ n→∞ 2π

in the L1 (T) norm. 2.4 The integrals in (2.5) have the formal appearance of a convolution although the operation involved, that is, vector integration, is different from the convolution as defined in section 1.7. The ambiguity, however, is harmless. Lemma. Let k be a continuous function on T and f ∈ L1 (T). Then Z 1 (2.6) k(τ )fτ dτ = k ∗ f. 2π

P ROOF : Assume first that f is continuous on T. We have, Appendix A, Z X 1 1 k(τ )fτ dτ = lim (τj+1 − τj )k(τj )fτj , 2π 2π j the limit being taken in the L1 (T) norm as the subdivision {τj } of [0, 2π) becomes finer and finer. On the other hand, X 1 lim (τj+1 − τj )k(τj )f (t − τj ) = (k ∗ f )(t) 2π j

uniformly and the lemma is proved for continuous f . For arbitrary f ∈ L1 (T), let ε > 0 be arbitrary and let g be a continuous function on T such that kf − gkL1 < ε. Then, since (2.6) is valid for g , Z Z 1 1 k(τ )fτ dτ − k ∗ f = k(τ )(f − g)τ dτ + k ∗ (g − f ) 2π 2π and consequently

1 Z

k(τ )fτ dτ − k ∗ f ≤ 2kkkL1 ε.

2π L1

Using lemma 2.4 we can rewrite (2.5): (2.5’)

f = lim kn ∗ f n→∞

in the L1 (T) norm.

J

12


2.5 One of the most useful summability kernels, and probably the best known, is Fejér’s kernel (which we denote by {Kn }) defined by

(2.7)

Kn (t) =

n X

1−

j=−n

|j| ijt e . n+1

The fact that Kn satisfies (S-1) is obvious from (2.7); that Kn (t) ≥ 0 and that (S-3) is satisfied is clear from Lemma. 1 Kn (t) = n+1

sin n+1 2 sin 12 t

2

.

P ROOF : Recall that (2.8)

sin2

t 1 1 1 1 = (1 − cos t) = − e−it + − eit . 2 2 4 2 4

A direct computation of the coefficients in the product shows that X n 1 1 1 |j| ijt − e−it + − eit e = 1− 4 2 4 n+1 j=−n 1 −i(n+1)t 1 1 i(n+1)t 1 = − e + − e . n+1 4 2 4

J

We adhere to the generally used notation and write σn (f ) = Kn ∗ f and σn (f, t) = (Kn ∗ f )(t). It follows from corollary 1.9 that (2.9)

σn (f, t) =

n X −n

1−

|j| ˆ f (j)eijt . n+1

2.6 The fact that σn (f ) → f in the L1 (T) norm for every f ∈ L1 (T), which is a special case of (2.5’), and the fact that σn (f ) is a trigonometric polynomial imply that trigonometric polynomials are dense in L1 (T). Other immediate consequences are the following two important theorems. 2.7 Theorem (The Uniqueness Theorem). Let f ∈ L1 (T) and assume that fˆ(n) = 0 for all n. Then f = 0.

P ROOF : By (2.9) σn (f ) = 0 for all n. Since σn (f ) → f , it follows that f = 0. J


13

An equivalent form of the uniqueness theorem is: Let f, g ∈ L1 (T) and assume fˆ(n) = gˆ(n) for all n, then f = g . 2.8 Theorem (The Riemann-Lebesgue Lemma). Let f ∈ L1 (T), then lim fˆ(n) = 0. |n|→∞

P ROOF : Let ε > 0 and let P be a trigonometric polynomial on T such that kf − P kL1 < ε. If |n| > degree of P , then |fˆ(n)| = |(f\ − P )(n)| ≤ kf − P kL1 < ε.

J

Remark: If K is a compact set in L1 (T) and ε > 0, there exist a finite number of trigonometric polynomials P1 . . . , PN such that for every f ∈ K there exists a j , 1 ≤ j ≤ N , such that kf − Pj kL1 < ε. If |n| is greater than max1≤j≤N (degree of Pj ) then |fˆ(n)| < ε for all f ∈ K . Thus, the Riemann-Lebesgue lemma holds uniformly on compact subsets of L1 (T). 2.9 is,

(2.10)

For ∈ L1 (T) we denote by Sn (f ) the nth partial sum of S[f ], that

(Sn (f ))(t) = Sn (f, t) =

n X

fˆ(j)eijt .

−n

If we compare (2.9) and (2.10) we see that (2.11)

σn (f ) =

1 (S0 (f ) + S1 (f ) + · · · + Sn (f )), n+1

in other words, the σn (f ) are the arithmetic means† of Sn (f ). It follows that if Sn (f ) converge in L1 (T) as n → ∞, then the limit is necessarily f. From corollary 1.9 it follows that Sn (f ) = Dn ∗ f where Dn is the Dirichlet kernel defined by (2.12)

Dn (t) =

n X −n

eijt =

sin(n + 12 )t . sin 21 t

† Often referred to as the Cesàro means or, especially in Fourier Analysis, as the Fejér means.

14


It is important to notice that {Dn } is not a summability kernel in our sense. It does satisfy condition (S-1); however, it does not satisfy either (S-2) or (S-3). This explains why the problem of convergence for Fourier series is so much harder than the problem of summability. We shall discuss convergence in chapter II. 2.10 D EFINITION : A homogeneous Banach space on T is a linear subspace B of L1 (T) having a norm k kB ≥ k kL1 under which it is a Banach space, and having the following properties:

(H-1) If f ∈ B and τ ∈ T, then fτ ∈ B and kfτ kB = kf kB (where fτ (t) = f (t − τ )). (H-2) For all f ∈ B,

τ, τ0 ∈ T,

limτ →τ0 kfτ − fτ0 k = 0.

Remarks: Condition (H-1) is referred to as translation invariance and (H-2) as continuity of the translation. We could simplify (H-2) somewhat by requiring continuity at one specific τ0 ∈ T, say τ0 = 0 rather than at every τ ∈ T, since by (H-1) kfτ − fτ0 kB = kfτ −τ0 − f kB

Also, the method of the proof of (H-2’) (see 2.1) shows that if we have a space B satisfying (H-1) and we want to show that it satisfies (H-2) as well, it is sufficient to check the continuity of the translation on a dense subset of B. An almost equivalent statement is Lemma. Let B ⊂ L1 (T) be a Banach space satisfying (H-1). Denote by Bc the set of all f ∈ B such that τ 7→ fτ is a continuous B -valued function. Then Bc is a closed subspace of B .

Examples of homogeneous Banach spaces on T . (a) C(T)–the space of all continuous 2π -periodic functions with the norm (2.13)

kf k∞ = max|f (t)| t

(b) C n (T)–the subspace of C(T) of all n-times continuously differentiable functions (n being a rational integer) with the norm (2.13’)

kf kC n =

n X 1 max|f (j) (t)| t j! j=0


15

(c) Lp (T), 1 ≤ p n0 implies (3.6)

supϑ 0 and n > n(ε) we have by (3.8) Φ(τ ) < ετ

in

0 < τ < ϑ = n−1/4

hence (3.11) is bounded by πεn 2πε + n+1 n+1

Z

ϑ

1/n

dτ < 3πε. τ2

J


21

Corollary. If the Fourier series of f ∈ L1 (T) converges on a set E of positive measure, its sum coincides with f almost everywhere on E . In particular, if a Fourier series converges to zero almost everywhere, all its coefficients must vanish.

Remark: This last result is not true for all trigonometric series. There are examples of trigonometric series converging to zero almost everywhere† without being identically zero. 3.3 The need to impose in Theorem 3.2 the strict condition (3.8) rather than the weaker condition Z h f (t0 + τ ) + f (t0 − τ ) (3.8’) Ψ(h) = − fˇ(t0 ) dτ = o(h) 2 0

comes from the fact that in order to carry the integration by parts we π2 have to replace Kn (t) by the monotonic majorant min n + 1, (n+1)τ . 2 If we want to prove the analogous result for P(r, t) rather that Kn (t), the condition (3.8’) is sufficient. Thus we obtain: Theorem (Fatou). If (3.8’) holds, then lim

r→1

∞ X

fˆ(j)r|j| eijt0 = fˇ(t0 ).

−∞

The condition (3.8’) with fˇ(t0 ) = f (t0 ) is satisfied at every point t0 where f is the derivative of its integral (hence almost everywhere). EXERCISES FOR SECTION 3 1. Let 0 < α < 1 and let f ∈ L1 (T). Assume that at the point t0 ∈ T, f satisfies a Lipschitz condition of order α, that is, |f (t0 + τ ) − f (t0 )| < K|τ |α for |τ | < π Prove that for α < 1 |σn (f, t0 ) − f (t0 )| ≤

π+1 Kn−α 1−α

while for α = 1 |σn (f, t0 ) − f (t0 )| ≤ 2πK

Hint: Use (3.10) and (3.4) with ϑ =

log n . n

1 n

† However, a trigonometric series converging to zero everywhere is identically zero (see [13], Chapter 5).

22

A N I NTRODUCTION TO H ARMONIC A NALYSIS If f ∈ Lipα (T), 0 < α ≤ 1, then kσn (f ) − f k∞ ≤

(

const kf kLipα n−α

const

kf kLip1 logn n

when 0 < α < 1, for α = 1.

3. Let f ∈ L∞ (T) and assume |fˆ(n)| < K|n|−1 . Prove that for all n and t, |Sn (f, t)| ≤ kf k∞ + 2K . Hint: n X |j| ˆ Sn (f, t) = σn (f, t) + f (j)eijt . n+1 −n

Pn

−1

4. Show that for all n and t, | 1 j sin jt| ≤ 12 π + 1. Hint: Consider f (t) = t/2 in [0, 2π). 5. Jackson’s kernel is Jn (t) = kKn k−2 K2 (t). Verify L2 n a. {Jn } is a positive summability kernel. b. For −π < t < π , Jn (t) < 2π 4 n−3 t−4 . c. If f ∈ Lip1 (T), then kJn ∗ f − f k∞ ≤ const kf kLip1 n−1 . Compare this to the corresponding estimate for kKn ∗ f − f k∞ in exercise 2 above. 4

THE ORDER OF MAGNITUDE OF FOURIER COEFFICIENTS

The only things we know so far about the size of Fourier coefficients ˆ {f (n)} of a function f ∈ L1 (T) is that they are bounded by kf kL1 , (1.4(e)) and that lim|n|→∞ fˆ(n) = 0 (the Riemann-Lebesgue lemma). In this section we discuss the following three questions: (a) Can the Riemann-Lebesgue lemma be improved to provide a certain rate of vanishing of fˆ(n) as |n| → ∞? We show that the answer to (a) is negative; fˆ(n) can go to zero arbitrarily slowly (see 4.1). (b) In view of the negative answer to (a), is it true that any sequence {an } which tends to zero as |n| → ∞ is the sequence of Fourier coefficients of some f ∈ L1 (T)? The answer to (b) is again negative (see 4.2). (c) How are properties like boundedhess, continuity, smoothness, etc. of a function f reflected by {fˆ(n)}? Question (c), in one form or another, is a recurrent topic in harmonic analysis. In the second half of this section we show how various smoothness conditions affect the size of the Fourier coefficients. “Order of magnitude” conditions on the Fourier coefficients are seldom necessary and sufficient for the function to belong to a given


23

function space. For example, a necessary condition for f ∈ C(T) is P ˆ P |f (n)|2 < ∞, a sufficient condition is |fˆ(n)| < ∞; in both cases the exponents are best possible. The only spaces, defined by conditions of size or smoothness of the functions, for which we obtain (in the following section) complete characterization, that is, a necessary and sufficient condition expressed in terms of order of magnitude, for a sequence {an } to be the Fourier coefficients of a function in the space, are L2 (T) and its “derivatives”† . 4.1 Theorem. Let {an }∞ n=−∞ be an even sequence of nonnegative numbers tending to zero at infinity. Assume that for n > 0 an−1 + an+1 − 2an ≥ 0.

(4.1)

Then there exists a nonnegative function f ∈ L1 (T) such that fˆ(n) = an . P P ROOF : We remark first that (an − an+1 ) = a0 and that the convexity condition (4.1) implies that (an − an+1 ) is monotonically decreasing with n, hence lim n(an − an+1 ) = 0, n→∞

and consequently N X

n(an−1 + an+1 − 2an ) = a0 − aN − N (aN − aN +1 )

n=1

converges to a0 as N → ∞. Put (4.2)

f (t) =

∞ X

n(an−1 + an+1 − 2an )Kn−1 (t),

n=1

where Kn denotes, as usual, the Fejér kernel. Since kKn kL1 = 1, the series (4.2) converges in L1 (T) and, all its terms being nonnegative, its limit f is nonnegative. Now fˆ(j) =

∞ X

ˆ n−1 (j) = n(an−1 + an+1 − 2an )K

n=1

=

∞ X

n=|j|+1

|j| = a|j| , n(an−1 + an+1 − 2an ) 1 − n

and the proof is complete. † Such

as the space of absolutely continuous functions with derivatives in L2 (T).

J

24


4.2 Comparing theorem 4.1 to our next theorem shows the basic difference between sine-series (a−n = −an ) and cosine-series (a−n = an ). Theorem. Let f ∈ L1 (T) and assume that fˆ(|n|) = −fˆ(−|n|) ≥ 0. Then X1 fˆ(n) < ∞. n n>0

P ROOF :R Without loss of generality we may assume that fˆ(0) = 0. Write t F (t) = 0 f (τ )dτ ; then F ∈ C(T) and, by theorem 1.6, 1 Fˆ (n) = fˆ(n), in

n 6= 0.

Since F is continuous, we can apply Fejér’s theorem for t0 = 0 and obtain (4.3)

lim 2

N →∞

N X

n=1

1−

n fˆ(n) = i F (0) − Fˆ (0) = −iFˆ (0), N +1 n

fˆ(n) J n ≥ 0, the theorem follows. P P Corollary. If an > 0, an /n = ∞, then an sin nt is not a Fourier series. Hence there exist trigonometric series with coefficients tending to zero which are not Fourier series.

and since

By Theorem 4.1, the series ∞ X X eint cos nt = log n 2 log|n| n=2 |n|≥2

is a Fourier series while, by theorem 4.2, its conjugate series ∞ X X sgn (n) sin nt = −i eint log n 2 log|n| n=2 |n|≥2

is not. 4.3 We turn now to some simple results about the order of magnitude of Fourier coefficients of functions satisfying various smoothness conditions. Theorem. If f ∈ L1 (T) is absolutely continuous, then fˆ(n) = o(1/n).


25

P ROOF : By theorem 1.6 we have fˆ(n) = (1/in)fb0 (n) and (the RiemannLebesgue lemma) fb0 (n) → 0. J Remark: By repeated application of Theorem 1.6 (i.e., by repeated integration by parts) we see that if f is k-times differentiable and f (k−1) is absolutely continuous‡ then (4.4) as |n| → ∞. fˆ(n) = o n−k 4.4 We can obtain a somewhat more precise estimate than the asymptotic (4.4). All that we have to do is notice that if 0 ≤ j ≤ k, then (j) (n) and hence fˆ(n) = (in)−j fd |fˆ(n)| ≤ |n|−j kf (j) kL1 .

(4.5)

We thus obtain Theorem. If f is k -times differentiable, and f (k−1) is absolutely continuous, then kf (j) kL1 . |fˆ(n)| ≤ min 0≤j≤k |n|j If f is infinitely differentiable, then |fˆ(n)| ≤ min 0≤j

kf (j) kL1 . |n|j

4.5 Theorem. lf f is of bounded variation on T, then |fˆ(n)| ≤

var(f ) . 2π|n|

P ROOF : We integrate by parts using Stieltjes integrals Z Z var(f ) 1 1 |fˆ(n)| = e−int f (t)dt = e−int df (t) ≤ . 2π 2πin 2π|n|

J

4.6 For f ∈ C(T) we denote by ω(f, h) the modulus of continuity of f , that is, ω(f, h) = sup|y|≤h kf (t + y) − f (t)k∞ .

For f ∈ L1 (T) we denote by Ω(f, h) the integral modulus of continuity of f , that is, (4.6)

Ω(f, h) = kf (t + h) − f (t)kL1 .

We clearly have Ω(f, h) ≤ ω(f, h). ‡ So

that f (k) ∈ L1 (T) and f (k−1) is its primitive,

26


π ). Theorem. For n 6= 0, |fˆ(n)| ≤ 12 Ω(f, |n|

P ROOF : fˆ(n) = of variable,

1 2π

R

−1 2π

f (t)e−int dt =

1 fˆ(n) = 4π

Z

f (t +

R

f (t)e−in(t+π/n) dt; by a change

π ) − f (t) e−int dt, n

hence |fˆ(n)| ≤

1 π Ω(f, ) 2 |n|

J

Corollary. lf f ∈ Lipα (T), then fˆ(n) = O (n−α ). 4.7 Theorem. Let 1 < p ≤ 2 and let q be the conjugate exponent, i.e., P p q = p−1 . If f ∈ Lp (T) then |fˆ(n)|q < ∞.

The case p = 2 will be proved in the following section. The case 1 2. Thus, if f ∈ Lp (T) P with p > 2, then f ∈ L2 (T) and consequently |fˆ(n)|2 < ∞. This is all that we can assert even for continuous functions. There exist P continuous functions f such that |fˆ(n)|2−ε = ∞ for all ε > 0, see IV.2. In fact, given any {cn } ∈ `2 , there exists a continuous function f such that |fˆ(n)| > |cn |, see Appendix B.2.1.

EXERCISES FOR SECTION 4 I. Given a sequence {ωn } of positive numbers such that ωn → 0 as |n| → ∞, show that there exists a sequence {an } satisfying the conditions of theorem 4.1 and an > ω n for all n. 2. Show that if |fˆ(n)||n|l < ∞, then f is l-times continuously differentiable. Hence, if fˆ(n) = O |n|−k where k > 2, and if

P

l=

(

k−2

k integer

[k] − 1

otherwise

then f is l-times continuously differentiable.


27

Remarks: Properly speaking the elements of L1 (T) are equivalence classes of functions any two of which differ only on a set of measure zero. Saying that a function f ∈ L1 (T) is continuous or differentiable etc. is a convenient and innocuous abuse of language with obvious meaning. Exercise 2 is all that we can state as a converse to theorem 4.4 if we look for continuous derivatives. It can be improved if we allow square summable derivatives (see exercise 5.5). 3. A function f is analytic on T if in a neighborhood of every t0 ∈ T, f (t) P∞ can be represented by a power series (of the form n=0 an (t − t0 )n ). Show that f is analytic if, and only if, f is infinitely differentiable on T and there exists a number R such that supt |f (n) (t)| ≤ n! Rn ,

n > 0.

4. Show that f is analytic on T if, and only if, there exist constants K > 0 and a > 0 such |fˆ(j)| ≤ Ke−a|j| . Hence show that f is analytic on T if, P ˆthat ijz and only if, f (j)e converges for |=(z)| < a for some a > 0. 5. Let f be analytic on T and let g(eit ) = f (t). What is the relation between the Laurent expansion of g about 0 (which converges in an annulus containing the circle |z| = 1) and the Fourier series of f ? 6. Let f be infinitely differentiable on T and assume that for some α > 0, and all n ≥ 0, supt |f (n) (t)| < Knαn . Show that α |fˆ(j)| ≤ K exp − |j|1/α . e

7. Assume |fˆ(j)| ≤ K exp −|j|1/α . Show that f is infinitely differentiable and

|f (n) (t)| ≤ K1 ean nan

for some constants a and P Kn1 . (n) Hint: |f (t)| ≤ 2K |j| exp(−|j|1/α ). Compare this last sum to the integral R∞ n 1/α x exp(−x )dx and change the variable of integration putting y = x1/α . 0 P∞ cos 3n t 8. Prove that if 0 < α < 1, then f (t) = belongs to Lipα (T); 3nα 1 hence corollary 4.6 cannot P be improved. sin nt 9. Show that the series ∞ converges for all t ∈ T. n=2 log n 5

FOURIER SERIES OF SQUARE SUMMABLE FUNCTIONS

In some respects the greatest success in representing functions by means of their Fourier series happens for square summable functions. The reason is that L2 (T) is a Hilbert space, its inner product being defined by Z 1 (5.1) f (t)¯ g (t)dt, hf, gi = 2π

28


and in this Hilbert space the exponentials form a complete orthogonal system. We start this section with a brief review of the basic properties of orthonormal and complete systems in abstract Hilbert space and conclude with the corresponding statements about Fourier series in L1 (T). 5.1 Let H be a complex Hilbert space. Let f, g ∈ H. We say that f is orthogonal to g if hf, gi = 0. This relation is clearly symmetric. If E is a subset of H we say that f ∈ H is orthogonal to E if f is orthogonal to every element of E . A set E ⊂ H is orthogonal if any two vectors in E are orthogonal to each other. A set E ⊂ H is an orthonormal system if it is orthogonal and the norm of each vector in E is one, that is, if, whenever f, g ∈ E , hf, gi = 0 if f 6= g and hf, f i = 1. Lemma. Let {ϕn }N n=1 be a finite orthonormal system. Let a1 , . . . , aN be complex numbers. Then N N

X

X

an ϕ n = |an |2 . 1

1

P ROOF : N N N N N X X X X

X

an ϕ n = h an ϕ n , an ϕ n i = an hϕn , am ϕ m i 1

1

1

=

1

X

an a ¯n =

1

X

|an |

2

J

Corollary. Let {ϕn }∞ in H and let {an }∞ 1 be an orthonormal system 1 P be a sequence of complex numbers such that |an |2 < ∞. Then P∞ n=1 an ϕn converges in H.

P ROOF : Since H is complete, all that we have to show is that the partial P sums SN = N 1 an ϕn form a Cauchy sequence in H. Now, for N > M , N N X

X

2 kSN − SM k2 = an ϕ n = |an |2 → 0 as M → ∞. M +1

M +1

J

5.2 Lemma. Let H be a Hilbert space. Let {ϕn } be a finite orthonormal system in H. For f ∈ H write an = hf, ϕn i. Then

(5.2)

N N X X

2

0≤ f− an ϕn = kf k2 − |an |2 . 1

1


29

P ROOF : N N N X X X

2

f − an ϕn = hf − an ϕ n , f − an ϕ n i = 1

= kf k2 −

1

N X 1

a ¯n hf, ϕn i −

1

N X 1

an hϕn , f i +

N X 1

|an |2 = kf k2 −

N X

|an |2 .

1

J

Corollary (Bessel’s inequality). Let H be a Hilbert space and {ϕα } an orthonormal system in H. For f ∈ H write aα = hf, ϕα i Then X (5.3) |aα |2 ≤ kf k2 .

The family {ϕα } in the statement of Bessel’s inequality need not be finite nor even countable. The inequality (5.3) is equivalent to saying that for every finite subset of {ϕα } we have (5.2). In particular aα = 0 P except for countably many values of α and the series |aα |2 converges. If H = L2 (T) all orthonormal systems in H are finite or countable (cf. exercise 2 at the end of this section) and we write them as sequences {ϕn }. 5.3 D EFINITION : A complete orthonormal system in H is an orthonormal system having the additional property that the only vector in H orthogonal to it is the zero vector. Lemma. Let {ϕn } be an orthonormal system in H. Then the following statements are equivalent: (a) {ϕn } is complete. (b) For every f ∈ H we have X (5.4) kf k2 = |hf, ϕn i|2 . P (c) f = hf, ϕn iϕn .

P ROOF : The equivalence of (b) and (c) follows immediately from (5.2). If f is orthogonal to {ϕn } and if (5.4) is valid, then kf k2 = 0, hence f = 0. Thus (b) ⇒ (a). We complete the proof by showing (a) ⇒ (c). P From Bessel’s inequality and corollary 5.1 it follows that hf, ϕn iϕn P converges in H. If we denote g = hf, ϕn iϕn we have hg, ϕn i = hf, ϕn i or, equivalently, g − f is orthogonal to {ϕn }. Thus if {ϕn } is complete f = g. J

30


5.4 Lemma (Parseval). Let {ϕn } be a complete orthonormal system in H. Let f, g ∈ H. Then hf, gi =

(5.5)

∞ X

hf, ϕn ihϕn , gi .

n=1

P ROOF : If f is a finite linear combination of {ϕn }, (5.5) is obvious. In the general case hf, gi = lim h N →∞

N X

hf, ϕn iϕn , gi = lim

N →∞

n=1

N X

hf, ϕn ihϕn , gi .

n=1

J

5.5 For H = L2 (T) the exponentials {eint }∞ n=−∞ form a complete orthonormal system. The orthonormality is evident: Z 1 int imt he , e i = ei(n−m)t dt = δn,m . 2π

The completeness is somewhat less evident; it follows from Theorem 2.7 since Z 1 int hf, e i = f (t)eint dt = fˆ(n) . 2π The general results about complete orthonormal systems in Hilbert space now yield Theorem. Let f ∈ L2 (T). Then

(a)

X

(b)

f = lim

|fˆ(n)|2 =

N →∞

N X

1 2π

fˆ(n)eint

Z

|f (t)|2 dt

in the L2 (T) norm.

−N

(c) For any square summable sequence {an }n∈Z of complex numbers, P that is such that |an |2 < ∞, there exists a unique f ∈ L2 (T) such that an = fˆ(n). (d) Let f, g ∈ L2 (T). Then 1 2π

Z

f (t)g(t)dt =

∞ X

n=−∞

fˆ(n)ˆ g (n) .


31

We denote by `2 the space of all square summable sequences {an }∞ −∞ , P (that is, such that |an |2 < ∞). With pointwise addition and scalar 1 P multiplication, and with the norm |an |2 2 and the inner product P∞ h{an }, {bn }i = −∞ an¯bn , `2 is a Hilbert space. Theorem 5.5 amounts to the statement that the correspondence f 7→ {fˆ(n)} is an isometry between L2 (T) and `2 . EXERCISES FOR SECTION 5 1. Let {ϕn }N n=1 be an orthogonal system in a Hilbert space H. Let f ∈ H. Show that

min f −

a1 ,...,aN

N X 1

aj ϕ j

is attained at the point aj = hf, ϕj i, j = 1, . . . , N , and only there. 2. A Hilbert space H is separable if it contains a dense countable subset. Show that an orthonormal system in a separable Hilbert space is either finite or countable. √ Hint: The distance between two orthogonal vectors of norm 1 is 2. 3. Prove that an orthonormal system {ϕn } in H is complete if, and only if, the set of finite linear combinations of {ϕn } is dense in H. 4. Let f be absolutely continuous on T and assume f 0 ∈ L2 (T); prove that

X

|fˆ(n)| ≤ kf kL1

v u ∞ u X n−2 kf 0 kL2 . + t2 1

Hint: |fˆ(0)| ≤ kf kL1 , and |nfˆ(n)|2 = kf 0 k2L2 ; apply the Cauchy-Schwarz inequality to the last identity, 5. Assume f ∈ L1 (T) and fˆ(n) = O(|n|−k ). Show that f is m-times differentiable with f (m) ∈ L2 (T) provided k − m > 21 .

P

6

ABSOLUTELY CONVERGENT FOURIER SERIES

We shall study absolutely convergent Fourier series in some detail later on: here we mention only some elementary facts. 6.1 We denote by A(T) the space of (continuous) functions on T having an absolutely convergent Fourier series, that is, the functions f for P 1 ˆ ˆ which ∞ −∞ |f (n)| < ∞. The mapping f 7→ {f (n)}n∈Z of A(T) into ` (the Banach space of absolutely convergent sequences) is clearly linP P ear and one-to-one. If ∞ an eint converges −∞ |an | < ∞ the series

32


uniformly on T and, denoting its sum by g , we have an = gˆ(n). It follows that the mapping above is an isomorphism of A(T) onto `1 . We introduce a norm to A(T) by kf kA(T) =

(6.1)

∞ X

|fˆ(n)| .

−∞

With this norm A(T) is a Banach space isometric to `1 ; we now claim it is an algebra. Lemma. Assume that f, g ∈ A(T). Then f g ∈ A(T) and kf gkA(T) ≤ kf kA(T) kgkA(T) . Pˆ P P ROOF : We have f (t) = f (n)eint , g(t) = gˆ(n)eint and since both series converge absolutely: XX f (t)g(t) = fˆ(k)ˆ g (m)ei(k+m)t . k

m

Collecting the terms for which k + m = n we obtain XX f (t)g(t) = fˆ(k)ˆ g (n − k)eint n

k

P so that fcg(n) = k fˆ(k)ˆ g (n − k); hence X XX X X |fcg(n)| ≤ |fˆ(k)||ˆ g (n − k)| = |fˆ(k)| |ˆ g (n)| . n

J

k

6.2 Not every continuous function on T has an absolutely convergent Fourier series, and those that have cannot† be characterized by smoothness conditions (see exercise 5 of this section). Some smoothness conditions are sufficient, however, to imply the absolute convergence of the Fourier series. Theorem. Let f be absolutely continuous on T and f 0 ∈ L2 (T). Then f ∈ A(T) and

(6.2)

∞ X 21 kf kA(T) ≤ kf kL1 + 2 n−2 kf 0 kL2 . 1

P ROOF : This is exercise 4 of the previous section and the hint given there is essentially the whole proof. J † See,

however, exercise 7.8.


33

?6.3 We refer to exercise 2.2 for the definitions of Lipα (T) and of its norm.

Theorem (Bernstein). If f ∈ Lipα (T) for some α > 12 , then f ∈ A(T) and kf kA(T) ≤ cα kf kLipα

(6.3)

where the constant cα depends only on α.

P ROOF : f (t − h) − f (t) ∼

X

(e−inh − 1)fˆ(n)eint .

if take h = 2π/(3·2m ) and 2m ≤ n ≤ 2m+1 we have |e−inh − 1| ≥ and consequently X X |fˆ(n)|2 ≤ |e−inh − 1|2 |fˆ(n)|2 = kfh − f k2L2 ≤ (6.4)

2m ≤n 0. Then f ∈ A(T).

We refer to [28], Vol. 1, p. 241, for the proof.

34


?6.5 Remark: There is a change of scene in this section compared with the rest of the chapter. We no longer talk about functions summable on T and their Fourier series–we discuss functions summable on Z (i.e., absolutely convergent sequences) and their "Fourier transforms" which happen to be continuous functions on T. Lemma 6.1, for instance, is completely analogous to theorem 1.7 with the roles of T and Z reversed.

EXERCISES FOR SECTION 6 1. For n = 1, 2, . . . let fn ∈ A(T) and kfn kA(T) ≤ 1. Assume that fn converge to f uniformly on T. Show that f ∈ A(T) and kf k ≤ 1. 2. Show that the conditions in exercise 1 do not imply limkf −fn kA(T) = 0; however, if we add the assumption that kf kA(T) = limn→∞ kfn kA(T) then we do have kf − fn kA(T) → 0. 3. For 0 < a < π define ∆a (t) =

(

1 − a−1 |t|

for |t| ≤ a

0

for a ≤ |t| ≤ π

Show that ∆a ∈ A(T) and k∆a kA(T) = 1. Hint: ∆â (n) ≥ 0 for all n. 4. Let f ∈ C(T) be even on (−π, π), decreasing on [0, π] and convex there (i.e., f (t + 2h) + f (t) > 2f (t + h) for 0 ≤ t ≤ t + 2h ≤ π ). Show that f ∈ A(T) and, if f ≥ 0, kf kA(T) = f (0). Hint: f can be approximated uniformly by positive combinations of ∆a . Compare with theorem 4.1. 5. Let ϕ be a "modulus of continuity," that is, an increasing concave function on [0, 1] with ϕ(0) = 0. Show P that if the sequence of integers {λn } increases fast enough and if f (t) = n−2 eiλn t , then ω(f, h) 6= O (ϕ(h)) as h → 0. ω(f, h) is the modulus of continuity of f (defined in 4.6). 6. (Rudin, Shapiro.) We define the trigonometric polynomials Pm and Qm inductively as follows: P0 = Q0 = 1 and m

Pm+1 (t) = Pm (t) + ei2

t

i2m t

Qm+1 (t) = Pm (t) − e

tQm (t) tQm (t).

(a) Show that |Pm+1 (t)|2 + |Qm+1 (t)|2 = 2 |Pm (t)|2 + |Qm (t)|2

hence and

|Pm (t)|2 + |Qm (t)|2 = 2m+1

kPm kC(T) ≤ 2(m+1)/2 . (b) For |n| < 2m , Pˆm+1 (n) = Pˆm (n), hence thereP exists a sequence {εn }∞ n=0 2m −1 such that εn is either 1 or -1 and such that Pm (t) = 0 εn eint .

I. F OURIER S ERIES ON T m−1

(c) Write fm = Pm − Pm+1 = ei2 that f ∈ Lip 1 (T) and f 6∈ A(T).

t

Qm−1 and f =

35

P∞ 1

2−m fm . Show

2

Hint: For 2−k ≤ h ≤ 21−k write k ∞ X X f (t + h) − f (t) = + 2−m (fm (t + h) − fm (t)) .

P∞

1

k+1

P∞

1

By part (a) the sum k+1 is bounded by 2 k+1 2−m 2m/2 < 5h 2 . Using part (a), exercise 2.12, and the fact that fm is aPtrigonometric polyomial of degree k 2m − 1, one obtains a similar estimate for 1 . 2 7. Let f, g ∈ L (T). Show that f ∗ g ∈ A(T). 7

FOURIER COEFFICIENTS OF LINEAR FUNCTIONALS

We consider a homogeneous Banach space B on T and assume, for simplicity, that eint ∈ B for all n. As usual, we denote by B ∗ the dual space of B . 7.1

(7.1)

The Fourier coefficients of a functional µ ∈ B ∗ are, by definition: µ ˆ(n) = heint , µi,

n ∈ Z;

and we call the trigonometric series S[µ] ∼

∞ X

µ ˆ(n)eint

−∞

the Fourier series† of µ. Clearly |ˆ µ(n)| ≤ kµkB ∗ keint kB .

The notation (7.1) is consistent with our definition of Fourier coefficients in case that µ is identified naturally with a summable function. For instance, if B = Lp (T), 1 < p < ∞, B ∗ is canonically identified with Lq (T) where q = p/(p − 1). To the function g ∈ Lq (T) corresponds the linear functional Z 1 f 7→ hf, gi = f ∈ Lp (T) f (t)g(t)dt, 2π Z Z and 1 1 heint , gi = eint g(t)dt = e−int g(t)dt 2π 2π thus gˆ(n) defined in (7.1) for the functional g coincides with the nth Fourier coefficient of the function g . † We keep, however, the convention of 1.3 that a Fourier series, without complements, is a Fourier series of a summable function.

36


Theorem (Parseval’s formula). Let f ∈ B , µ ∈ B ∗ ; then N X |n| ˆ (7.2) hf, µi = lim µ(n). 1− f (n)ˆ N →∞ N +1 −N PN ˆ int P ROOF : (a) For polynomials P (t) = we clearly have −N P (n)e PN ˆ hP, µi = −N P (n)ˆ µ(n). (b) Since, by theorem 2.11, f = limN →∞ σN (f ) in the B norm, it follows from (a) and the continuity of µ that hf, µi = limhσN (f ), µi = lim

N →∞

N X

1−

−N

|n| ˆ f (n)ˆ µ(n). N +1

J

Remark: The fact that the limit in (7.2) exists is an implicit part of the theorem. It is equivalent to the C-1 summability‡ of the series Pˆ f (n)ˆ µ(n). If this last series converges then clearly hf, µi =

(7.3)

∞ X

fˆ(n)ˆ µ(n)

−∞

We shall sometimes refer to (7.3) as Parseval’s formula, keeping in mind that if the series on the right does not converge then (7.3) is simply an abbreviation for (7.2). Corollary (Uniqueness theorem). If µ ˆ(n) = 0 for all n, then µ = 0. P P 7.2 We shall write µ ∼ µ ˆ(n)eint , and may write µ = µ ˆ(n)eint if the series converges in some sense (which should be clear from the context). This is an abuse of language which, if used with caution, presents no risk of misunderstanding and obviates tedious repetitions. In accordance with our abuse of language we define, for µ ∈ B ∗ , the elements Sn (µ) and σn (µ) of B ∗ by Sn (µ) =

n X

σn (µ) =

n X

−n

(7.4)

−n

‡ Cesàro

µ ˆ(j)eijt

of order 1

1−

|j| µ ˆ(j)eijt n+1


37

We shall also write Sn (µ, t) =

(7.5) σn (µ, t) =

n X

µ ˆ(j)eijt

−n n X −n

1−

|j| µ ˆ(j)eijt n+1

The correspondence between the functionals (7.4) and the functions (7.5) is clearly Z n X 1 hf, Sn (µ)i = fˆ(j)ˆ µ(j) f (t)Sn (µ, t)dt = 2π −n for all f ∈ B ; similarly for σn (µ). The mapping Sn : f 7→ Sn (f ) on B is clearly a bounded linear operator, and so is Sn : µ 7→ Sn (µ) on B ∗ . It follows from Parseval’s formula that Sn on B ∗ is the adjoint of Sn on B and consequently has the same norm. Similarly, σ n : µ 7→ σn (µ) on B ∗ is the adjoint of ∗ σ n : f 7→ σn (f ) on B and consequently§ kσ n kB = 1. We remark that by Parseval’s formula, for every µ ∈ B ∗ , σn (µ) converges weak-star to µ. 7.3 Parseval’s formula enables us to characterize sequences of Fourier coefficients of linear functionals. Theorem. Let B be a homogeneous Banach space on T. Assume that eint ∈ B for all n. Let {an }∞ n=−∞ be a sequence of complex numbers. Then the following two conditions are equivalent: (a) There exists µ ∈ B ∗ , kµk ≤ C , such that µ ˆ(n) = an for all n. (b) For all trigonometric polynomials P X Pˆ (n)an ≤ CkP kB .

P ROOF : The implication (a) ⇒ (b) follows immediately from Parseval’s formula. If we assume (b) then X (7.6) P 7→ Pˆ (n)an is a linear functional on the space of all trigonometric polynomials, bounded in the B norm, and therefore (theorem 2.12) admits a unique extension µ of norm < C to B . Since µ extends (7.6) we have µ ˆ(n) = heint , µi = an . § kσ

B∗ nk

denotes the norm of σ n as operator on B ∗ .

J

38


P Corollary. A trigonometric series S ∼ an eint is the Fourier series of some µ ∈ B ∗ , kµk ≤ C , if, and only if, kσN (S)k ≤ C for all N . Here σN (S) denotes the element in B* the Fourier series of which is PN ijt N (1 − |j|/(N + 1))aj e .

P ROOF : The necessity follows from 7.2; the sufficiency from the current theorem and the observation that for trigonometric polynomials P X Pˆ (n)an = lim hP, σN (S)i . J N →∞ 7.4 In the case B = C(T ) the dual space B ∗ is identified R with the space M (T) of all (Borel) measures on T (we set hf, µi = f dµ) We shall refer to Fourier coefficients of measures as Fourier-Stieltjes coefficients and to Fourier series of measures as Fourier-Stieltjes series. The mapping f 7→ (1/2π)f (t)dt is an isometric embedding of L1 (T) in M (T). The Fourier coefficients of (1/2π)f (t)dt are precisely fˆ(n), hence a Fourier series is a Fourier-Stieltjes series. An example of a measure that is not obtained as (1/2π)f (t)dt is the so-called Dirac measure; it is the measure δ of mass one concentrated at t = 0. δ can also be defined by hf, δi = f (0) for all f ∈ C(T). We denote by δτ , τ ∈ T, the unit mass concentrated at τ . Thus δ = δ0 and hf, δτ i = f (τ ) for all τ ∈ T. From (7.1) it follows that δˆτ (n) = e−inτ ˆ and in particular δ(n) = 1. This shows that Fourier-Stieltjes coefficients need not tend to zero at infinity (however, by 7.1, |ˆ µ(n)| ≤ kµkM (T) ). 7.5 We recall that a measure µRis positive if µ(E) ≥ 0 for every measurable set E , or equivalently, if f dµ ≥ 0 whenever f ∈ C(T) is nonnegative. If µ is absolutely continuous, that is, if µ = (1/2π)g(t)dt with g ∈ L1 (T), then µ is positive if and only if g(t) ≥ 0 almost everywhere. P Lemma. A series S ∼ an eint is the Fourier-Stieltjes series of a positive measure if, and only if, for all n and t ∈ T, σn (S, t) =

n X −n

1 − |j|/(n + 1) aj eijt ≥ 0.

P ROOF : If S = S(µ) for a positive µ ∈ M (T) and if f ∈ C(T) is nonnegative, we have 1 2π

Z

f (t)σn (S, t)dt =

n X −n

1−

|j| ˆ µ(j) = σn (f )dµ ≥ 0 f (j)ˆ n+1


39

since µ ≥ 0 and, by 3.1, σn (f, t) > 0. Since this is true for arbitrary nonnegative f , σn (S, t) ≥ 0 on T. Assuming σn (S, t) ≥ 0 we obtain Z 1 kσn (S)kM (T) = σn (S, t)dt = a0 2π and, by Corollary 7.3, RS = S(µ) for some µ R∈ M (T). For arbitrary nonnegative f ∈ C(T), f dµ = limn→∞ (l/2π f (t)σn (S, t)dt ≥ 0 and it follows that µ is a positive measure. J Remark: The condition “σn (S, t) ≥ 0 for all n” can clearly be replaced by “σn (S, t) ≥ 0 for infinitely many n’s”. 7.6 We are now able to characterize Fourier-Stieltjes coefficients of positive measures as positive definite sequences. D EFINITION : A numerical sequence {an }n∈Z is positive definite if for any sequence {zn } having only a finite number of terms different from zero we have X (7.7) an−m zn z¯m ≥ 0. n,m

Theorem (Herglotz). A numerical sequence {an }n∈Z is positive definite if, and only if, there exists a positive measure µ ∈ M (T) such that an = µ ˆ(n) for all n.

P ROOF : Assume an = µˆ(n) with positive µ. Then Z X 2 X XZ (7.8) an−m zn z¯m = e−int eimt zn z¯m = zn e−int dµ ≥ 0. n,m

n,m

n

If, on the other hand, we assume that {an } is positive definite, we write P S∼ an eint and, for arbitrary N and t ∈ T we choose ( eint |n| ≤ N zn = 0 |n| > N P P We have n,m an−m zn z¯m = j Cj,N aj eijt where Cj,N is the number of ways to write j in the form n − m where |n| ≤ N and |m| ≤ N , that is, Cj,N = max(0, 2N + 1 − |j|). It follows that X 1 Cj,N aj eijt ≥ 0 σ2N (S, t) = 2N + 1 j and the theorem follows from 7.5.

J

40


7.7

If {an } is positive definite, then |an | ≤ a0 ,

(7.9)

n+1 )} is positive definite. This can be and the sequence {(an − an−1 +a 2 seen directly by checking condition (7.7), or deduced from Herglotz’ theorem and the observations that if µ is the positive measure such that an = µ ˆ(n), then a0 = µ ˆ(0) = kµk, ν = (1 − cos t)µ is nonnegative, and n+1 νˆ(n) = an − an−1 +a . 2 Also, since an = a−n , we have a0 − ε, is arbitrarily small. In particular, for every ε > 0 and positive N there exist in any sufficiently large interval on Z, intervals of length N , on which |ˆ µ| < ε.

EXERCISES FOR SECTION 7 1. Let B be homogeneous on T and B ∗ its dual. Show that S ∼ an eint ∗ is the Fourier series of some µ ∈ B if and only if kσN (S)k is bounded as N → ∞. 2. Denote Kn,τ (t) = KN (t − τ ). Show that for every µ ∈ M (T)

P

σn (µ, τ ) = hKn,τ , µi .

46


Deduce that σn (µ, τ ) ≥ 0 if µ is positive. P PN 3. Show that a trigonometric series an eint such that −N an eint ≥ 0 for all N and t ∈ T is a Fourier-Stieltjes series of a positive measure. 4. We shall prove later (see IV.2.1) that if f ∈ Lp (T) with 1 < p ≤ 2, then kfˆk`q =

X

|fˆ(n)|q

1/q

≤ kf kLp (T)

q=

p ,. p−1

Assuming this, show that if {an } ∈ `p is a numerical sequence then there exists a function g ∈ Lq (T) such that gˆ(n) = an , and kgkLq (T) ≤ k{an }k`p . 5. The elements of the dual space of C m (T) are called distributions of order m on T. We denote by Dm (T) = (C m (T))∗ the space of distributions of order m on T. Since C m+1 (T) ⊂ C m (T) we have Dm (T) ⊂ Dm+1 (T). Write D(T) = ∪m Dm (T). (a) Prove that if µ ∈ Dm then |ˆ µ(n)| ≤ const |n|m

n 6= 0.

(b) Given a numerical sequence {an } satisfying an = O (|n|m ), there exists a distribution µ ∈ Dm+1 such that an = µ ˆ(n) for all n. P Hint: If f ∈ C m+1 (T) then |nm fˆ(n)| < ∞. P Thus a trigonometric series an eint is the Fourier series of a distribution on T if and only if, for some m, an = O (|n|m ) n 6= 0. Let µ ∈ D and let O be an open subset of T. We say that µ vanishes on O if hϕ, µi = 0 for all ϕ ∈ C ∞ (T) such that the support of ϕ (i.e., the closure of the set {t : ϕ(t) 6= 0}) is contained in O. (c) Prove that if µ vanishes on the open sets O1 and O2 , then it vanishes on O1 ∪ O2 . Hint: Show that if the support of ϕ ∈ C ∞ (T) is contained in O1 ∪ O2 then there exist ϕ1 , ϕ2 ∈ C ∞ (T), with supports contained in O1 , O2 respectively, such that ϕ = ϕ1 + ϕ2 . (d) Extend the result of (c) to any finite union of open sets; hence, using the compactness of the support of the test functions ϕ, show that if µ vanishes in S the open sets Oα , α running over some index set I , then µ vanishess on Oα . α∈I Thus the union of all the open subsets of T on which µ vanishes is again such a set. This is clearly the largest open set on which µ vanishes.

D EFINITION : The support of µ is the complement in T of the largest open set O ⊂ T on which µ vanishes. (e) Show that if µ ∈ Dm and if f ∈ C m (T) vanishes on a neighborhood of the support of µ, then hf, µi = 0. The same conclusion holds if for some homogeneous Banach space B , the distribution µ belongs to B ∗ and f ∈ B (see exercise 2.11).


47

(f) We define the derivative µ0 of a distribution µ ∈ Dm by hf, µ0 i = −hf 0 , µi

for f ∈ C m+1 (T).

Show that µ0 ∈ Dm+1 and µb0 (n) = in µ ˆ(n).

(g) Show that support(µ0 ) ⊂ support(µ).

(h) Show that the map µ 7→ µ0 maps Dm onto the subspace of Dm+1 consisting of all µ ∈ Dm+1 satisfying µ ˆ(0) = 0. Hence, every µ ∈ Dm can be written in the form µ ˆ(0)dt + µ1 where µ1 is the mth derivative of a measure. (i) A distribution µ on T is positive if hf, µi ≥ 0 for every nonnegative test function f ∈ C ∞ (T). Show that a positive distribution is a measure. Hint: Positivity implies, for real-valued f , hf, µi ≤ max f (t)h1, µi. 6. The dual space of A(T) is commonly denoted by P M (T) and its elements referred to as pseudo-measures. Show that with the natural identifications M (T) ⊂ P M (T) ⊂ D1 (T), and P M (T) consists precisely of those µ for which {ˆ µ(n)} is bounded. Moreover, the correspondence µ ↔ {ˆ µ(n)} is an isometry of P M (T) onto `∞ . 7. Let α, β ∈ T, let N be an integer, and let µ be the measure carried by the arithmetic progression {α + jβ}N j=−N , which places the mass zero at α and the mass j −1 at α + jβ, 1 ≤ |j| ≤ N . Show that kµkP M (T) ≤ π + 2. Hint: See exercise 3.4. 8. Let f ∈ A(T) be real valued and monotonic in a neighborhood of t0 ∈ T. Show that |f (t) − f (t0 )| = O (log|t − t0 |−1 )−1 as t → t0 . 9. Let µ, µn ∈ M (T), n = 1, 2, . . . . Prove that µn → µ in the weak-star topology if, and only if, kµn kM (T) = O (1) and µ ˆn (j) → µ ˆ(j) for all j . 10. By definition, a sequence {ξn }∞ ⊂ T is uniformly distributed if for n=1 PN any arc I ⊂ T we have limN →∞ N −1 n=1 11I (ξn ) = (2π)−1 |I|. Prove the following statements: Pn −1 a. {ξn }∞ δ n=1 ⊂ T is uniformly distributed if, and only if, µn = n 1 ξj converge in the weak-star topology to (2π)−1 dt, i.e., if for all integers l 6= 0, Pn n−1 1 eilξj → 0. (Weyl’s criterion). b. if α is an irrational multiple of π , the sequence {nα} is uniformly distributed on T. 11. Show that a measure µ ∈ M (T) is absolutely continuous if, and only if, limτ →0 kµτ − µk = 0, where µτ is the translate of µ by τ (defined by µτ (E) = µ(E − τ )). 12. Let µ ∈ M (T). Prove: σn (µ, t) converge to zero at every t 6∈ support(µ), the convergence uniform on every closed set disjoint from support(µ). 13. Let µ ∈ M (T) be singular with respect to dt (that is, there exists a Borel set E0 of Lebesgue measure zero, such that µ(E) = µ(E ∩ E0 ) for every Borel set E ). Show that σn (µ, t) → 0 almost everywhere (dt).

48


14. Show that the conclusion of exercise 12 is false if we assume µ ∈ D1 instead of µ ∈ M (T); however, if we replace Fejér’s kernel by Poisson’s, the conclusion is valid for every µ ∈ D. m = 0 uniformly Hint: For 0 < δ < π , and positive integers m, limr→1 ∂ ∂tP(r,t) m in (δ, 2π − δ). 15. (Bernstein’s inequality) Let tj,n = (2j+1)π , and δj,n = δtj,n . Write 4n 2n−1 1 X δj,n , 2n

µn =

νn = Kn (t)µn .

0

(a) Check that kνn k = νˆn (0) = 1. (b) Write νn∗ = neint νn , then kνn∗ k = n, and νbn∗ (j) is periodic of period 4n and νn∗ (j)

b

=

(

j

for |j| ≤ n,

2n − j

for n ≤ j ≤ 3n.

(c) Prove Bernstein’s inequality: if P =

Pn

−n

aj eijt , then

kP 0 k∞ ≤ n kP k∞ .

(7.25) Hint: P 0 = νn∗ ∗ P . (d) Prove that if P =

Pn

−n

aj eijt , then kPe0 k∞ ≤ n kP k∞ .

Hint: Find a measure µn of norm n such that µ ˆn (j) = |j| for |j| ≤ n. (e) Let B be a homogeneous Banach space on T, and P = Prove that kP 0 kB ≤ nkP kB , and kPe0 kB ≤ n kP kB . 8

Pn

−n

aj eijt ∈ B .

ADDITIONAL COMMENTS AND APPLICATIONS

8.1 Approximation by trigonometric polynomials. The order of magnitude of the Fourier coefficients of a function f gives some indication of the smoothenss of the function. We get more precise information from the rapidity of the approximation of f by trigonometric polynomials (as a function of their degree), or from the decomposition of f into a series of polynomials given by (8.8) below. For ϕ ∈ C(T) denote En (ϕ) = infkϕ − P k∞ , the infimum for all trigonometric polynomials P of degree ≤ n. If m is a positive integer and 0 < η < 1, C m+η (T) denotes the space {f ∈ C m (T) : f (m) ∈ Lipη }, endowed with the norm kf kC m+η = kf kC m (T) + kf (m) kLipη .


49

Theorem (Jackson). Let m be a non-negative integer, and 0 ≤ η < 1. If f ∈ C m+η (T), then: (8.1) En (f ) = O n−m−η .

The converse of the statement is true provided η > 0. See 8.3 below. The proof will use the information we have in the exercises for Section 3, Bernstein’s inequality for derivatives of polynomials (exercise 7.15 above), and the “reverse inequality” which is discussed in the following subsection. 8.2

The “reverse Bernstein inequality”.

Theorem. Let m be a positive integer. There exist a constant Cm such P that if f = |j|≥n aj eijt ∈ C m (T), then kf k∞ ≤ Cm |n|−m kf (m) k∞ .

(8.2)

P ROOF : We prove the following statement: given positive integers m and n, there exist measures µm,n such that −m (8.3) µd m,n (j) = j

for

|j| ≥ n,

and

kµm,n kM (T) ≤ Cm n−m .

Since f = im µm,n ∗ f (m) , this clearly implies (8.2). Denote by φm,n the positive function in L1 (T) whose Fourier coefficients are given by ( j −m for |j| ≥ n d φm,n (j) = −m −m −m n + (n − |j|)(n − (n + 1) ) for |j| < n.

The coefficients in the range |j| < n are chosen so as to fulfill the conditions of Theorem 4.1: they are symmetric, linear in [0, n + 1] with slope matching that of {j −m } on [n, n + 1], so that φd m,n (j) is convex on [0, ∞). It follows that −m kφm,n kL1 = φd . m,n (0) < (m + 1)n

For even m set µm,n = φm,n dt and obtain Cm = m + 1. For m = 1, we use the polynomials Ψn,k = ei2

k

nt

1 k+1 K2k (n−1) + ei2 nt K2k (n−1) , 2

50


Clearly kΨn,k kL1 (T) ≤ 3/2, Ψd n,k (j) = 0 for negative j , while for j ≥ n P∞ d Ψ (j) = 1 (with one or two non-zero summands for each j ). It k=0 n,k follows that the function Φ∗n (t) =

∞ X

Ψn,k ∗ φ1,n2k

k=0

c∗n (j) = 0 for j ≤ 0, Φ c∗n (j) = 1/j for all j ≥ n, and satisfies: Φ

X

∗ (8.4) kφ1,n2k k ≤ 6n−1 .

Φn 1 ≤ 3/2 L (T)

We set µ1,n = Φ∗n (t) − Φ∗n (−t) dt, and (8.2) holds with C1 = 12. Finally, for m = 2l + 1 we set µm,n = µ1,n ∗ µ2l,n and Cm = 12m.

For polynomials of the form P =

P

n≤|j|≤4n

J

aj eijt we have

−1 m Cm n kP k∞ ≤ kP (m) k∞ ≤ (4n)m kP k∞ .

(8.5)

Moreover, the same holds if we replace k k∞ by the norm of any homogeneous Banach space on T. 8.3 Lemma. Let Pk be trigonometric polynomials of degrees bounded P by 2k and kPk k∞ = O 2−(m+η)k . Then f = Pk ∈ C m+η (T). P ROOF : By Bernstein’s inequality, we have kPk kC m (T) = O 2−ηk so the series converges in C m (T), and f ∈ C m . Focusing on Pk(m) we have (m) (m+1) (8.6) k∞ = O 2(1−η)k , kPk k∞ = O 2−ηk , and kPk

which imply (8.7)

(m) |Pk (t

+ h) −

(m) Pk (t)|

≤

(

C|h|2k(1−η) ) C2

−kη

for |h| ≤ 2−kη for |h| ≥ 2−kη

the constant C coming from the O bound. For any h and any t ∈ T, X X (m) f (t+h)−f (m) (t) ≤ C |h|2k(1−η) + 2−kη ≤ C1 |h|η , |h|≤2−kη

and f ∈ C m+η (T).

|h|≥2−kη

J


51

If f ∈ C(T), n ≥ 2, the polynomial Wn (f ) = (V2n − Vn ) ∗ f is of the P form n≤|j|≤4n aj eijt . We set W1 (f ) = V2 ∗ f , and have X (8.8) f = lim V2k ∗ f = W2k (f ). k≥1

Notice that, since kV2n − Vn kL1 (T) < 6, we have, for any homogenous Banach space B on T, (8.9)

kWn (f )kB ≤ 6 kf kB .

Theorem. Let m ≥ 0 be an integer and 0 < η < 1. A necessary and sufficient condition for f to be in C m+η (T) is (8.10) kWn (f )k∞ = O n−m−η .

For η = 0, the condition (8.10) is necessary, but not sufficient.

P ROOF : For any g , Wn (g) = 2(K4n ∗ g − g) − 3(K2n ∗ g − g) + (Kn ∗ g − g), and hence (8.11)

kWn (g)k∞ ≤ 6

max kσk (g) − gk∞ .

k=n,2n,4n

Combine this (for g = f (m) , where f ∈ C m+η , and 0 < η < 1,) with exercise 2 at the end of Section 3, and obtain (8.12)

kWn (f (m) )k∞ ≤ Ckf (m) kLipη n−η ,

where C is a universal constant. By (8.5), and the fact that convolution commutes with differentiation, thus kWn (f )k∞ ≤ Ckf kC m+η n−(m+η) , and we proved the “necessary” part of the theorem. For the proof that condition (8.10) is sufficient, write Pk = W2k (f ) and apply the lemma. J This is essentially Jackson’s theorem. Our estimate of Wn (f ) was based on the estimate kσn (f (m) ) − f (m) k∞ ≤ Ckf kC m+η n−η , which implies the same (with a different constant) for kVn ∗ f (m) − f (m) k∞ , and the “reverse Bernstein inequality” gives† En (f ) ≤ kVn ∗ f − f k∞ ≤ Ckf kC m+η n−(m+η) .

This is the “direct” side of Jackson’s theorem. The “converse” follows from the current theorem combined with the observation that for any polynomial P of degree less than 2k we have W2k (f ) = W2k (f − P ), hence kW2k (f )k∞ ≤ E2k−1 (f ). † This

is the reason we use Vn rather than σn : for f = cos t, σn (f ) − f = n−1 f .

52


8.4 One application of Theorem 8.2 is the fact that if f ∈ C m+η , e ], the conjugate where m ≥ 0 is an integer, and 0 < η < 1, then S[f Fourier series of f , is also the Fourier series of a function in C m+η . In the terminology of the next chapter, C m+η admits conjugation. This follows from P Lemma. Let P be a polynomial of the form P = n≤|j|≤4n aj eijt , and P sgn (j)aj eijt . Then Pe its conjugate, Pe = n≤|j|≤4n

kPek∞ ≤ 7kP k∞ .

P ROOF : Pe = P − 2(e−i 3nt V2n ∗ P ).

J

Theorem. C m+η admits conjugation. P P ROOF : With some abuse of notation we may write f˜ = n≥1 W^ 2n (f ), use the lemma, and invoke Theorem 8.2. J

8.5 Multipliers on Fourier coefficients. Let B and B1 be homogeneous Banach spaces on T and let T be an operator from B into B1 which commutes with translations. Then, for all n, T eint = t(n)eint P and for f ∈ B the Fourier series of T f is t(n)fˆ(n)eint . In other words, T is expressed as a multiplier on the Fourier coefficients of f . We have seen concrete examples of this when T was differentiation, or convolution by a fixed measure, and just in the previous subsection— conjugation, for which the multiplier is t(n) = sgn (n). The proof of Theorem 8.4 can be imitated for other multipliers. ForP mally, if {t(n)} is given, the operator T : f 7→ t(n)fˆ(n)eint is well defined for polynomials f . If we denote kT kn the norm of the multiplier restricted to the sup-normed space of trigonometric polynomials of the P form 2n ≤|j|≤2n+2 aj eijt , then the proof of Theorem 8.4 consisted in the observation that kT kn is uniformly bounded when t(n) = sgn (n). The same argument, using Theorem 8.2, proves r1 r2 Theorem. A sequence {t(n)} is a multiplier C 7→ C , r2 6∈ Z, if and ∗ n(r2 −r1 ) only if kT kn = O 2 . In particular, this is true if

(8.13)

X

2n ≤|j|≤2n+2

|t(n)|2 = O 22n(r2 −r1 ) .


53

P ROOF : We check that (8.13) implies the estimate kT kn = O 2n(r2 −r1 ) . P For polynomials of the form 2n ≤|j|≤2n+2 aj eijt the operator T is the P convolution with the kernel Tn (t) = 2n ≤|j|≤2n+2 t(j)eijt , and its norm is bounded by kTn kL1 (T) ≤ kTn kL2 (T) . J

8.6 The difference equation. Given a function f ∈ L1 (T) and α ∈ T, we are asked to find g such that

(8.14)

g(t + α) − g(t) = f (f ).

Under what condition can this be done and what can be said about the “solution” g ? Formally the solution is obvious, at least if g ∈ L1 (T): (8.14) is satisfied if, and only if, (8.15)

for all n ∈ Z

gˆ(n)(einα − 1) = fˆ(n),

in other words, we need to set gˆ(0) = 0 and (8.16)

gˆ(n) = (einα − 1)−1 fˆ(n),

and the question becomes that of identifying (pairs of) spaces on which the sequence (einα − 1)−1 is a multiplier. The answer depends on the diophantine properties of α, i.e., on the rate of growth of (einα − 1)−1 . Theorem. If |einα − 1| ≥ C|n|−γ then the sequence {(einα − 1)−1 } is a multiplier C r1 (T) 7→ C r2 (T) whenever r1 − r2 > γ . If r2 is not an integer, the same holds for r1 − r2 = γ . Lemma. Let {zj }M −M ⊂ T be such that for j 6= k , |zj − zk | ≥ a, and P |zj − 1| ≥ a. Then |zj − 1|−2 ≤ 4a−2 .

P ROOF : The worst estimate is obtained when the points are packed as close to 1 as the condition permits, that is, for zj = eija , j 6= 0, and z0 = ei(M +1)a . J P ROOF OF THE THEOREM . The lemma, with M = 2n+2 and a = C2−nγ , implies X |(einα − 1)|−2 = O 22nγ . 2n ≤|j|≤2n+2

Now apply theorem 8.5.

J

54


EXERCISES FOR SECTION 8 1. Show that if m is an integer, the condition En (f ) = o n−m log n is necessary for f ∈ C m (T) but is not sufficient. Show also that the condition P 2km E2k (f ) < ∞ is sufficient; is it necessary? k 2. Let B be a homogenoeous Banach space on T. For f ∈ B consider the B -valued function ϕ(τ ) = fτ . Prove that if ϕ is differentiable at some τ0 ∈ T, it is uniformly differentiable on T, and that this happens if, and only if, f 0 ∈ B . 3. Let B be a homogeneous Banach space on T and assume that, for some 1 ≤ p < ∞, k · kB is equivalent to k{kW2n ( · )kB }k`p . Prove that B admits conjugation

Chapter II

The Convergence of Fourier Series

We have mentioned already that the problems of convergence of Fourier series, that is, the convergence of the (symmetric) partial sums, Sn (f ), are far more delicate than the corresponding problems of summability with respect to "good" summability kernels such as Fejér’s or Poisson’s. As in the case of summability, problems of convergence "in norm" are usually easier than those of pointwise convergence. Many problems, concerning pointwise convergence for various spaces, are still unsolved and the convergence almost everywhere of the Fourier series of square summable functions was proved only recently (L. Carleson 1965). Convergence is closely related to the existence and properties of the conjugate function. In this chapter we give only a temporary incomplete definition of the conjugate function. A proper definition and the study of the basic properties of conjugation are to be found in chapter III. 1

CONVERGENCE IN NORM

1.1

(1.1)

Let B be a homogeneous Banach space on T. As usual we write Sn (f ) = Sn (f, t) =

n X

fˆ(j)eijt .

−n

We say that B admits convergence in norm if (1.2)

lim kSn (f ) − f kB = 0.

n→∞

Our purpose in this section is to characterize the spaces B which have this property. We have introduced the operators Sn : f 7→ Sn (f ) in chapter I. Sn is well defined in every homogeneous Banach space B ; we denote its norm, as an operator on B , by kSn kB . 55

56


Theorem. A homogeneous Banach space B admits convergence in norm if, and only if, kSn kB are bounded (as n → ∞), that is, if there exist a constant K such that kSn (f )kB ≤ Kkf kB

(1.3) for all f ∈ B and n > 0.

If Sn (f ) converge to f for all f ∈ B , then Sn (f ) are bounded for every f ∈ B . By the uniform boundedness theorem, it follows that kSn kB = O(1). On the other hand, if we assume (1.3), let f ∈ B, ε > 0, and let P be a trigonometric polynomial satisfying kf − P kB ≤ ε/2K . For n greater than the degree of P , we have Sn (P ) = P and hence PROOF :

kSn (f ) − f kB =kSn (f ) − Sn (P ) + P − f kB ≤kSn (f − P )kB + kP − f kB ≤ K

1.2

ε ε + ≤ ε. 2K 2K

J

The fact that Sn (f ) = Dn ∗ f , where Dn , is the Dirichlet kernel

(1.4)

Dn (t) =

n X −n

eijt =

sin(n + 1/2)t sin t/2

yields a simple bound for kSn kB . In fact, kDn ∗ f kB ≤ kDn kL1 kf kB , so that kSn kB ≤ kDn kL1 .

(1.5)

The numbers Ln = kDn kL1 are called the Lebesgue constants; they tend to infinity like a constant multiple of log n (see exercise 1 at the end of this section). In the case B = L1 (T) the inequality (1.5) becomes an equality. This can be seen as follows: denote by KN the Fejér kernel and remember 1 that kKN kL1 = 1. We have kSn kL ≥ kSn (KN )kL1 = kσN (Dn )kL1 and since σN (Dn ) → Dn , as N → ∞, we obtain kSn kL

1

(T)

≥ kDn kL1 ;

hence kSn kL

1

(T)

= kDn kL1 .

It follows that L1 (T) does not admit convergence in norm.

II. T HE C ONVERGENCE OF F OURIER S ERIES

57

1.3 In the case B = C(T) convergence in norm is simply uniform convergence. We show that Fourier series of continuous functions need not converge uniformly by showing that kSn kC(T) are unbounded; more precisely we show that kSn kC(T) = Ln . For this, we consider continuous functions ψn satisfying kψn k∞ = supt |ψn (t)| ≤ 1

and such that ψn (t) = sgn (Dn (t)) except in small intervals around the points of discontinuity of sgn (Dn (t)). If the sum of the lengths of these intervals is smaller than ε/2n, we have Z 1 kSn kC(T) ≥ |Sn (ψn , 0)| = Dn (t)ψn (t) dt > Ln − ε 2π which, together with (1.5), proves our statement. 1.4 For a class of homogeneous Banach spaces on T, the problem of convergence in norm can be related to invariance under conjugation. In chapter I we defined the conjugate series of a trigonometric series P P an eint to be the series −i sgn (n)an eint . If f ∈ L1 (T) and if the Pˆ series conjugate to f (n)eint is the Fourier series of some function 1 g ∈ L (T), we call g the conjugate function of f and denote it by f˜. This definition is adequate for the purposes of this section; however, it does not define f˜ for all f ∈ L1 (T) and we shall extend it later. D EFINITION : A space of functions B ⊆ L1 (T) admits conjugation if for every f ∈ B, f˜ is defined and belongs to B . If B is a homogeneous Banach space which admits conjugation, then the mapping f 7→ f˜ is a bounded linear operator on B . The linearity is evident from the definition and in order to prove the boundedness we apply the closed graph theorem. All that we have to do is show that the operator f 7→ f˜ is closed, that is, that if lim fn = f and lim fñ = g in B , then g = f˜. This follows from the fact that for every integer j ˆ gˆ(j) = lim fñ (j) = lim −i sgn (j)fˆn (j) = −i sgn (j) lim fˆn (j) n→∞

n→∞

n→∞

ˆ = −i sgn (j)fˆ(j) = f˜(j).

If B admits conjugation then the mapping ∞

(1.6)

f 7→ f [ =

X 1ˆ 1 f (0) + (f + if˜) ∼ fˆ(j)eijt 2 2 0

58


is a well-defined, bounded linear operator on B . Conversely, if the mapping f 7→ f [ is well-defined in a space B , then B admits conjugation since f˜ = −i(2f [ − f − f (0)). Theorem. Let B be a homogeneous Banach space on T and assume that for f ∈ B and for all n, eint f ∈ B and keint f kB = kf kB .

(1.7)

Then B admits conjugation if, and only if, B admits convergence in norm.

P ROOF : By Theorem 1.1 and the foregoing remarks, it is clearly sufficient to prove that the mapping f 7→ f [ is well defined in B if, and only if, the operators Sn are uniformly bounded on B . Assume first that there exists a constant K such that kSn kB ≤ K . Define (1.8)

Sn[ (f ) =

2n X

fˆ(j)eijt = eint Sn (e−int f );

0

by (1.7) we have kS[ kB ≤ K . let f ∈ B and ε > 0; let P ∈ B be a trigonometric polynomial satisfying kf − P kB ≤ ε/2K . We have (1.9)

kSn[ (f ) − Sn[ (P )kB = kSn[ (f − P )kB ≤

ε . 2

[ If n and m are both greater than the degree of P , Sn[ (P ) = Sm (P ) and it follows from (1.9) that [ kSn[ (f ) − Sm (f )kB ≤ ε.

The sequence {Sn[ (f )} is thus a Cauchy sequence in B ; it converges and P ˆ ijt [ [ its limit has the Fourier series ∞ 0 f (j)e . So f = lim Sn (f ) ∈ B . [ Assume conversely that f 7→ f is well defined, hence bounded, in B . Then Sn[ f = f [ − ei(2n+1)t (e−i(2n+1)t f )[ which means that kS[n kB is bounded by twice the norm over B of the mapping f 7→ f [ . Since, by (1.7) and (1.8), kSn kB = kS[n kB , the theorem follows. J


59

1.5 We shall see in chapter III that, for 1 < p < ∞, Lp (T) admits conjugation, hence: Theorem. For 1 < p < ∞, the Fourier series of every f ∈ Lp (T) converges to f in the Lp (T) norm.

EXERCISES FOR SECTION 1 1. Show that the Lebesgue constants Ln = kDn kL1 (T) , satisfy Ln = 4/π 2 log n + O (1). Hint:† Z π n−1 Z (j+1)π X sin(n + 21 )t n+1/2 |sin(n + 1 )t| 1 2 dt = 2 Ln = dt + O (1) ; π 0 sin t/2 π t jπ j=1

remember that

Z

n+1/2

(j+1)π n+1/2

|sin(n + 1/2)t|dt = jπ n+1/2

2 . n+1

2. Show that if the sequence {Nj } tends to infinity fast enough, then the Fourier series of the function f (t) =

∞ X

2−j KNj (t)

1

does not converge in L1 (T). 3. Let {an } be an even sequence of positive numbers, convex on (0, ∞) and at infinity (cf. I.4.1). Prove that the partial sums of the series P vanishing an eint are bounded in L1 (T) if, and only if, an log n = O (1) and the series converges in L1 (T) if, and only if, lim an log n = 0. 4. Show that B = C m (T) does not admit convergence in norm. Hint: Sn commute with derivation. 5. Let ϕ be a continuous, concave (i.e., ϕ(h) + ϕ(h + 2δ) ≤ 2ϕ(h + δ)), and increasing function on [0, 1], satisfying ϕ(0) = 0. Denote by Λϕ the subspace of C(T) consisting of the functions f for which, as h → 0, ω(f, h) = O (ϕ(h)). Denote by λϕ the subspace of Λϕ consisting of the functions f for which ω(f, h) = o(ϕ(h)) as h → 0. (ω(f, h) is the modulus of continuity of f ; see I.4.6.) Consider the followingstatements: (a) ϕ(h) = O −(log h)−1 as h → 0. (b) For every f ∈ λϕ S[f ] is uniformly convergent. (c) ϕ(h) = o(−(log h)−1 ) as h → 0. (d) For every f ∈ Λϕ , S[f ] is uniformly convergent. Show that (a) is equivalent to (b) and that (c) is equivalent to (d). † For

another way, see [16].

60 2

A N I NTRODUCTION TO H ARMONIC A NALYSIS CONVERGENCE AND DIVERGENCE AT A POINT

We have seen in the previous section that the Fourier series of a continuous function need not converge uniformly. In this section we show that it may even fail to converge pointwise, and then give two criteria for the convergence of Fourier series at a point. 2.1 Theorem. There exists a continuous function whose Fourier series diverges at a point.

We give two proofs which are in fact one; the first is "abstract" based on the Uniform Boundedness Principle, and is very short. The second is a construction of a concrete example in essentially the way one proves the Uniform Boundedness Principle.

P ROOF A: The mappings f 7→ Sn (f, 0) are continuous linear functionals on C(T), We saw in the previous section that these functionals are not uniformly bounded and consequently, by the Uniform Boundedness theorem, there exists an f ∈ C(T) such that {Sn (f, 0)} is not bounded. In other words, the Fourier series of f diverges unboundedly at t = 0. J P ROOF B: As we have seen in section 1, there exists a sequence of functions ψn ∈ C(T) satisfying: kψn k∞ ≤ 1,

(2.1)

|Sn (ψn , 0)| >

(2.2)

1 1 kDn kL1 > log n. 2 10

We put ϕn (t) = σn2 (ψn , t) and notice that ϕn is a trigonometric polynomial of degree n2 satisfying kϕn k∞ ≤ 1,

(2.1’)

|Sn(ϕn ,t) − Sn(ψn ,t) | < 2

and hence

|Sn (ϕn , 0)| >

(2.2’)

1 log n − 2. 10

n

With λn = 23 we define (2.3)

f (t) =

X 1 ϕλ (λn t) n2 n


61

and claim that f is a continuous function whose Fourier series diverges at t = 0. The continuity of f follows immediately from the uniform convergence of the series in (2.3); to show the divergence of the Fourier P iλj t series of f at zero, we notice that ϕλj (λj t) = m ϕ d ; hence λj (m)e |S λ2n (f, 0)| = |Sλ2n

(2.4)

∞ n X X 1 1 ϕ (λ t), 0 + ϕ d (0)| j 2 λj 2 λj j j n+1 1

∞ n−1 X 1 1 X 1 = ϕ (0) + S (ϕ , 0) + ϕ d (0) λ λ λ λ j n n j 2 2 2 j n j 1 n+1

≥

K log λn − 3, n2

which tends to ∞, and the theorem follows. Remark: f (t) =

m−1 X 1

J

∞

X 1 1 ϕ (λ t) + ϕλ (λn t). λ n n n2 n2 n n

The first sum is a trigonometric polynomial and so does not affect the convergence of the Fourier series of f . The second sum is periodic with period 2π/λm (since λm divides λk for k ≥ m); consequently the partial sums of the Fourier series of f are unbounded at every point of the form 2πj/λm for any positive integers j and m. If we want to obtain divergence at every rational multiple of 2π , all that we have to do is put n λn = n!23 . 2.2 Our first convergence criterion is really a simple Tauberian theorem due to Hardy. Theorem. Let f ∈ L1 (T) and assume 1 fˆ(n) = O (2.5) as |n| → ∞. n

Then Sn (f, t) and σn (f, t) converge for the same values of t and to the same limit. Also, if σn (f, t) converges uniformly on some set, so does Sn (f, t).

P ROOF : The condition (2.5) implies the following weaker condition which is really all that we need: for every ε > 0 there exists a λ > 1 such that X (2.5’) |fˆ(j)| < ε. lim sup n→∞

n≤|j|≤λn

62


Let ε > 0 and let λ > 1 be such that (2.5’) is valid. We have Sn (f, t) =

(2.6)

[λn] + 1 n+1 σ[λn] (f, t) − σn (f.t) [λn] − n [λn] − n |j| ˆ [λn] + 1 X 1− − f (j)eijt . [λn] − n [λ] + 1 n≤|j|≤λn

(where [λn] denotes the integral part of λn). By (2.5’) there exists an n0 such that if n > n0 , the last term in (2.6) is bounded by ε. If σn (f, t0 ) converge to a limit σ(f, t0 ), it follows from (2.6) that for n1 sufficiently large, n > n1 implies |Sn (f, t0 ) − σ(f, t0 )| < 2ε,

(2.7) in other words, (2.8)

lim Sn (f, t0 ) = σ(f, t0 ).

The choice of n1 depends only on the rate of convergence of σn (f, t0 ) to σ(f, t0 ) so that if this convergence is uniform on some set, so is (2.8). J Corollary. Let f be of bounded variation on T; then the partial sums Sn (f, t) converge to 12 (f (t + 0) + f (t − 0)) and in particular to f (t) at every point of continuity. The convergence is uniform on closed intervals of continuity of f .

P ROOF : By Fejér’s theorem the foregoing holds true for σn (f, t), and the statement follows from the fact that for functions of bounded variation, (2.5) is valid (cf. Theorem I.4.5). J R1 2.3 Lemma. Let f ∈ L1 (T) and assume −1 f (t) t dt < ∞. Then lim Sn (f, 0) = 0.

P ROOF : Sn (f, 0) =

(2.9)

Z

f (t) sin(n + 1/2)t dt = sin 2t Z Z 1 1 f (t) cos t/2 f (t) cos ntdt + sin tdt 2π 2π sin t/2 1 2π

cos t/2 By our assumption f (t) ∈ L1 (T); hence, by the Riemann-Lebesgue sin t/2 lemma, all the integrals in (2.9) tend to zero. J


63

2.4 Theorem (Principle of localization). Let f ∈ L1 (T) and assume that f vanishes in an open interval I . Then Sn (f, t) converge to zero for t ∈ I , and the convergence is uniform on closed subsets of I .

P ROOF : The convergence to zero at every t ∈ I is an immediate consequence of Lemma 2.3. If I0 is a closed subinterval of I , the functions f (t − t0 ) cos t/2 ϕt0 (t) = , t0 ∈ I0 , form a compact family in L1 (T), sin t/2 hence by Remark I.2.8, the integrals in (2.9) corresponding to f (t − t0 ), t0 ∈ I0 , tend to zero uniformly. J The principle of localization is often stated as follows: let f, g ∈ L1 (T) and assume that f (t) = g(t) in some neighborhood of a point t0 . Then the Fourier series of f and g at t0 are either both convergent and to the same limit or both divergent and in the same manner. 2.5

Another immediate application of Lemma 2.3 yields

Theorem (Dini’s test). Let f ∈ L1 (T). If 1

f (t + t ) − f (t ) 0 0 dt < ∞ t −1

Z

then Sn (f, t0 ) → f (t0 ).

EXERCISES FOR SECTION 2 1. Show that if a sequence of continuous functions on some interval is unbounded on a dense subset of the interval, then it is bounded only on a set of the first category. Use that to show that the Fourier series of f (defined in (2.3)) converges only on a set of the first category. 2. Show that for every given (countable) sequence {tn } there exists a continuous function whose Fourier series diverges at every tn . 3. Let g be the 2π -periodic function defined by: g(0) = 0, g(t) = t − π for 0 < t < 2π . (a) Discuss the convergence of the Fourier series of g . (b) Show that |Sn (g, t)| ≤ π + 2 for all n and t. (c) Put ϕn (t) = (π+2)−1 eint Sn (g, t); show that kϕn k∞ ≤ 1 and |Sn (ϕn , 0)| > K log n for some constant K > 0. (d) Show that for |t| < π/2, some constant K1 , and all n and m, |Sm (ϕn , t)| ≤

K1 |t|

64


(e) Show that for a proper choice of the integers nj and λj , the Fourier series of the continuous function f (t) =

∞ X 1 j=1

j2

eiλj t ϕnj (t)

diverges for t = 0 and converges for all other t ∈ T. ?3

SETS OF DIVERGENCE

We consider a homogeneous Banach space B on T.

3.1

D EFINITION : A set E ⊂ T is a set of divergence for B if there exists an f ∈ B whose Fourier series diverges at every point of E . D EFINITION :

3.2

For f ∈ L1 (T) we put Sn∗ (f, t) = supm≤n |Sm (f, t)|

(3.1)

S ∗ (f, t) = supn |Sn (f, t)|.

Theorem. E is a set of divergence for B if, and only if, there exists an element f ∈ B such that S ∗ (f, t) = ∞

(3.2)

for t ∈ E.

The theorem is an easy consequence of the following: Lemma. Let g ∈ B . There exist an element f ∈ B , and a positive even sequence {Ωj } such that limj→∞ Ωj = ∞ monotonically, and such that fˆ(j) = Ωj gˆ(j) for all j ∈ Z.

Let λ(n) be such that kσλ(n) (g) − gkB < 2−n . We write f = g + (g − σλn (g)). The series converges in norm; hence P∞ f ∈ B . Also fˆ(j) = Ωj gˆ(j) where Ωj = 1 + n=1 min(1, |j|/(λn + 1)). J PROOF OF THE LEMMA :

P

PROOF OF THE THEOREM : Condition (3.2) is clearly sufficient for the P divergence of fˆ(j)eijt for all t ∈ E . Assume, on the other hand, that P for some g ∈ B , gˆ(j)eijt diverges at every point of E . Let f ∈ B and {Ωj } be the function and the sequence corresponding to g by the lemma. We claim that (3.2) holds for f .


65

This follows from: for n > m, Sn (g, t) − Sm (g, t) =

n X

m+1

Sj (f, t) − Sj−1 (f, t) Ω−1 j

−1 =Sn (f, t)Ω−1 n − Sm (f, t)Ωm+1

(3.3)

+

n−1 X

−1 (Ω−1 j − Ωj+1 )Sj (f, t),

m+1

hence Sn (f, t) − Sm (f, t) ≤ 2S ∗ (f, t)Ω−1 . m+1

It follows that if S ∗ (f, t) < ∞, the Fourier series of g converges and t 6∈ E . J Remark: Let ωn , n ≥ 1, be positive numbers such that ωj = O (Ωj ), P −1 −1 and ∞ 1 (Ωj − Ωj+1 )ωj < ∞. Then, for all t ∈ E , Sj (f, t) 6= o(ωj ). This follows immediately from (3.3). 3.3 For the sake of simplicity we assume throughout the rest of this section that

(3.4)

If f ∈ B and n ∈ Z then eint f ∈ B and keint f kB = kf kB .

Lemma. Assume (3.4); then E is a set of divergence for B if, and only if, there exists a sequence of trigonometric polynomials Pj ∈ B such that X kPj kB < ∞ and sup S ∗ (Pj , t) = ∞ on E. (3.5)

P ROOF : Assume the existence of a sequence {Pj } satisfying (3.5). Denote by mj the degree of Pj and let νj be integers satisfying νj > νj−1 + mj−1 + mj .

Put f (t) =

P

eiνj t Pj (t). For n ≤ mj we have Sνj +n (f, t) − Sνj −n−1 (f, t) = eiνj t Sn (Pj , t);

P hence fˆ(j)eijt diverges on E . Conversely, assume that E is a set of divergence for B . By Remark 3.2 there exists a monotone sequence ωn → ∞ and a function f ∈ B

66


such that |Sn (f, t)| > ωn infinitely often for every t ∈ E . We now pick a sequence of integers {λj } such that (3.6)

kf − σλj (f )kB < 2−j

and then integers µj such that (3.7)

ωµj > 2 supt S ∗ (σλj (f ), t)

and write Pj = Vµj+1 ∗ (f − σλj (f )) where as usual Vµ denotes de la Vallée Poussin’s kernel (see 1.2.13). It follows immediately from (3.6) P that kPj kB < ∞. If t ∈ E and n is an integer such that |Sn (f, t)| > ωn , then for some j , µj < n ≤ µj+1 and Sn (Pj , t) = Sn (f − σλj (f ), t) = Sn (f, t) − Sn (σλj (f ), t).

Hence, by (3.7), |Sn (Pj , t)| > 12 ωn , and (3.5) follows.

J

Theorem. Assume (3.4). Let Ej , j = 1, 2, . . . , be sets of divergence for B . Then E = ∪Ej is a set of divergence for B .

P ROOF : Let {Pnj } be the sequence of polynomials corresponding to Ej . Omitting a finite number of terms for each j does not change (3.5), but P permits us to assume j,n kPnj k < ∞ which shows, by the lemma, that E is a set of divergence for B . J 3.4

We turn now to examine the sets of divergence for B = C(T).

Lemma. Let E be a union of a finite number of intervals on T; denote the measure of E by δ . There exists a trigonometric polynomial ϕ such that 1 1 S ∗ (ϕ, t) > on E log 2π 3δ (3.8) kϕk∞ ≤ 1 .

P ROOF : It will be convenient to identify T with the unit circumference {z : |z| = 1}. Let I be a (small) interval on T, I = {eit |t − t0 | ≤ ε : }; the function ψI = (1 + ε − ze−it0 )−1 has a positive real part throughout the unit disc, its real part is larger than 1/3ε on I , and its value at the origin (z = 0) is (1 + ε)−1 . We now write E ⊂ ∪N 1 Ij , the lj being small intervals of equal length 2ε such that N ε < δ , and consider the function ψ(z) =

1+εX ψtj (z). N


67

ψ has the following properties: 0

|z| ≤ 1

for

ψ(0) =1

(3.9)

|ψ(z)| ≥

1 1 > 3N ε 3δ

on E .

The function log ψ which takes the value zero at z = 0 is holomorphic in a neighborhood of {z : |z| ≤ 1} and has the properties |=(log ψ(z))| < π

(3.10)

on T −1

|log ψ(z)| > log(3δ)

on E

Since the Taylor series of log ψ converges uniformly on T, we can take P n a partial sum Φ(z) = M 1 an z of that series such that (3.10) is valid for Φ in place of log ψ . We can now put M

ϕ(t) =

M

X X 1 −iM t 1 e =(Φ(eit )) = e−iM t an eint − a ¯n e−int π πi 1 1

and notice that |SM (ϕ, t)| =

1 |Φ(eit )|. 2π

J

Theorem. Every set of measure zero is a set of divergence for C(T).

P ROOF : If E is a set of measure zero, it can be covered by a union P ∪In , the In being intervals of length |In | such that |In | < 1 and such that every t ∈ E belongs to infinitely many ln ’s. Grouping finite sets of intervals we can cover E infinitely often by ∪En such that every n En is a finite union of intervals and such that |En | 2n−1 /2πn2 on En . Since every t ∈ E belongs to infinitely many En ’s, our theorem follows from Lemma 3.3. J 3.5 Theorem. Let B be a homogeneous Banach space on T satisfying the condition (3.4). Assume B ⊃ C(T); then either T is a set of divergence for B or the sets of divergence for B are precisely the sets of measure zero.

68


P ROOF : By Theorem 3.4 it is clear that every set of measure zero is a set of divergence for B . All that we have to show in order to complete the proof is that, if some set of positive measure is a set of divergence for B , then T is a set of divergence for B . Assume that E is a set of divergence of positive measure. For α ∈ T denote by Eα the translate of E by α; Eα is clearly a set of divergence for B . Let {αn } be the sequence of all rational multiples of 2π and put ˜ = ∪Eα . By Theorem 3.2 E ˜ is a set of divergence, and we claim that E ˜ is a set of measure zero. In order to prove that, we denote by χ T\E the indicator function of E˜ and notice that χ(t − α) = χ(t)

for all t and αn .

This means X

−iαn j ijt χ(j)e ˆ e =

j

X

ijt χ(j)e ˆ

j

or −iαn j χ(j)e ˆ = χ(j) ˆ

(all αn )

If j 6= 0, this implies χ(j) ˆ = 0; hence χ(t) =constant almost everywhere and, since χ is an indicator function, this implies that the measure of E˜ is either zero or 2π . Since E˜ ⊃ E , E˜ is almost all of T. Now T \ E˜ is a set of divergence (being of measure zero) and E˜ is a set of divergence, hence T is a set of divergence. J 3.6 Thus, for spaces B satisfying the conditions of Theorem 3.5, and in particular for B = Lp (T), 1 < p < ∞, or B = C(T), either there exists a function f ∈ B whose Fourier series diverges everywhere, or the Fourier series of every f ∈ B converges almost everywhere. In the case B = L1 (T) it was shown by Kolmogorov that the first possibility holds. The case of B = L2 (T) was settled only recently by L. Carleson [4], who proved the famous "Lusin conjecture"; namely that the Fourier series of functions in L2 (T) converge almost everywhere. This result was extended by Hunt [12] to all Lp (T) with p > 1. The proof of these results is still rather complicated and we do not include it. We finish this section with Kolmogorov’s theorem. Theorem. There exists a Fourier series diverging everywhere.

P ROOF : For arbitrary κ > 0 we shall describe a positive measure µκ , of total mass one having the property that for almost all t ∈ T (3.11)

S ∗ (µκ , t) = supn |Sn (µκ , t)| > κ.


69

Assume for the moment that such µκ exist; it follows from (3.11) that there exists an integer Nκ and a set Eκ of (normalized Lebesgue) measure greater than 1 − 1/κ, such that for t ∈ Eκ supn κ

(3.12)

If we write now ϕκ = µκ ∗ VNκ , (VNκ being de la Vallée Poussin’s kernel), then ϕκ is a trigonometric polynomial, kϕκ kL1 (T) < 3 and S ∗ (ϕκ , t) ≥ supn 0 and α1 , . . . , αN be real numbers, then there exists an integer n such that |einxj − eiαj | < ε

j = 1, . . . , N.

We construct now the measures µκ as follows: let N be an integer, let x1 , . . . , xN be real numbers such that x1 , . . . , xN , π are linearly independent over the rationals and such that |xj − (2πj/N )| < 1/N 2 , and let P µ = 1/N δxj . For t ∈ T we have Sn (µ, t) =

Z

Dn (t − x)dµ(x) =

N 1 X Dn (t − xj ) = N 1

N 1 X sin(n + 12 )(t − xj ) = . N 1 sin(t − xj )

For almost all t ∈ T, the numbers t − x1 , . . . , t − xN , π are linearly independent over the rationals. By Kronecker’s theorem there exist, for each such t, integers n such that t − xj 1 i(n+ 12 )(t−xj ) − i sgn sin j = 1, . . . , N : < e 2 2

70


hence sin(n + 21 )(t − xj ) 1 t − xj −1 > sin 2 2 sin 12 (t − xj )

for all j .

It follows that (3.13)

Sn (µ, t) >

N 1 X t − xj −1 sin 2N j=1 2

and since the xj ’s are so close to the Rroots of unity of order N , the π sum in (3.13) is bounded below by 12 1/N |sin t/2|−1 dt > log N > κ, provided we take N large enough. J

EXERCISE FOR SECTION 3 1. Let B be a homogeneous Banach space on T. Show that for every f ∈ B there exist g ∈ B and h ∈ L1 (T) such that f = g ∗ h. Hint: Use Lemma 3.2 and Theorem I.4.1.

Chapter III

The Conjugate Function and Functions Analytic in the Unit Disc

We defined the conjugate function for some summable functions by means of their conjugate Fourier series. Our first purpose in this chapter is to extend the notion to all summable functions and to study the basic properties of the conjugate function for various classes of functions. This is done mainly in the first two sections. In section 1 we use the "complex variable" approach to define the conjugate function and obtain some basic results about the distribution functions of conjugates to functions belonging to various classes. In section 2 we introduce the Hardy-Littlewood maximal functions and use them to obtain results about the so-called maximal conjugate function. We show that the conjugate function can also be defined by a singular integral and use this to obtain some of its local properties. In section 3 we discuss the Hardy spaces H p . As further reading we mention [11]. 1

THE CONJUGATE FUNCTION

1.1 We identify T with the unit circumference {z : z = eit } in the complex plane. The unit disc {z : |z| < 1} is denoted by D and the ¯ . For f ∈ L1 (T) we denote by closed unit disc, {z : |z| ≤ 1}, by D it f (re ), r < 1, the Poisson integral of f ,

(1.1)

f (reit ) = (P(r, ·) ∗ f )(t) =

∞ X

r|n| fˆ(n)eint .

−∞

In chapter I we have considered P(r, ·) ∗ f as a family of functions on T, depending on the parameter r, 0 ≤ r < 1. The main idea in this section is to consider it as a function of the complex variable z = reit in D. 71

72


The functions r|n| eint , −∞ < n < ∞, are harmonic in D, and, since the series in (1.1) converges uniformly on compact subsets of D, it follows that f (reit ) is harmonic in D. We saw in I.3.3 that at every point t where f is the derivative of its integral (hence almost everywhere) f (eit ) = limr→1 f (reit ). Actually it is not very hard to see that for almost all t, f (z) → f (eit ) as z → eit nontangentially (i.e., if z → eit , remaining in a sector of the form {ζ : |arg(1 − ζe−it )| ≤ α < π}. (See [28], Vol. 1, p. 101.) The harmonic conjugate to (1.1) is the function (1.2)

f˜(reit ) = −i

∞ X

sgn (n)(n)r|n| fˆ(n)eint = (Q(r,·) ∗ f )(t)

−∞

where (1.2’)

Q(r, t) = −i

∞ X −∞

sgn (n)(n)r|n| )eint =

2r sin t 1 − 2r cos t + r2

is the harmonic conjugate of Poisson’s kernel P (r, t) (normalized by the condition Q(0, t) = sgn (0) = 0). We shall show that f˜(reit ) has a radial limit for almost all t. Denoting this radial limit by f˜(eit ) we shall show that if f has a conjugate in the sense of section II.1, then this conjugate is f˜(eit ). We may therefore call f˜ the conjugate function of f . 1.2 Lemma. Every function harmonic and bounded in D is the Poisson integral of some bounded function on T.

P ROOF : Let F be harmonic and bounded in D. Let rn ↑ 1 and write fn (eit ) = F (rn eit ). The sequence {fn } is a bounded sequence in L∞ (T); hence for some sequence nj → ∞, fnj converges in the weak-star topology (L∞ (T) being the dual of L1 (T)) to some function F (eit ). Let ρeiτ ∈ D, then Z Z 1 1 it P(ρ, t − τ )F (e )dt = lim P(ρ, t − τ )fnj (eit )dt j→∞ 2π 2π = lim F (rnj ρeiτ ) = F (ρeiτ ). J j→∞ 1.3 Lemma. Assume f ∈ L1 (T) and let f˜(reit ) be defined by (1.2) . Then, for almost all t, f (reit ) tends to a limit as r → 1.

P ROOF : Since the mapping f 7→ f˜(reit ) is clearly linear and since any f in L1 (T) can be written as f1 − f2 + if3 − if4 with fj ≥ 0 in L1 (T),

III. T HE C ONJUGATE F UNCTION

73

there is no loss of generality in assuming f > 0. The function F (z) = ˜ e−f (z)−if (z) is holomorphic (hence harmonic) in D. Since the Poisson integral of a nonnegative function is nonnegative f (z) > 0, and since f˜ is real valued (being the harmonic conjugate of the real valued f ), it follows that |F (z)| ≤ 1 in D. By Lemma 1.2 (and I.3.3) F has a radial limit of modulus e−f (t) almost everywhere. Since f ∈ L1 (T), f (eit ) < ∞, hence limr→1 F (reit ) 6= 0 almost everywhere; and at every point where F (eit ) exists and is nonzero, f˜(reit ) has a finite radial limit. J 1.4 D EFINITION : The conjugate function of a function f ∈ L1 (T) is the function f˜(eit ) = limr→1 f˜(reit ). If the series conjugate to the Fourier series of f is the Fourier series of some g ∈ L1 (T), then the Poisson integral of g is clearly f˜(reit ), which converges radially to g(eit ) for almost all t (theorem I.3.3). It follows that in this case f˜ = g and our new definition of the conjugate function extends that of II.1. P cos nt We have seen in I.4.2 that ∞ n=2 log n is a Fourier series while P P∞ sin nt sin nt the conjugate series, ∞ n=2 log n , is not. Since n=2 log n converges P cos nt everywhere, its sum is the conjugate function of f = ∞ n=2 log n and P sin nt 1 we can check that ∞ n=2 log n 6∈ L (T). Thus the conjugate function of a summable function need not be summable. P sin nt 1 Remark: At this point we cannot deduce that ∞ n=2 log n 6∈ L (T) from the mere fact that the series is not a Fourier series. However, we shall prove in section 3 that if f˜ ∈ L1 (T), for some f ∈ L1 (T), then f˜(reit ) is the Poisson integral of f˜. From that we can deduce that if ˜ ] so that if S[f ˜ ] is not a Fourier f˜ ∈ L1 (T) then its Fourier series is S[f 1 ˜ series then f 6∈ L (T). The difficulty in asserting immediately that f˜(reit ) is the Poisson integral of f˜ stems from the fact that we have only established pointwise convergence almost everywhere of f˜(reit ) to f˜(eit ) and this type of convergence is not sufficient to imply convergence of integrals. 1.5 We denote the (Lebesgue) measure of a measurable set E ⊂ T by |E|. D EFINITION : The distribution function of a measurable, real-valued function f on T is the function m(x) = mf (x) = |{t : f (t) ≤ x}|,

−∞ < x < ∞.

74


Distribution functions are clearly continuous to the right and monotone, increasing from zero at x = −∞ to 2π as x → ∞. The basic property of distribution functions is: for every continuous function F on R Z Z (1.3) F (f (t))dt = F (x)dmf (x). T

D EFINITION : A measurable function f is of weak Lp type, 0 0 mf (λ) ≥ 2π − Cλ−p

(1.4)

(or equivalently, |{t : |f (t)| ≥ λ}| ≤ Cλ−p ). Every f ∈ Lp (T) is clearly of weak Lp type. In fact, for all λ > 0 Z ∞ Z ∞ 1 1 p p kf kLp = x dm|f | (x) ≥ xp dm|f | (x) 2π 0 2π λ Z 1 p ∞ λp ≥ dm|f | (x) = λ (2π − m|f | (λ)) 2π 2π λ hence (1.4) is satisfied with C = 2πkf kpLp . It is equally clear that there are functions of weak Lp type which are not in Lp (T); |sin t|−1/p is a simple example. 0

Lemma. lf f is of weak Lp type then f ∈ Lp (T) for every p0 < p.

P ROOF : Z Z 0 |f |p dt =

∞

0

Z

∞

xp dm|f | (x) = 1 0 Z ∞ 0 p0 ∞ = m|f | (1) − [x (2π − m|f | (x))]1 + (2π − m|f | (x))d(xp ) 1 Z ∞ Z ∞ 0 −p p0 xp −p−1 dx < ∞. x d(x ) = 2π + C ≤ 2π + C J 1 1 xp dm|f | (x) ≤ m|f | (1) +

1.6 Theorem. lf f ∈ L1 (T) then f˜ is of weak L1 type.

P ROOF : We assume first that f > 0; also, we normalize f by assuming kf kL1 = 1. We want to evaluate the measure of the set of points where z−iλ 1 1 ˜ |f | > λ. The function Hλ (z) = 1 + π arg z+iλ = 1 + π = log z−iλ is z+iλ clearly harmonic and nonnegative in the half plane 0, and its level lines are circular arcs passing through the points iλ and −iλ. The


75

level line Hλ (z) = 12 is the half circle z = λeiϑ , −π/2 < ϑ < π/2, hence if |z| > λ then Hλ (z) > 12 . Also it is clear that Hλ (1) = 1 − (2π) arctan λ < 2/πλ.

Now Hλ (f (z) + if˜(z)) is a well-defined positive harmonic function in D, hence Z 1 2 (1.5) Hλ f (reit ) + if˜(reit ) dt = Hλ (f (0)) = Hλ (1) < , 2π πλ and remembering that Hλ (f + if˜) ≥

1 2

if |f + if˜| > λ, we obtain,

|{t : |f˜(reit )| > λ}| ≤

8 . λ

Since the mapping f 7→ f˜ is linear it is clear that if we omit the normalization kf kL1 = 1 we obtain, letting r → 1, that for f ≥ 0 in L1 (T) |{t : |f˜(eit )| > λ}| ≤ 8kf kL1 λ−1 .

Every f ∈ L1 (T) can be written as f = f1 − f2 + if3 − if4 where fj ≥ 0 and kfj kL1 ≤ kf kL1 . We have f˜ = f˜1 − f˜2 + if˜3 − if˜4 and consequently {t : |f˜(eit )| > λ} ⊂

4 [

{t : |f˜j (eit )| > λ/4}.

j=1

It follows that for c = 128 and every f ∈ L1 (T) (1.6)

|{t : |f˜(reit )| > λ}| ≤ ckf kL1 λ−1 .

J

Corollary. If f ∈ L1 (T) then f ∈ Lα (T) for all α < 1.

P ROOF : Lemma 1.5.

J

1.7 The method of proof of Theorem 1.6 can be used for bounded functions as well. Theorem. lf f is real valued and |f | ≤ 1, then for 0 ≤ α < π/2 Z 1 2 ˜ it (1.7) eα|f (e )| dt ≤ . 2π cos α

76


P ROOF : Put F (z) = f˜(z) − if (z). Since cos(αf (z)) ≥ cos α, we have ˜

2− 2 λ

Now, by (1.7) with α =

√

or

2,

1 {t : |f˜j | > 2− 2 λ}
2− 2 λ

4π −λ √ e , cos 2

j = 1, 2 J

?1.8 We shall see in chapter VI that a finite Borel measure on R is completely determined by its Fourier-Stieltjes transform (just as measures on T are determined by their Fourier-Stieltjes coefficients). This means that two distribution functions, 2 (x), of real-valued funcR iξx m1 (x) andRmiξx tions on T are equal if e dm1 (x) = e dm2 (x) for all ξ ∈ R. Using this remark we shall show now that if f is the indicator function of some set U ⊂ T, then mf˜(λ) depends only on the measure of U and not on the particular structure of U . Thus we can compute mf˜(λ) explicitly by replacing U by an interval of the same measure.

Theorem. Let U ⊂ T be a set of measure 2α. Let f be the indicator function of U and let χα be the indicator function of (−α, α). Write mα (λ) = mχ˜α (λ). Then mf˜(λ) = mα (λ).


77

P ROOF : Apply Cauchy’s formula on z = reit to the analytic functions ˜ F ξ (z) = eξ(f (z)+if (z)) , let r → 1 and remember that f = 0 on T \ U and f = 1 on U ; this gives Z Z ˜ it iξ f˜(eit ) ξ (1.9) e dt + e eiξf (e ) dt = 2πF ξ (0) = 2πeξα/π . U

T\U

Rewriting (1.9) for −ξ instead of ξ and then taking complex conjugates, we obtain Z Z ˜ it iξ f˜(eit ) −ξ (1.10) e dt + e eiξf (e ) dt = 2πe−ξα/π . U

T\U

From (1.9) and (1.10) we obtain Z

˜

it

eiξf (e ) dt = 2π

U

(1.11) Z

iξ f˜(eit )

e

T\U

sinh ξα π sinh ξ

sinh ξ 1 − dt = 2π sinh ξ

α π

.

We write now mf˜(λ) = n1 (λ) + n2 (λ) where n1 (λ) = |U ∩ {t : f˜(eit ) ≤ λ}|

and

n2 (λ) = |(T \ U ) ∩ {t : f˜(eit ) ≤ λ}|

and we can rewrite (1.11) as Z

eiξx dn1 (x) = 2π

sinh ξα π sinh ξ

Z

eiξx dn2 (x) = 2π

sinh ξ 1 − sinh ξ

(1.12)

α π

.

We see that n1 (x) and n2 (x) are uniquely determined by α and so they are the same for f˜ and χ˜α We thus obtain that f˜ and χ˜α have the same distribution of values not only on T but also on U for f˜ and (−α, α) for χ ˜α . J The Fourier series of χa is ∞ X sin nα −∞

πn

∞

eint =

X sin nα α +2 cos nt π πn 1

78


hence χ ˜α (reit ) =2

∞ X

∞

rn

1

X cos n(t − α) − cos n(t + α) sin nα sin nt = rn πn πn 1

∞

∞

1 X 1 n in(t−α) X 1 n in(t+α) = < r e − r e π n n 1 1 reit − eiα 1 = log it π re − e−iα

and finally χ ˜α (eit ) =

(1.13)

eit − eiα 1 1 1 − cos(t − α) log it log . = −iα π e −e 2π 1 − cos(t + α)

It follows from (1.13) that for λ > 1 the set {t : χ˜α (eit ) > λ} is an interval containing t = −α and contained in (−α − β1 , −α + β2 ) where β2 β1 = = e−πα , hence 2α + β 2α − β 1

(1.14)

2

mα (λ) ≥ 2π − 5αe−πλ .

Corollary. Let f be the indicator function of a set U of measure 2α on T. Then, for λ > 1 {t : |f˜(eit )| > λ} < 10αe−πλ . (1.15)

1.9 Returning to L1 (T), we use Theorem 1.6 and the fact that conjugation is an operator of norm 1 on L2 (T) to obtain the following theorem. The method applies in a general context which we discuss briefly in the following subsection. Theorem. If † f log+ |f | ∈ L1 (T), then f˜ ∈ L1 (T).

P ROOF : We shall use the fact that for g ∈ L2 (T) we have g˜ ∈ L2 (T) and k˜ g kL2 ≤ kgkL2 . This is an immediate corollary of Theorem I.5.5. As we have seen in 1.5, this implies m|˜g| (λ) ≥ 2π(1 − kgk2L2 λ−1 ). R∞ We have to prove that 1 λdm|f˜| (λ) < ∞ which is the same thing as RR λdm|f˜| (λ) = O (1) as R → ∞. Integrating by parts and remembering 1

(1.16)

† log+

x = sup(log x, 0) for x ≥ 0.


79

(1.6) we see that the theorem is equivalent to R

Z

(1.17)

2π − m|f˜| (λ) dλ = O (1)

1

as R → ∞.

In order to estimate 2π − m|f˜| (λ) we write f = g + h, where g = f when ˜ and consequently |f | ≤ λ and h = f when |f | > λ. We have f˜ = g˜ + h ˜ {t : |f˜(t)| > λ} ⊂ {t : |˜ g (t)| > λ/2} ∪ {t : |h(t)| > λ/2}.

(1.18) By (1.16) (1.19)

|{t : |˜ g (t)| > λ/2}| ≤ 8πλ−2 kgk2L2 = 8πλ−2

Z 0

λ

x2 dm|f˜|

and by (1.6) Z

˜ |{t : |h(t)| > λ/2}| ≤ 2cλ−1 khkL1 = 2cλ−1

∞

λ

xdm|f˜| ;

1

1

for x ≥ λ, (log x) 2 ≥ (log λ) 2 and we obtain 2c ˜ |{t : |h(t)| > λ/2}| ≤ √ λ log λ

(1.20)

Z

∞

λ

p x log xdm|f˜|

By (1.18), (1.19), and (1.20) we have 2π − m|f˜| (λ) ≤ 8πλ−2

Z

λ

0

2c x2 dm|f˜| + √ λ log λ

Z

∞

λ

p x log xdm|f˜| .

Thus (1.17), and hence the theorem, will follow if we show that as R → ∞, Z R Z λ −2 x2 dm|f˜| dλ = O (1) λ 0

1

(1.21)

Z 1

R

1 λ log λ √

Z

∞

λ

p x log xdm|f˜| dλ = O (1)

The information that we have concerning m|f˜| is that it is a monotonic function tending to 2π at infinity and such that Z ∞ (1.22) x log xdm|f˜| < ∞ 1

80


In order to derive (1.21) from (1.22) we apply Fubini’s theorem. The domain for the first integral is the trapezoid {(x, λ) : 1 < λ < R, 0 < x < λ}

and integrating first with respect to λ we obtain Z 1 Z R Z R Z λ 1 2 1 1 2 2 −2 1− x dm|f˜| dλ = λ x dm|f˜| + − x dm|f˜| R x R 0 0 1 1 Z R xdm|f˜| = O (1) ≤2π + 1

The domain for the second integral is the strip {(x, λ) : 1 < λ < R, λ < x}.

Integrating first with respect to λ we obtain Z

R

1

Z ∞ p 1 x log xdm|f˜| dλ = λ log λ λ Z Z R p x log xdm|f˜| + 2 log R 2 √

∞

R

1

p x log xdm|f˜| = O (1)

and the proof is complete.

J

1.10 When the underlying measure space is infinite, e.g. the line R rather than T, we can use µ({x : f (x)} > λ) instead of the distribution function. For postive integrable functions it gives the complete information about the distribution of f . A slightly coarser gauge, which is often more transparent and easier than the distribution function to work with, even when the underlying measure is finite, is the “lumping” of dmf , defined (for arbitrary measure space {X, B, µ}, finite or infinite), as follows:

For a measurable real-valued f and n ∈ Z set

D EFINITION :

mn = mn (f ) = µ({x : 2n−1 < |f (x)| ≤ 2n }).

Observe that:

(1.23)

a) f is of weak type p if, and only if, mn (f ) = O (2−np ), P∞ np b) f ∈ Lp if and only if −∞ 2 mn (f ) < ∞, in fact kf kpLp ≤

∞ X −∞

2np mn (f ) ≤ 2p kf kpLp .


81

1.11 What we have done in 1.9 is interpolate, using what we know about the properties of an operator (the conjugation operator f 7→ f˜) on L1 (T) and on L2 (T), to prove that it maps the intermediate space L log L(T) into L1 (T). The same method can be used to prove M. Riesz’ theorem below. We use the parameters mn rather than the distribution functions, and the reader should compare the first proof below to that of 1.9. Riesz’ original proof is given as second proof. Theorem (M. Riesz). For 1 < p < ∞, the mapping f 7→ f˜ is a bounded linear operator on Lp (T).

We have mentioned already that for p = 2 the theorem is obvious (from Theorem I.5.5). From Parseval’s formula (I.7.1) it follows that if p and q are conjugate exponents, the mappings f 7→ f˜ in Lp (T) and in Lq (T) are, except for a sign, each other’s adjoints and consequently if one is bounded, so is the other and by the same bound. Thus it is enough to prove the theorem for 1 < p < 2.

F IRST P ROOF : Assume 1 < p < 2. We need to show that there exists a constant Cp such that if f ∈ Lp (T) then f˜ ∈ Lp (T), and kf˜kp ≤ Cp kf kp . P np ˜ ˜ Since kf˜kpLp ≤ ∞ −∞ 2 mn (f ), we estimate mn (f ). Given n, we write f = f0,n + f1,n where f0,n (t) = f (t) if |f (t)| ≥ 2n (and is zero elsewhere) and f1,n (t) = f (t) if |f (t)| < 2n , (and is zero elsewhere). Since 1 2n implies at least one of the inequlities |f˜0,n (t)| > 2n−1 or |f˜1,n (t)| > 2n−1 , so that (1.25) mn+1 (f˜) ≤ µ({t : |f˜0,n (t)| > 2n−1 }) + µ({t : |f˜1,n (t)| > 2n−1 }) By 1.6 we have (1.26)

µ({t : |f˜0,n (t)| > 2n−1 }) ≤

∞ X kf0,n kL1 −n ≤ c 2 mj (f )2j , 1 2n−1 n+1

and since conjugation has norm 1 on L2 (T), n

(1.27)

µ({t : |f˜1,n (t)| > 2n−1 }) ≤

X kf1,n k2L2 ≤ c2 2−2n mj (f )22j . 2n−2 2

82


It follows that mn (f˜) ≤ c1 2−n

∞ X

mj (f )2j + c2 2−2n

n

and

kf˜kpLp ≤

∞ X

n X

mj (f )22j

2np mn (f˜)

−∞

∞ n X X 2pn 2−n mj (f )2j + 2−2n mj (f )22j

≤c3

X

=c3

X

n+1

(n−j)(p−1)

2

mj (f )2jp +

n≤j

X

2(p−2)(n−j) mj (f )2jp

n≥j

the sums with respect to both n and j . Summing first with respect to n produces constants which depend only on p and we have X kf˜kpLp ≤ cp mj (f )2jp ≤ C(p)kf kpLp . j J

S ECOND P ROOF : Let f ∈ Lp (T), f ≥ 0. Let f (reit ) be its Poisson integral, f˜(reit ) the harmonic conjugate, and H(reit ) = f (reit ) + if˜(reit ). We may clearly assume that f does not vanish identically, and, since f ≥ 0, it follows that f (reit ) > 0, hence H(reit ) 6= 0 in D. Let G(reit ) be the branch of (H(reit ))p which is real at r = 0. Let γ be a real number satisfying (1.28)

γ

π , 2

For 0 < r < 1 we have Z Z Z 1 1 1 |G(reit )|dt = |G(reit )|dt + |G(reit )|dt, 2π 2π I 2π II R R where I is taken over the set where |arg(H(z))| < γ and II is taken over the complementary setR (defined by the condition γ ≤ |arg(H(z))| < π/2, where z = reit ). In I we have |H(z)| < f (z)(cos γ)−1 ,

hence (1.29)

1 2π

Z I

|G(reit )|dt ≤ (cos γ)−p kf kplp


83

and, in particular, (1.30)

1 2π

Z

0,

{t : |f˜(eit )| > λ} < cp kf kpLp λ−p .

Remark: This is an immediate consequence of 1.11; try, however, to prove it by using 1.8. Hint: Assume that f is real valued. Denote Uλ = {t : f˜(eit ) > λ} and Vλ = {t : f˜(eit ) < −λ}. Denote by gλ the indicator function of Uλ ; deduce from Parseval’s formula that (with q = p/(p − 1)) λ|Uλ | ≤

Z

f˜(eit )gλ (eit )dt = −

Z

f (eit )geλ (eit )dt ≤ 2πkf kLp kgeλ kLq

and use (1.15) to evaluate kgeλ kLq . Repeat for Vλ .

2 THE MAXIMAL FUNCTION OF HARDY AND LITTLEWOOD

2.1 D EFINITION : the function

(2.1)

The maximal function of a function f ∈ L1 (T) is

1 Z t+h f (τ )dτ . Mf (t) = sup0 Iα . 4 α n=1


85

P ROOF : Denote a1 = supI∈Ω |I| and let I1 be any interval of Ω satisfying |I1 | > 34 a1 ; let Ω2 be the subfamily of all the intervals in Ω which do not intersect I1 . Denote a2 = supI∈Ω2 |I| and let I2 ∈ Ω2 be such that |I2 | > 34 a2 . We continue by induction; having picked I1 , . . . , Ik we consider the family Ωk+1 of the intervals of Ω which intersect none of Ij , j ≤ k, and pick Ik+1 ∈ Ωk+1 such that |Ik+1 | > 43 ak+1 where ak+1 = supI∈Ωk+1 |I|. We claim that the sequence {In } so obtained satisfies (2.2). In fact, denoting by Jn the interval of length 4|In | of which S S In is the center part, we claim that Jn ⊃ Iα which clearly implies (2.2). We notice first that ak → 0 and consequently ∩Ωk = ∅. For I ∈ Ω let k denote the first index such that I 6∈ Ωk ; then I ∩ Ik−1 6= ∅ and, since |Ik−1 | ≥ 43 |I|, I ⊂ Jk−1 and the lemma is proved. J 2.2 Theorem. For f ∈ L1 (T), Mf is of weak L1 type.

P ROOF : Since Mf (t) ≤ M|f | (t), we may assume that f ≥ 0. Let λ > 0; if Mf (t) > λ let It be an interval centered at t such that Z (2.3) f (t)dt > λ|It |. It

Thus we cover the set {t : Mf (t) > λ} by a family of intervals {It }. Let {In } be a pairwise disjoint subsequence of {It } satisfying (2.2). Then, by (2.2) and (2.3), R R S S (2.4) |{t : Mf (t) > λ}| ≤ | It | ≤ 4| In | ≤ λ4 ∪In f (t)dt ≤ λ4 T f (t)dt. J 2.3 The maximal function of a bounded function is clearly bounded by the same bound so that the map f 7→ Mf has norm 1 in L∞ (T). The map is subliniear† rather then linear, but we can still interpolate between L1 (T) and L∞ (T). Lemma. Let f ∈ L1 (T) and let m(λ) = m|f | (λ) be the distribution function of |f |. Then Z 4 ∞ (2.5) |{t : Mf (t) > 2λ}| ≤ ydm(y). λ λ † An operator S is sublinear if S(f + f ) is defined whenever Sf and Sf are both 1 2 1 2 defined, and if |S(f1 + f2 )| ≤ |Sf1 | + |Sf2 | a.e.

86


P ROOF : Write f = g + h where g = f when |f | ≤ λ and h = f when |f | > λ. We have Mf (t) ≤ Mg (t) + Mh (t) ≤ λ + Mh (t); hence, by (2.4), |{t : Mf (t) > 2λ}| ≤ |{t : Mh (t) > λ}| ≤

Z

4 λ

|h(τ )|dτ ≤

4 λ

Z

∞

ydm(y).

λ

J

In terms of the “lumped” distribution this says, (with n shifted to n − 2), X X (2.6) mn (Mf ) ≤ 4·22−n 2j mj (f ) = 16 2(j−n) mj (f ) j≥n−1

j≥n−1

Theorem. (a) For 1 < p < ∞ there exists a constant cp such that if f ∈ Lp (T), then Mf ∈ Lp (T) and kMf kLp ≤ cp kf kLp . (b) If f log+ |f | ∈ L1 (T) then Mf ∈ L1 (T) and Z kMf kL1 ≤ 2 + 4 |f | log+ |f |dt. T

P ROOF : (a) If f ∈ Lp (T) then kMf kpLp ≤ = 16

X

X 2

P

j

2jp mj (f ) ≤ 2p kf kpLp . By (2.6),

2np mn (Mf ) ≤ 16

X

2np+(j−n) mj (f )

j+1≥n

(n−j)(p−1)

jp

·2 mj (f ) = 16

j+1≥n

X

2(p−1)n

≤ 16

2jp mj (f )

j

n≤1

X

X

(p−1)n

2

·kf kpLp

= cpp kf kpLp .

n≤1

(b) If f log+ |f | ∈ L1 (T) then kMf kL1 ≤ 1 +

X

= 1 + 16

X

j>0 1≤n≤j+1

by (2.6).

j>0

R 1 jmj (f ) ≤ log 2· 2π |f | log+ |f |.

2n mn (Mf ) ≤ 1 + 16

n≥1

X

P

X

2n+(j−n) mj (f )

j+1≥n≥1 j

2 mj (f ) = 1 + 16

X

(j + 1)2j mj (f ).

j>0

J

The use of the “lumped distribution” necessarily gives somewhat worse constants than the same proof done with the distribution functions. Here is the proof done “properly”.


87

P ROOF : Denote by m(λ) and n(λ) the distribution functions of |f | and Mf respectively. We can rewrite (2.5) in the form (2.7)

2π − n(2λ) ≤

4 λ

Z

∞

y dm(y) ≤

λ

4 λp

Z

∞

y p dm(y);

λ

hence if f ∈ Lp (T), 1 ≤ Rp < ∞, we have λRp (2π − n(λ)) → 0 as λ → ∞. ∞ p 2p ∞ p 1 We have kMf kpLp = 2π λ dn(λ) = 2π λ dn(2λ); integrating by 0 0 parts we obtain (1 ≤ p < ∞) Z ∞ Z ∞ λp dn(2λ) = [λp (2π − n(2λ))]∞ (2π − n(2λ))pλp−1 dλ + 0 0 0 ( R∞ R∞ 8p 0 λp−2 λ ydm(y)dλ if p > 1 ≤ R ∞ −1 R ∞ ydm(y)dλ if p = 1 2π + 8 1 λ λ and integrating by parts again we finally obtain: Z ∞ Z ∞ 8p 8p (for p>1) λp dn(2λ) ≤ λp dm(λ) = 2πkf kpLp ; p−1 0 p−1 0 Z

(for p=1)

0

∞

λdn(2λ) ≤ 2π + 8

Z

∞

λ log λdm(λ) Z = 2π + 8 |f | log+ |f |dt. 1

T

J

2.4 Lemma. Let k be a nonnegative even R π function on (−π, π), monotone nonincreasing on (0, π), such that −π k(t)dt = 1. Then for all f ∈ L1 (T) Z (2.8) k(t − τ )f (τ )dτ ≤ Mf (t).

P ROOF : The definition (2.1) is equivalent to Z Mf (t) = sup0 0 and α > 0 |f (t + h) − f (t)| ≤ K|h|α

for all t ∈ E ,

then the integrals (2.21) are uniformly bounded and 1 2π

Z ε

π

τ f (t − τ ) − f (t + τ ) cot dτ → f˜(eit ) 2

uniformly in t ∈ E as ε → 0. It follows, reexamining the proof of 2.7, that f˜(reit ) → f˜(eit ) uniformly for t ∈ E as r → 1. In particular, if E is an interval, it follows that f˜(eit ) is continuous on E . 2.10 Conjugation is not a local operation; that is, it is not true that if f (t) = g(t) in some interval I , then f˜(t) = g˜(t) on I , or equivalently, that if f (t) = 0 on I , then f˜(t) = 0 on I . However, Theorem. lf f (t) = 0 on an interval I , then f˜(t) is analytic on I .

P ROOF : By the previous remarks f is continuous on I . Thus the function F = f + if˜ is analytic in D and is continuous and purely imaginary on I . By Schwarz’s reflection principle F admits an analytic extension through I , and since F (eit ) = if˜(eit ) on I , the theorem follows. J

Remark: Using (2.21) we can estimate the successive derivatives of f at points t ∈ I and show that f is analytic on I without the use of the "complex" reflection principle.

EXERCISES FOR SECTION 2 The first three exercises were covered already in Theorem I.8.4. The main point here is the localization (exercise 4). 1. Assume f ∈ Lipα (T), 0 < α ≤ 1. Show that f˜ ∈ Lipα0 (T) for all α0 < α. 2. Assume f ∈ C n (T), n ≥ 1. Show that f˜ ∈ C n−1 (T) and f˜(n−1) ∈ Lipα (T) for all α < 1. 3. Assume f ∈ Lipα (T), 0 < α < 1. Show that f˜ ∈ Lipα (T)


93

Hint: 1 f˜(t + h) − f˜(t) = 2π

Z

f (t + h − τ ) − f (t + h) cot

1 − 2π

Z

α

=O (h ) + −

Z

f (t − τ ) − f (t) cot

Z

τ dτ 2

2π−2h

f (t − τ ) − f (t + h) cot

2h 2π−2h

f (t − τ ) − f (t) cot

2h α

=O (h ) +

Z

τ dτ 2

τ dτ 2

2π−2h

f (t − τ ) − f (t)

2h

− f (t + h) − f (t)

Z

2π−2h

2h

cot

τ +h dτ − 2

cot

τ +h τ − cot dτ 2 2

τ +h dτ. 2

4. Localize exercises 1-3, that is, assume that f satisfies the respective conditions on an an interval I ⊂ T and show that the conclusions hold in I . 3 THE HARDY SPACES

In this section we study some spaces of functions holomorphic in the unit disc D. These spaces are closely related to spaces of functions on T and we obtain, for example, a characterization of Lp functions and of measures whose Fourier coefficients vanish for negative values of n. We also prove that if for some f ∈ L1 (T), f˜(eit ) is summahie ˜ ], and, finally, we obtain results concerning the absolute then S[f˜] = S[f convergence of some classes of Fourier series. We start with some preliminary remarks about products of Moebius functions. ¯

ζ(ζ−z) 3.1 Let 0 < |ζ| < 1; the function b(z, ζ) = |ζ|(1−z ¯ defines, as is ζ) well known, a conformal representation of D onto itself, taking ζ into zero and zero into |ζ|. The important thing for us now is that b(z, ζ) vanishes only at z = ζ and |b(z, ζ)| = 1 on |z| = 1. If 0 < |ζ| < r, then ¯ ζ(ζ−z) b zr , ζr = r |ζ|(r 2 ¯ is holomorphic in |z| < r , vanishes only at z = ζ , z ζ −zζ) and b , = 1 on |z| = r. For ζ = 0 we define b(z, 0) = z . r

r

Let f be holomorphic in |z| < r and denote its zeros there by ζ1 , . . . , ζk (counting each zero as many times as its multiplicity). The Q −1 function f1 (z) = f (z) b zr , ζrn is holomorphic in |z| < r, is zero-free and satisfies |f1 (z)| = |f (z)| for |z| = r. Since log|f1 (z)| is

94


harmonic in |z| < r we have log|f1 (0)| =

1 2π

Z

log|f1 (reit )|dt

and if we assume, for simplicity, that f (0) 6= 0, the formula above is equivalent to Poisson-Jensen’s formula: (3.1)

log|f (0)| + log

k Y

r|ζn |−1 =

n=1

1 2π

Z

log|f (reit )|dt.

We implicitly assumed that f has no zeros of modulus r; however, since both sides of the formula depend continuously on r, the above is valid even if f vanishes on |z| = r. The reader should check the form that Poisson-Jensen’s formula takes when f vanishes at z = 0. Q The term log kn=1 r|ζn |−1 is positive, and removing it from (3.1) we obtain Jensen’s inequality Z 1 (3.2) log|f (0)| ≤ log|f (reit )|dt. 2π or, if f has a zero of order s at z = 0, log| lim z z→0

−s

1 f (z)| + log(r ) ≤ 2π s

Z

log|f (reit )|dt.

Another form of Jensen’s inequality is: let f be holomorphic in 0 |z| < r and let ζ10 , . . . , ζm , be (some) zeros of f in |z| < r, counted each one at most as many times as its multiplicity. Then Z m X 1 (3.3) log|f (0)| + log|f (reit )|dt. log(r|ζn0 |−1 ) ≤ 2π 1 Inequality (3.3) is obtained from (3.1) by deleting some (positive) terms of the form log(r|ζn |−1 ) from the left-hand side. 3.2 Let p > 0 and let f be holomorphic in D. We introduce the notation Z 1 (3.4) hp (f, r) = |f (reit )|p dt. 2π

If 0 < r 1 we have hp (f, rρ) = kf (rρeit )kpLp ≤ kf (reit )kpLp = hp (f, r)


95

or, in other words, hp (f, r) is a monotone nondecreasing function of r. P The case p = 2 is particularly obvious since for f (z) = an z n we have P h2 (f, r) = |an |2 r2n . We show now that the same is true for all p > 0. Lemma. Let f be holomorphic in D and p > 0. Then hp (f, r) is a monotone nondecreasing function of r.

P ROOF : We reduce the case of an arbitrary positive p to the case p = 2. Let r1 < r < 1. Assume first that f has no zeros on |z| ≤ r and consider the function† g(z) = (f (z))p/2 ; then Z Z Z 1 1 1 |f (r1 eit )|p dt = |g(r1 eit )|2 dt ≤ |g(reit )|2 dt 2π 2π 2π Z 1 = |f (reit )|p dt 2π or hp (f, r1 ) ≤ hp (f, r). If f has zeros inside |z| < r but not on |z| = r, we denote the zeros, repeating each according to its multiplicity, by ζ1 , . . . , ζk , and write k Y z ζn −1 f1 (z) = f (z) b , r r 1

For |z| < r we have |f (z)| < |f1 (z)|, for |z| = r we have |f (z)| = |f1 (z)| and f1 is zero-free in |z| ≤ r. It follows that hp (f, r1 ) < hp (f1 , r1 ) ≤ hp (f1 , r) = hp (f, r).

Since hp (f, r) is a continuous function of r, the same is true even if f does have zeros on |z| = r, and the lemma is proved, J 3.3 Lemma. Let {ζn } be a sequence of complex numbers satisfying P |ζn | < 1 and (1 − |ζn |) < ∞. Then the product

(3.5)

B(z) =

∞ Y 1

b(z, ζn ) = z m

ζ¯n (ζn − z) |ζ |(1 − z ζ¯n ) 6 0 n =

Y

ζn

converges absolutely and uniformly in every disc Dr = {z : |z| ≤ r}, r < 1. † Any

branch of (f (z))p/2 .

96


P ROOF : It is sufficient to show that formly in |z| ≤ r < 1. But

P

1−

ζ¯n (ζn −z) |ζn |(1−z ζ¯n )

converges uni-

ζ¯n (ζn − z) |ζn | + zζn 1 − |ζn | 1 + r (1 − |ζn |) 1 − = ≤ 1−r |ζn |(1 − z ζ¯n ) |ζn |(1 − zζn ) P and the series converges (1 − |ζn |) < ∞. J

The product (3.5), often called the Blaschke product corresponding to {ζn }, is clearly holomorphic in D and it vanishes precisely at the points ζn . Nothing prevents, of course, repeating the same complex number a (finite) number of times in {ζn }, so that we can prescribe not only the zeros but their multiplicities as well. Since all the terms in (3.5) are bounded by 1 in modulus, we have |B(z)| < 1 in D. 3.4 We now introduce the spaces H p (H for Hardy) and N (N for Nevanlinna).

D EFINITION : The space H p , p > 0, is the (linear) space of all functions f holomorphic in D, such that (3.6)

kf kpH p = lim hp (f, r) = sup0 0, let f = BF be the canonical factorization of f , and write G(z) = (F (z))p/2 . Then G belongs to H 2 and consequently G(reit ) → G(eit ) for almost all t ∈ T; at every such t, F (reit ) converges to some F (eit ) such that |F (eit )|p/2 = |G(eit )|. Since B has radial limit of absolute value one almost everywhere we see that it f (eit ) = lim f (reit ) exists and |f (eit )|p/2 )| almostR everywhere. R = |G(e p p 1 1 2 it 2 Now kf kH p = kF kH p = kGkH 2 = 2π |G(e )| dt = 2π |f (eit )|p and the proof is complete. J


99

3.9 The convergence assured by Theorem 3.8 is pointwise convergence almost everywhere. For p ≥ 2 we know that f is the Poisson integral of f (eit ) and consequently f (reit ) converges to f (eit ) in the Lp (T) norm. We shall show that the same holds for p = 1 (hence for p ≥ 1); first, however, we use the case p ≥ 2 to prove: 0

Theorem. Let 0 < p < p0 and suppose f ∈ H p and f (eit ) ∈ Lp (T). 0 Then f ∈ H p .

P ROOF : As before, if we write f = BF , G(z) = (F (z))p/2 , then G ∈ 0 H 2 and G(eit ) ∈ L2p /p (T). G is the Poisson integral of G(eit ) and 0 0 0 consequently G ∈ H 2p /p which means F ∈ H p , hence f ∈ H p . J Corollary. Let f ∈ L1 (T) and assume f˜ ∈ L1 (T): then (f + if˜) ∈ H 1 .

P ROOF : We know (Corollary 1.6) that (f + if˜) ∈ H p for all p < 1 and by the assumption (f + if˜)(eit ) ∈ L1 (T). J 3.10 Theorem. Every function f in H 1 can be factored as f = f1 f2 with f1 , f2 ∈ H 2 .

P ROOF : Let f = BF be the canonical factorization of f . We can take f1 = F 1/2 , f2 = BF 1/2 . J 3.11 We can now prove: Theorem. Let f ∈ H 1 and let f (eit ) be its boundary value. Then f is the Poisson integral of f (eit ).

P ROOF : We prove the theorem by showing that f (reit ) converges to P f (eit ) in the L1 norm. This implies that if f (z) = an z n , then an are the Fourier coefficients of f (eit ) which is clearly equivalent to f being the Poisson integral of f (eit ). Write f = f1 f2 with fj ∈ H 2 , j = 1, 2. f (reit ) − f (eit ) = f1 (reit )f2 (reit ) − f1 (eit )f2 (eit );

adding and subtracting f1 (eit )f2 (reit ) and using the Cauchy-Schwarz inequality, we obtain kf (reit ) − f (eit )kL1 ≤ kf2 kL2 kf1 (reit ) − f1 (eit )kL2 + kf1 kL2 kf2 (reit ) − f2 (eit )kL2

As r → 1, kfj (reit ) − fj (eit )kL2 → 0, and the proof is complete.

J

100


Remark: See exercise 2 at the end of the section for an extension of the theorem to the case 0 < p < 1. ˜ ]. Corollary. Let f ∈ L1 (T) and f˜ ∈ L1 (T). Then S[f˜] = S[f

P ROOF : From 3.9 and the theorem above follows that f˜(reit ) is the Poisson integral of f˜(eit ) (see Remark 1.4). J 3.12 Theorem. Assume p ≥ 1. A function f belongs to H p if, and only if, it is the Poisson integral of some f (eit ) ∈ Lp (T) satisfying

(3.8)

fˆ(n) = 0

for all n < 0.

P ROOF : Let f ∈ H p ; by Theorem 3.11, f is the Poisson integral of f (eit ) and (3.8) is clearly satisfied. On the other hand, let f ∈ Lp (T) and assume (3.8); the Poisson integral of f , f (reit ) =

∞ X

fˆ(n)rn eint =

0

∞ X

fˆ(n)z n

0

it

is holomorphic in D, and since kf (re )kLp ≤ kf (eit )kLp , it follows that f (z) ∈ H p . J 3.13 For p > 1, we can prove that every f ∈ H p is the Poisson integral of f (eit ) without appeal to Theorem 3.11 or any other result obtained in this section. We just repeat the proof of Lemma 1.2 (which is the case p = ∞ of 3.12): if f ∈ Lp (R), kf (reit )kp is bounded as r → 1; we can pick a sequence rn → 1 such that fn (eit ) = f (rn eit ) converge weakly in Lp (T) to some f (eit ). Since weak convergence in Lp (T) implies convergenee of Fourier coefficients, it is clear that (3.8) is satisfied and that the function f with which we started is the Poisson integral of f (eit ). For p = 1 the proof as given above is insufficient. L1 (T) is a subspace of M (T), the space of Borel measures on T, which is the dual of C(T), and the argument above can be used to show that every f ∈ H 1 is the Poisson integral of some measure µ on T. This measure has the property

(3.9)

µ ˆ(n) = 0,

for all n < 0.

All that we have to do in order to complete the (alternative) proof of Theorem 3.12 in the case p = 1 is to prove that the measures satisfying (3.9), often called analytic measures, are absolutely continuous with respect to the Lebesgue measure on T.


101

Theorem (F. and M. Riesz). Let µ be a Borel measure on T satisfying

(3.9)

µ ˆ(n) = 0,

for all n < 0.

Then µ is absolutely continuous with respect to Lebesgue measure. We first prove: Lemma. Let E ⊂ T be a closed set of measure zero. There exists a ¯ such that: function ϕ holomorphic in D and continuous in D

(3.10)

(i) ϕ(eit ) = 1

on E , ¯ \ E. (ii) |ϕ(e )| < 1 on D it

P ROOF : Since E is closed and of measure zero we can construct a function ψ on T such that ψ(eit ) > 0 everywhere, ψ(eit ) is continuously differentiable in each component of T \ E , ψ(eit ) → ∞ as t approaches E , and ψ(eit ) ∈ L2 (T). The Poisson integral ψ(z) of ψ(eit ) is positive on D and ψ(z) → ∞ as z approaches E . The conjugate function is ¯ \ E (see the end of section 2) and consequently, if we continuous in D ˜ ψ(z)+iψ(z) put ϕ(z) = ψ(z)+iψ(z)+1 then ϕ is holomorphic in D and continuous in ˜ ¯ D \ E . At every point where ψ(z) < ∞ we have |ϕ(z)| < 1, and as ψ(z) → ∞, ϕ(z) → 1. If we define ϕ(z) = 1 on E then ϕ satisfies (3.10). J P ROOF OF THE THEOREM : Assume that µ satisfies the condition (3.9). We can assume µ ˆ(0) = 0 as well (otherwise consider µ − µ ˆ(0)dt) and it then follows from Parseval’s formula that Z ¯ (3.11) hf , µi = f dµ = 0 for every f ∈ C(T) which is the boundary value of a holomorphic function in D or, equivalently, such that fˆ(n) = 0 for all negative n. Let E ⊂ T be closed and of (Lebesgue) measure zero. Let ϕ be a function satisfying (3.10). Then, by (3.11) Z ϕm dµ = 0 for all m > 0 and by (3.10) lim

m→∞

Z

ϕm dµ = µ(E).

Thus µ(E) = 0 for every closed set E of Lebesgue measure zero and, since µ is regular, the theorem follows. J

102


3.14 Theorem 3.13 can be given a more complete form in view of the following important Theorem. Let f ∈ H p , p > 0; then log|f (eit )| ∈ L1 (T)

Remarks: The same conclusion holds under the weaker assumption f ∈ N . We state it for H p since we did not prove the existence of f (eit ) for f ∈ N (cf. [28], Vol. 1, p. 276). Since p log+ |f | < |f |p we already know that log+ |f (eit )| ∈ L1 (T). Thus the content of the theorem is that f (eit ) cannot be too small on a large set.

P ROOF : Replacing f by z −m f , if f has a zero of order m at z = 0, we may assume f (0) 6= 0. Let r < 1; then, by Jensen’s inequality Z 1 log|f (0)| − kf kN ≤ − log− |f (reit )|dt ≤ 0 2π (where log− x = − log x if x < 1 and zero otherwise). It follows that R |log|f (reit )||dt is bounded as r → 1 and the theorem follows from Fatou’s lemma. J Corollary. If f 6= 0 is in H p , f (eit ) can vanish only on a set of measure zero.

Combining Theorem 3.13 with our last corollary we obtain that analytic measures are equivalent to Lebesgue’s measure (i.e., they all have the same null sets). 3.15 Theorem. Let E be a closed proper subset of T. Any continuous function on E can be approximated uniformly by Taylor polynomials‡ .

P ROOF : We denote by C(E) the algebra of all continuous functions on E endowed with the supremum norm. The theorem claims that the restrictions to E of Taylor polynomials are dense in C(E). If a measure µ carried by E is orthogonal to all z n , n = 0, 1, . . . , it is analytic: hz n , µi = µ ˆ(n) = 0, and hence µ = f dt with f ∈ H 1 , f carried by E . By Theorem 3.14, f = 0. Hence there is no nontrivial functional on C(E), which is orthogonal to all Taylor polynomials, and the theorem follows from the Hahn-Banach theorem. J ‡ We

form

the term "Taylor polynomial" to designate trigonometric polynomials of the Puse N int 0

an e

.


103

3.16 We finish this section with another application of 3.10. P P∞ n 1 −1 Theorem (Hardy). Let f (z) = ∞ < ∞. 0 an z ∈ H . Then 1 |an |n

Remark: The theorem can also be stated: Let f ∈ L1 (T) satisfy (3.8), P ˆ −1 then ∞ ≤ kf kL1 . 1 |f (n)|n

P ROOF : If F (eit ) is a primitive of f (eit ) then F is continuous on T and consequently its Fourier series is Abel summable to F at every t ∈ T. P n In particular ∞ tends to a finite limit as r → 1. If we assume 1 (an /n)r P∞ an ≥ 0 for all n then 1 an /n is clearly convergent (compare with I.4.2). P In the general case we write f = f1 f2 with fj = Aj,n z n ∈ H 2 , P P ∗ N ∗ n ∗ ∗ ∗ j = 1, 2. Write fj (z) = |Aj,n |z , and f (z) = f1 (z)f2 (z) = an z . ∗ 2 ∗ 1 The functions fj are clearly in H , hence f ∈ H and, since a∗n ≥ 0, P it follows from the first part of the proof that (a∗n /n) < ∞. But n n X X |an | = A1,k A2,n−k ≤ |A1,k ||A2,n−k | = a∗n k=0

k=0

and the theorem follows.

J

3.17 Let f ∈ H 1 and assume that f (eit ) is of bounded variation on P∞ P∞ T. If f ∼ 0 an eint then 1 inan eint is the Fourier-Stieltjes series of df . Thus the measure df satisfies the condition of Theorem 3.13 and consequently df = f 0 dt and f 0 (z) is in H 1 . Combining this with 3.16 we obtain: Theorem. Let f ∈ H 1 and assume that f (eit ) is of bounded variation P ˆ on T. Then f (eit ) is absolutely continuous and ∞ −∞ |f (n)| < ∞.

An equivalent form of the theorem is (see 3.9): Theorem. Let f, f˜ ∈ L1 (T) and assume that both f and f˜ are of bounded variation. Then both f and f˜ are absolutely continuous. and P∞ ˆ −∞ |f (n)| < ∞.

EXERCISES FOR SECTION 3 1. Deduce Theorem 3.13 (F. and M. Riesz) from R Theorem 3.11. 2. Show that for all p > 0, if f ∈ H p , then |f (eit ) − f (reit )|p dt → 0 as r → 1.

104


Hint: Reduce the general case to the case in which f is zero-free. In the case 1 that f is zero-free, write f1 = f 2 ; then f1 ∈ H 2p and f (eit ) − f (reit ) = (f1 (eit ) − f1 (reit ))(f1 (eit ) + f1 (reit ));

hence show that if the statement is valid for 2p is also valid for p. Use the fact that it is valid for p ≥ 2. 3. Let E be a closed set of measure zero on T. Let ϕ be a continuous function on E . (a) Show that there exists a function Φ, holomorphic in D and continuous ¯ such that Φ(eit ) = ϕ(eit ) on E . on D (b) Show that Φ can be chosen satisfying the additional condition supz∈D¯ |Φ(z)| = supeit ∈E |ϕ(eit )|. Hint: Construct Φ by successive approximation using 3.15 and Lemma 3.13. 4. Let f ∈ L1 (T) be absolutely continuous and assume f 0 log+ |f 0 | ∈ L1 (T). P Prove that |fˆ(n)| < ∞.

Chapter IV

Interpolation of Linear Operators and the Theorem of Hausdorff-Young

Interpolation of norms and of linear operators is really a topic in functional analysis rather than harmonic analysis proper; but, though less so than ten years ago, it still seems esoteric among authors in functional analysis and we include a brief account. The interpolation theorems that are the most useful in Fourier analysis are the Riesz-Thorin theorem and the Marcinkiewicz theorem. We give a general description of the complex interpolation method and prove the Riesz-Thorin theorem in section 1. In the second section we use Riesz-Thorin to prove the Hausdorff-Young theorem. We do not discuss the Marcinkiewicz theorem although it appeared implicitly in the proof of theorem III.1.9. We refer the reader to Zygmund ([28] chap. XII) for a complete account of Marcinkiewicz’s theorem. 1 INTERPOLATION OF NORMS AND OF LINEAR OPERATORS

1.1 Let B be a normed linear space and let F be defined in some domain Ω in the complex plane, taking values in B . We say that F is holomorphic in Ω if, for every continuous linear functional µ on B , the numerical function h(z) = hF (z), µi is holomorphic in Ω. Assume now that B is a linear space with two norms k k0 and k k1 defined on it. We consider the family B of all B -valued functions which are holomorphic and bounded, with respect to both norms, in a neighborhood of the strip Ω = {z : 0 ≤ 0 whenever |f | > 0. † If

we write f = |f |eiϕ with real-valued ϕ, we may take g = min(1, |f |p0 )eiϕ .

108


Theorem. Let (X, dx) be a measure space, B = Lp0 ∩ Lp1 (dx) (with 1 ≤ p0 < p1 ≤ ∞). Denote by k kj , the norms induced by Lpj (dx), and by k kα the interpolating norms. Then k kα coincides with the norm induced on B by Lpα (dx) where p0 p1 p0 (1.6) pα = = if p1 = ∞ . p0 α + p1 (1 − α) 1−α

P ROOF : Let f ∈ B and kf kpα ≤ 1. Consider F (z) = |f |a(z+α)+1 eiϕ where f = |f |eiϕ and a=

p0 − p1 p0 α + p1 (1 − α)

=

−1 if p1 = ∞ . 1−α

We have F (α) = f and consequently kf kα ≤ kF k. Notice now that |F (iy)| = |f |1−aα = |f pα /p0 | so that Z 1/pα kF (it)k0 = |f |pα dx ≤ 1; similarly kF (1 + it)k1 ≤ 1 (use the same argument if p1 < ∞ and check directly if p1 = ∞); hence kf kα ≤ 1. This proves k kα ≤ k kLpα . In order to prove the reverse inequality, we denote by q0 , q1 the conjugate exponents of p0 and p1 and notice that the exponent conjugate to pα is q q0 q1 1 (1.6’) qα = = if q0 = ∞ q0 α + q1 (1 − α) α We now set B 0 = Lq0 ∩ Lq1 (dx) and denote by B0 the corresponding space of holomorphic B 0 -valued functions. Let f B and assume kf kLpα > 1; then, since B 0 Ris dense in Lqα , there exists a g ∈ B 0 such that kgkLqα ≤ 1 and such that f gdx > 1. As in the first part of this proof, there exists a function G ∈ B0 such that G(α) = g and kGk (with respect to q0 , q1 ) is bounded by 1. Let F ∈ B such that F (α) = f . The function h(z) = f F (z)G(z)dx (remember that for each z ∈ Ω, F (z) ∈ B and G(z) ∈ B 0 ) is holomorphic and bounded in Ω (see Appendix A). Now h(α) > 1, hence, by the Phragmèn-Lindelöff theorem, |h(z)| must exceed 1 on the boundary. However, on the boundary |h(z)| ≤ kF kkGk ≤ kF k so that kF k > 1. This proves kf kα ≥ 1 and it follows that k kα and k kLpα are identical. J 1.4 As a corollary to Theorems 1.2 and 1.3, we obtain the RieszThorin theorem.

IV. I NTERPOLATION OF L INEAR O PERATORS

109

Theorem. Let (X, x) and (Y, y) be measure spaces. Let B = Lp0 ∩ 0 0 Lp1 (dx) and B 0 = Lp0 ∩ Lp1 (dy), and let S be a linear transformation from B to B 0 , continuous as S : (B, k kj ) 7→ (B 0 , k k0j ), j = 0, 1, where 0 k kj (resp. k k0j ) is the norm induced by Lpj (dx) (resp. Lpj (dy)). Then S is continuous as S : (B, k kα ) 7→ (B 0 , k k0α ) 0

where k kα , (resp. k k0α ) is the norm induced by Lpα (dx) (resp. Lpα (dy)), pα and p0α are defined in (1.6)). A bounded linear transformation S from one normed space B to another can be completed in one and only one way, to a transformation having the same norm, from the completion of B into the completion of the range space of S. Thus, under the assumption of 1.4, S can 0 be extended as a transformation from Lpα (x) into Lpα (y) with norm satisfying (1.4). The same remark is clearly valid for Theorem 1.2. 1.5 Our first application of the Riesz-Thorin theorem is Bochner’s proof of M. Riesz’ Theorem III.1.11. We show that Lp (T) admits conjugation if p is an even integer. It then follows by interpolation that the same is true for all p > 2, and by duality, for all p > 1. Let f be a real-valued trigonometric polynomial and assume, for simplicity, f (0) = 0. As usual we denote the conjugate by f˜ and put f [ = 12 (f + if˜). f [ is a Taylor polynomial‡ and its constant term is zero; the same is clearly true for (f [ )p , p being any positive integer. Consequently Z 1 (f [ (t))p dt = 0 . 2π

Assume now that p is even, p = 2k, and consider the real part of the identity above; we obtain: Z Z Z 1 2k 1 2k 1 (f˜)2k dt − (f˜)2k−2 f 2 dt + (f˜)2k−4 f 4 dt 2π 2 2π 4 2π − · · · = 0.

By Hölder’s inequality Z 1 (f˜)2k−2m f 2m dt ≤ kf˜k2k−2m kf k2m L2k ; L2k 2π ‡ We

form

the term "Taylor polynomial" to designate trigonometric polynomials of the Puse N int 0

e

.

110


hence kf˜k2k L2k ≤

2k ˜ 2k−2 2k ˜ 2k−4 kf kL2k kf k2L2k + kf kL2k kf k4L2k + . . . , 2 4

or, denoting Y = kf˜kL2k kf k−1 L2k

we have Y

2k

2k 2k 2k−2 ≤ Y + Y 2k−4 Y 2k−4 + · · · + 1 2 4

which implies that Y is bounded by a constant depending on k (i.e., on p). Thus the mapping f 7→ f˜ is bounded in the Lp (T) norm for all polynomials f , and, since polynomials are dense in Lp (T), the theorem follows. EXERCISES FOR SECTION 1 1. Prove inequality (1.5). P 2. Let sequence of numbers. Find min( |an |) under the conP {an2} be a P ditions |an | = 1, |an |4 = a. 3. Let B be a vector space with consistent e a space of linear functionals on B which are continunorms k k0 , k k1 , and B e induced by ous with respect to both k k0 and k k1 . Let k k∗j be the norm on B ∗ the duality with (B, k kj ), j = 0, 1, and k kα the interpolating norms. Let k kα∗ e, e induced by the dual of (B, k kα ). Prove that for f ∈ B be the norm on B kf kα∗ ≤ kf k∗α .

4. Let X, B be a measurable space, and let µ and ν be positive measures on it. Let B = L2 X, B, µ ∩ L2 X, B, ν . i. What are necessary and sufficient conditions for the consistency of the norms (on B ): k k0 = k kL2 (µ) , and k k1 = k kL2 (ν) .

ii. When the norms above are consistent, what are the interpolating norms k kα ? 5. Let 0 < a < b. For f ∈ C ∞ (T), define kf k0 = kf k1 = ∞

P

|fˆ(n)|2 |n|2b

12

P

|fˆ(n)|2 |n|2a

21

, and

. Show that the norms so defined are consistent on

C (T) and find the interpolating norms k kα . 6. Assume 0 < a < b. What are the interpolating norms between the ones induced on C ∞ (T) by C a (T) and by C b (T)? Hint: For f ∈ L1 (T) define kf kW,k = kW2k (f )k∞ ; the notation is that of I.8.2. For 0 < a < b, and for f ∈ C ∞ (T), define kf k0 = supk kf kW,k 2ak and kf k1 = supk kf kW,k 2bk .


111

2 THE THEOREM OF HAUSDORFF-YOUNG

The theorem of Riesz-Thorin enables us to prove now a theorem that we stated without proof at the end of 1.4 (Theorem I.4.7); it is known as the Hausdorff-Young theorem: 2.1 Theorem. Let 1 ≤ p ≤ 2 and let q be the conjugate exponent, P that is, q = p/(p − 1). If f ∈ Lp (T) then |fˆ(n)|q < ∞. More precisely P ˆ 1/q |f (n)|q ≤ kf kLp

P ROOF : The mapping F : f 7→ {fˆ(n)} is a transformation of functions on the measure space (T, dt) into functions on (Z, dn), Z being the group of integers and dn the so-called counting measure, that is, the measure that places a unit mass at each integer. We know that the norm of the mapping as L1 (T) 7→ L∞ (Z) = `∞ is 1 (I.1.4) and we know that it is an isometry of L2 (T) onto L2 (Z) = `2 (I.5.5). It follows from the RieszThorin theorem that F is a transformation of norm ≤ 1 from Lp (T) into Lq (Z) = `q , which is precisely the statement of our theorem. We can add that since the exponentials are mapped with no loss in norm, the norm of F on Lp (T) into `q is exactly 1. J 2.2 Theorem. Let 1 ≤ p ≤ 2 and let q be the conjugate exponent. If {an } ∈ `p then there exists a function f ∈ Lq (T) such that an = fˆ(n). P Moreover, kf kLq ≤ ( |an |p )1/p .

P ROOF : Theorem 2.2 is the exact analog to 2.1 with the roles of the groups T and Z reversed. The proof is identical: if {an } ∈ `1 then P f (t) = an eint is continuous on T and fˆ(n) = an . The case p = 2 is again given by Theorem I.5.5 and the case 1 2 since there exist continuous funcP tions f such that |fˆ(n)|2−ε = ∞ for all ε > 0. An example of such P ein log n int a function is f (t) = ∞ (see [28], vol. I, p. 199); n=2 n1/2 (log n)2 e P −2 −m/2 another example is g(t) = m 2 fm (t) where fm are the RudinShapiro polynomials (see exercise 6, part c of I.6). We can try to explain the phenomenon by a less explicit but more elementary construction. The first remark is that this, like many problems in analysis, is a problem of comparison of norms. It is sufficient, we claim, to show

112


that, given p < 2, there exist functions g such that kgk∞ ≤ 1 and P |ˆ g (n)|p is arbitrarily big. If we assume that, we may assume that our functions g are polynomials (replace g by σn (g) with sufficiently big n) and then, taking a sequence pj → 2, gj satisfying kgj k∞ ≤< 1 and P −1 imj t P |ˆ gj (n)|pj > 2j we can write f = j e gj (t) where the integers imj t mj increase fast enough to ensure that e gj (t) and eimk t gk (t) have no frequencies in common if j 6= k. The series defining f converges uniformly and for any p < 2 we have X 1 X XX 1 |ˆ gj (n)|p ≥ |ˆ g (n)|p = ∞ . |fˆ(n)|p = 2 2 j j j p >p n j j

One way to show the existence of the functions g above is to show that, given ε > 0, there exist functions g satisfying (2.1)

kgk∞ ≤ 1,

kgkL2 ≥

1 , 2

supn |ˆ g (n)| ≤ ε .

In fact, if (2.1) is valid then X X 1 |ˆ g (n)|p ≥ εp−2 |ˆ g (n)|2 ≥ εp−2 , 4 and if ε can be chosen arbitrarily small, the corresponding g will have P |ˆ g (n)|p arbitrarily large. Functions satisfying (2.1) are not hard to find; however, it is important to realize that when we need a function satisfying certain conditions, it may be easier to construct an example rather than look for one in our inventory. We therefore include a construction of functions satisfying (2.1). The key remark in the construction is simple yet very useful: if P is a trigonometric polynomial of degree N , f ∈ L1 (T) and λ > 2N is an integer, then the Fourier coefficients of ϕ(t) = f (λt)P (t) are either zero or have the form fˆ(m)Pˆ (k). This follows from the idenP tity ϕ(n) ˆ = λm+k=n fˆ(m)Pˆ (k) and the fact that there is at most one way to write n = λm + k with integers m, k such that |k| < N < λ/2. Consider now any continuous function of modulus 1 on T, which is not an exponential (of the form eint ); for example the function ψ(t) = P ˆ2 ei cos t . Since |ψ| = kψk2L2 = 1 and the sum contains more than one ˆ term, it follows that sup|ψ(n)| = ρ < 1. Let M be an integer such that M ρ < ε. Let η < 1 be such that η M > 21 . Let ϕ = σN (ψ), where the order N is high enough to ensure η < |ϕ(t)| < 1. It follows from the Q j preceding remark that if we set λ = 3N and g(t) = M j=1 ϕ(λ t), the Fourier coefficients of g are products of M Fourier coefficients of ϕ;


hence |ˆ g (n)| ≤ ρM < ε. On the other hand (2.1) is valid.

1 2

113

< η M < |g(t)| < 1 and

2.4 We can use the polynomials satisfying (2.1) to show also that Theorem 2.2 does not admit an extension to the case p > 2. In fact, we P can construct a trigonometric series an eint which is not a FourierP p Stieltjes series, and such that |an | < ∞ for all p > 2. Let gj be a trigonometric polynomial satisfying (2.1) with ε = 2−j . Since now p > 2 we have X X |ˆ gj (n)|p ≤ εp−2 |ˆ gj (n)|2 ≤ 2−j(p−2) P imj t je gj (t) = and consequently, for any choice of the integers mj , P P an eint does satisfy |an |p < ∞ for all p > 2. We now choose the integers mj increasing very rapidly in order to well separate the blocks corresponding to jeimj t gj (t) in the series above. If we denote by Nj the degree of the polynomial gj , we can take mj so that mj − 3Nj > P mj−1 + 3Nj−1 . If an eint is the Fourier-Stieltjes series of a measure µ then µ ∗ eimj t VNj = jeimj t gj

(VNj being de la Vallée Poussin’s kernel)

and consequently 3kµkM (T) > jkgj kL1 >

j 4

which is impossible. We have thus proved Theorem. (a) There exists a continuous function f such that for all P p < 2, |fˆ(n)|p = ∞. P (b) There exists a trigonometric series, an eint , which is not a FourierP Stieltjes series, such that |fˆ(n)|p < ∞ for all p > 2.

Both statements can be improved. See Appendix B. 2.5 We finish this section with another construction: that of a set E of positive measure on T which carries no function with Fourier coefficients in `p for any p < 2. Such a set clearly must be totally disconnected and therefore carries no continuous functions. Its indicator function, however, is a bounded function whose Fourier coefficients belong to no `p , p < 2. Theorem. There exists a compact set E on T such that E has positive measure and such that, the only function f carried by E with P ˆ p |f (j)| < ∞ for some p < 2, is f = 0.

114


First, we introduce the notation† kf kF `∞ = sup|fˆ(j)|

(2.2)

kf kF `p =

X

|fˆ(j)|p

1/p

;

and prove: Lemma. Let ε > 0, 1 ≤ p < 2. There exists a closed set Eε,p ⊂ T having the following properties: (1) The measure of Eε,p is > 2π − ε. (2) If f is carried by Eε,p then kf kF `∞ ≤ εkf kF `p .

P ROOF : Let γ > 0. Put ( ϕγ (t) =

γ−2π γ

for 0 < t < γ

1

for γ ≤ t ≤ 2π mod 2π.

mod 2π

Then, by Theorem 2.1 (2.3)

1

kϕγ kF `q ≤ kϕγ kLp ≤ 2πγ p −1 ,

where

1 1 + = 1. p q

We notice that ϕˆγ (0) = 0 so that, if we choose the integers λ1 , λ2 , . . . , λN P increasing fast enough, every Fourier coefficient of N 1 ϕγ (λj t) is essentially a Fourier coefficient of one of the summands. It then follows that (2.4)

N

1 X

1

ϕγ (λj t) ≤ N q −1 kϕγ kF `q .

q N 1 F`

We take a large value for N and put γ = ε/N and Φ(t) =

N 1 X ϕγ (λj t). N 1

Then, by (2.3) and (2.4), it follows that 1

1

1

1

1

kΦkF `q ≤ 4πγ p −1 N q −1 = 4πε p −1 N q − p

so that if N is large enough kΦkF `q ≤ ε. We can take Eε,p = {t : Φ(t) = 1} =

N \ 1

† Notice

that k kF `1 is the same as k kA(T) .

{t : ϕγ (λj t) = 1}.


115

Since ϕγ (λj t) 6= 1 on a set of measure γ , it follows that |Eε,p | ≥ 2π − N γ = 2π − ε.

Now if f is carried by Eε,p , then for arbitrary n, Z Z 1 1 e−int f (t)dt = e−int f (t)Φ(t)dt. fˆ(n) = 2π 2π It follows from Parseval’s formula that X ˆ |fˆ(n)| = | fˆ(n − m)Φ(m)| ≤ kΦkF `q kf kF `p ≤ εkf kF `p ; and the proof of the lemma is complete.

J

−n P ROOF OF THE THEOREM : Take E = ∩∞ , and n=1 Eεn ,pn where εn = 3 pn = 2 − εn . The measure of E is clearly positive, and if f is carried by E and kf kF `q < ∞, it follows that for all n large enough

kf kF `∞ ≤ εn kf kF `pn ≤ εn kf kF `p ,

hence fˆ = 0 and so f = 0.

J

EXERCISES FOR SECTION 2 1. Verify that 2−(m+1)/2 fm (fm , as defined in exercise 6 part c) of I.6, satisfy (2.1) when 2−(m+1)/2 < ε. 2. Show that ifN > ε−1 and if mn increases fast enough, then g , defined by: g(t) = eiN mn t for 2πn/N ≤ t ≤ 2π(n + 1)/N, n = 0, . . . , N , satisfies (2.1). 3. Let {an } be an even sequence of positive numbers. A closed set E ⊂ T is a set of type U (an ) if the only distribution µ carried by E and satisfying µ ˆ(n) = o(an ) as |n| → ∞, is µ = 0. Show that if an → 0 there exist sets E of positive measure which are of type U (an ). Hint: For 0 < a < π we write (see exercise 3 of I.6): ∆a (t) =

(

1 − a−1 |t|

|t| ≤ a

0

a ≤ |t| ≤ π .

ˆ a (0) = a/2π . Choose nj so that We have ∆a ∈ A(T), k∆a kA(T) = 1, and ∆ −j |n| > nj implies an < 10 ; put Ej = {t : ∆2−j (2nj t) = 0} and E = ∩∞ j=1 Ej . P Notice that |E| ≥ 2π − 21−j > 0. If µ is carried by E we have, for all m and j , heimt ∆3−j (2nj t) , µi = 0 since ∆3−j (2nj t) vanishes in a neighborhood of E . By Parseval’s formula 0 = heimt ∆3−j (2nj t) , µi =

X 3−j \ µ ˆ(m) + ∆ µ(m + 2nj k). 3−j (k)ˆ 2π k6=0

116


If nj > |m| and if |ˆ µ(n)| ≤ an we have for k ≥ 0, |ˆ µ(m + 2nj k)| < 10−j , hence −j −j (3 /2π)|ˆ µ(m)| < 10 . Letting j → ∞ we obtain µ ˆ(m) = 0, and, m being arbitrary, µ = 0.

Chapter V

Lacunary Series and Quasi-analytic Classes

The theme of this chapter is that of I.4, namely, the study of the ways in which properties of functions or of classes of functions are reflected by their Fourier series. We consider important special cases of the following general problem: let Λ be a sequence of integers and B a homogeneous Banach space on T; denote by BΛ the closed subspace of B spanned by {eiλt }λ∈Λ or, equivalently, the space of all f ∈ B with Fourier series of the form P iλt . Describe the properties of functions in BΛ in terms of λ∈Λ aλ e their Fourier series (and Λ). An obvious example of the above is the case of a finite Λ in which all the functions in BΛ are polynomials. If Λ is the sequence of nonnegative integers and B = Lp (T), 1 < p < ∞, then BΛ is the space of boundary values of functions in the corresponding H p . In the first section we consider lacunary sequences Λ and show, for instance, that if Λ is lacunary à la Hadamard then (L1 (T))Λ = (L1 (T))Λ and every bounded function in (L1 (T))Λ has an absolutely convergent Fourier series. In the second section we prove the Denjoy-Carleman theorem on the quasi-analyticity of classes of infinitely differentiable functions and discuss briefly some related problems. 1 LACUNARY SERIES

1.1 A sequence of positive integers {λn } is said to be Hadamard lacunary, or simply lacunary, if there exists a constant q > 1 such that P an z λn is lacunary if the seλn+1 > qλn for all n. A power series quence {λn } is, and a trigonometric series is lacunary if all the frequencies appearing in it have the form ±λn where {λn } is lacunary. The reason for mentioning Hadamard’s name is his classical theo-

117

118


rem stating that the circle of convergence of a lacunary power series is a natural boundary for the function given by the sum of the series within its domain of convergence. The general idea behind Hadamard’s theorem and behind most of the results concerning lacunary series is that the sparsity of the exponents appearing in the series forces on it a certain homogeneity of behavior. 1.2 Lacunarity can be used technically in a number of ways. Our first example is "local"; it illustrates how a Fourier coefficient that stands apart from the others is affected by the behavior of the function in a neighborhood of a point. Lemma. Let f ∈ L1 (T) and assume that fˆ(j) = 0 for all j satisfying 1 ≤ |n0 − j| ≤ 2N . Assume that f (t) = O (t) as t → 0. Then (1.1) |fˆ(n0 )| ≤ 2π 4 N −1 sup|t| N2 , we obtain −N 1 − N +1

JN (t) < 2π 4 N −3 t−4 .

We now write 1 2π

|fˆ(n0 )| ≤ =

Z

Z

|f (t)|JN (t)dt = +

|t| 4λn and f (t) = 1 an cos λn t is real-valued, then 1X |an |. 2 Hint: Consider the sets {t : an cos λn t > |an /2|}. 5. If dn → 0 as n → ∞ we can write dn = δn ψn where {δn } is bounded ˆ and ψn = ψ(n) for some ψ ∈ L1 (T). (See theorem 1.4.1 and exercise 1.4.1.) Deduce that if Λ is a lacunary sequence and dλ → 0 as |λ| → ∞, there exists a function g ∈ L1 (T) such that

sup|f (t)| ≥

dλ = gˆ(λ) for λ ∈ Λ

6. Use Theorem I.4.1 to show thatPif |dn |2 < ∞, there exist sequences ˆ {δn } and {ψn } such that dn = δn ψn , |δn |2 < ∞, and ψn = ψ(n) for some 1 ψ ∈ L (T). Remembering that the convolution of a summable function with a bounded function is continuous, prove Theorem 1.6. 7. Assume λj+1 /λj > q > 1. There exists a number M = Mq such that P every integer n has at most one representation of the form n = ηj λmj +jM where ηj = −1, 0, 1, and 1 ≤ mj ≤ M . Use this to show that the product Q∞ PM (1 + m=1 dm+jM cos(λm+jM t + ϕm+jM )) has (formally) the Fourier co1 efficient 12 dk eiϕk at the point λk ). Show that if 0 < dk < I/Mq for all k, then the product above is the Fourier Stieltjes series of a positive measure which interpolates { 12 dk eiϕk } on {λk }. P 8. Assume λj+1 /λj > q > 1 and |dj |2 < ∞. Find a product analogous to that of exercise 7), which is the Fourier series of a bounded function, and which interpolates {dj } on {λj }. *9. Show that the following condition is sufficient to imply that the sequence Λ is a Sidon set: to every sequence {dλ } such that |dλ | ≤ 1 there exists a measure µ ∈ M (T) such that |ˆ µ(λ) − dλ | < 12 . 10. Show that a finite union of lacunary sequences is a Sidon set. *11. Let λn be positive integers such that λn+1 /λn ≥ 3. Show that for every P 2U > 0 and ε > 0 there exists a > 0 such that if bn are real numbers, |bn | = 2 and sup |bn | ≤ a, then, if |u| < U , we have

P

(1.9)

Z P u2 1 eiu bn cos λn t − e− 2 < ε. 2π

V. L ACUNARY S ERIES AND Q UASI - ANALYTIC C LASSES

125

eiubn cos λn t = 1 + iubn cos λn t − 21 u2 b2n cos2 λn t (1 + cn ) Hint: Write Q 3 3 3 0, is log-convex if the sequence {log cn } is a convex function of n. This amounts to saying that, given k < l < m in the range of n, we have (2.3)

log cl ≤

m−l l−k log ck + log cm m−k m−k

or equivalently (2.4)

(m−l)/(m−k) (l−k)/(m−k) cm

cl ≤ ck

P 1 2.2 The identity kf (n) kL2 = ( fˆ(j)2 j 2n ) 2 allows an expression of condition (2.2) directly in terms of the Fourier coefficients of f . Also it implies

126


Lemma. Let f be N times differentiable on T. Then the sequence {kf (n) kL2 } is monotone increasing and log-convex for 1 ≤ n ≤ N . P 1 P ROOF : The fact that kf (n) kL2 = ( fˆ(j)2 j 2n ) 2 is monotone increasing is obvious. In order to prove (2.4) we write p = (m − k)/(m − l), and q = (m − k)/(l − k); then 1/p + l/q = 1 and by Hölder’s inequality P ˆ 2 2l P 2/p 2/q |f (j)| j = (|fˆ(j)|2/p j 2k/p )(|fˆ(j)|2/q j 2m/q ) ≤ kf (k) kL2 kf (m) kL2

which is exactly (2.4).

J

It follows from lemma 2.2 (cf. exercise 2 at the end of this section) that for every sequence {Mn0 } there exists a sequence {Mn } which is monotone increasing and log-convex such that C # {Mn } = C # {Mn0 }. Thus, when studying classes C # {Mn } we may assume without loss of generality that {Mn } is monotone increasing and log-convex; throughout the rest of this section we always assume that, for k < l < m, (2.5)

(m−l)/(m−k)

Ml ≤ Mk

(l−k)/(m−k) Mm

2.3 For a (monotone increasing and log-convex) sequence we define the associated function τ (r) by

(2.6)

τ (r) = inf Mn r−n . n≥0

We consider sequences Mn which increase faster that Rn for all R > 0; the infimum in (2.6) is attained and we can write τ (r) = minn≥0 Mn r−n . If we write µ1 = M1−1 , and µn = Mn−1 /Mn for n > 1; then µn is monotone-decreasing since by (2.5), µn+1 /µn = Mn2 /Mn−1 Mn+1 < 1; Q we have Mn r−n = n1 (µj r)−1 and consequently (2.6’)

τ (r) =

Y

(µj r)−1 .

µj r>1

The function τ (r) was implicitly introduced in 1.4; thus it follows from I.4.4 that if f ∈ C # {Mn } then, for the appropriate R > 0 |fˆ(j)| ≤ τ (jR−1 ),

and exercise 1.4.6 is essentially an estimate for τ (r) in the case Mn = nαn .


127

2.4 An analytic function on T is completely determined by its Taylor expansion around any point to t0 ∈ T, that is, by the sequence (n) {f (n) (t0 )}∞ (t0 ) = 0, n = 0, 1, 2, . . . , it follows n=0 . In particular if f that f = 0 identically. D EFINITION : A class of infinitely differentiable functions on T is quasi-analytic if the only function in the class, which vanishes with all its derivatives at some t0 ∈ T, is the function which vanishes identically. The main result of this section is the so-called Denjoy-Carleman theorem which gives a necessary and sufficient conditions for the quasianalyticity of classes C # {Mn }. Theorem. Let {Mn } be monotone increasing and log-convex. Let τ (r) be the associated function (2.6). The following three conditions are equivalent: (i) (ii) (iii)

C # {Mn } is quasi-analytic Z ∞ log τ (r) dr = −∞ 1 + r2 1 X Mn =∞ Mn+1

The proof will consist in establishing the three implications (ii) ⇒ (i) (Theorem 2.4 below), (i) ⇒ (iii) (Theorem 2.8), and (iii) ⇒ (ii) (Lemma 2.9). We begin with: 2.5 Lemma. Let ϕ(z) 6≡ 0 be holomorphic and bounded in the half plane 0 and continuous on −∞ 1 + y2 0 1+ζ P ROOF : The function F (ζ) = ϕ 1−ζ is holomorphic and bounded in ¯ the unit disc D (and R π is continuous on D except possibly at ζ = 1). By III.3.14 we have 0 log|F (eit )|dt > −∞. The change of variables that we have introduced gives for the boundaries eit = (iy − 1)/(iy + 1), or t = 2 arc cot y . Consequently dt = −2dy 1+y 2 and Z ∞ Z 1 π dy log|ϕ(iy)| = log|F (eit )|dt > −∞; 1 + y2 2 0 0

128

similarly


R∞ 0

dy log|ϕ(−iy)| 1+y 2 > −∞ and the lemma is proved.

J

2.6 Theorem. A sufficient for the quasi-analyticity of the R ∞ condition τ (r) class C # {Mn } is that 1 log 1+r 2 dr = −∞ where τ (r) is defined by (2.6).

P ROOF : Let f ∈ C # {Mn } and assume that f (n) (0) = 0 n = 0, 1, . . . . Define Z 2π 1 e−zt f (t)dt. ϕ(z) = 2π 0 Integrating by parts we obtain, z 6= 0, ϕ(z) =

2π 1 −1 −zt 1 e f (t) 0 + 2π z 2πz

Z

2π

e−zt f 0 (t)dt

0

and since f (0) = f (2π) = 0 the first term vanishes for all z 6= 0 (we have used the same integration by parts in I.4.5; there we did not assume f (0) = 0 but considered only the case z = im, that is, e−zt f is 2π periodic.) Repeating the integration by parts n times (using f (j) (0) = f (j) (2π) = 0 for j ≤ n), we obtain ϕ(z) =

1 2πz n

Z

2π

e−zt f (n) (t)dt.

0

For 0, ∞ 0 µj ≤ 1. Write ϕ(k) = 0 µj k . P∞ ikt Then f (t) = −∞ ϕ(k)e is carried by [−1, 1] ( mod 2π ), it is inQ finitely differentiable and kf (n) k ≤ 2 n0 µ−1 j .


129

PROOF : All the factors in the product defining ϕ(k) are bounded by 1 so that the product either converges or diverges to zero (actually it converges for all k) and ϕ(k) is well defined. ϕ(k) clearly tends to zero faster than any power of k so that the series defining f converges uniformly and f isninfinitely o∞differentiable. We have ϕ(0) = 1 so that f 6≡ 0. sin µj k The sequence is the sequence of Fourier coefficients of µj k k=−∞ QN sin µj k the function Γj (t) = πµ−1 j 11[−µj ,µj ] . If we write ϕN (k) = 0 µj k , we have ∞ X fN (t) = ϕN (k)eikt = Γ0 ∗ Γ1 ∗ . . . ΓN −∞

P PN and the support of fN is equal to [− N 2π . Since fN 0 µj , 0 µj ] mod P P∞ converges uniformly to f , the support of f is equal to [− ∞ µj , 0 µj ] 0 P mod 2π . Finally, since kf (n) k2L2 = |ϕ(k)|2 k 2n and |ϕ(k)| ≤

n Y 0

we obtain kf (n) kL2 plete.

n Y (µj k)−1 = ( µj )−1 k −n−1 , 0

Qn P ≤ ( 0 µj )−1 ( k6=0 k −2 )1/2 and the proof is comJ

2.8 Theorem. A necessary condition for the quasi-analyticity of the P Mn class C # {Mn } is that Mn+1 = ∞. P Mn P ROOF : Assume that Mn+1 < ∞. Without loss of generality we may P Mn 1 assume (replacing Mn by Mn0 = Mn Rn does not change < Mn+1 2 P Mn P Mn0 the class C # {Mn } while = R−1 0 Mn+1 Mn+1 ). Write µ0 = µ1 = 1/4, µj = Mj−1 /Mj , j ≥ 2. Then the function f defined by Lemma 2.7 has a zero of infinite order (actually vanishes outside of [−1, 1]), is not identically zero, and f ∈ C # {Mn }. J 2.9 Lemma. Under the assumption of theorem 2.4 we have Z ∞ X Mn log τ (r) 4 ≤ 2e dr. Mn+1 1 + r2 2 e PROOF :

As before we write µn = tion M(r) of {µn } by:

Mn−1 Mn .

We define the counting func-

M(r) = the number of elements µj such that µj r > e,

130


and recall that τ (r) = − log τ (r) =

Q

−1 ; µj r>1 (µj r)

X

log(µj r) ≥

hence X

log(µj r) ≥ M(r).

µj r>e

µj r>1

Thus for k = 2, 3, . . . (2.7)

Z

ek+1 ek

M(ek ) − log τ (r) dr ≥ 2k+2 2 1+r 2e

Z

ek+1

dr > ek

1 M(ek ) ; 2e2 ek

on the other hand, (2.8)

X

e1−k 1. 6. Let τ (r) be the function associated with a sequence {Mn }. (a) Show that (τ (r))−1 is log-convex function of r. (b) Show that Mn = maxr rn τ (r). 7. Let {ωn } be a log-convex sequence, n = 0, 1, .P . . , ωn ≥ 1, and let A{ωn } be the space of all f ∈ C(T) such that kf k{ωn } = |fˆ(n)|ω|n| < ∞. Show that with the norm k k{ωn } , A{ωn } is a Banach space. Show that a necessary and sufficient P∞condition for A{ωn } to contain functions with arbitrarily small support is 1 logn2ωn < ∞. 8. Let {ωn } be log-convex, ωn ≥ 1, and assume that ωn → ∞ faster than k ∞ any power of n. The sequences σn = ωn n=−∞ tend to zero as |n| → ∞. |n| Show that the subspace that σn , k = 0, 1, . . . generate in c0 (the space of sequences tending to zero at ∞) is uniformly dense in c0 if, and only if, σ logn2ωn = ∞ Hint: The dual space is `1 . If {an } ∈ `1 is orthogonal to σk , k = 0, . . . , Pofanc0int the function f (t) = , which clearly belongs to A{ωn }, has a zero of e ωn infinite order. 9. Let f be as in exercise 1.3. Show that f = 0 identically.

Chapter VI

Fourier Transforms on the Line

In the preceding chapters we studied objects (functions, measures, and so on) defined on T. Our aim in this chapter is to extend the study to objects defined on the real line R. Much of the theory, especially the L1 theory, extends almost verbatim and with only trivial modifications of the proofs; such results, analogous in statement and in proof to theorems that we have proved for T, often are stated without a proof. The difference between the circle and the line becomes more obvious when we try to see what happens for Lp with p > 1. The (Lebesgue) measure of Rbeing infinite entails that, unlike L1 (T) which contains most of the "natural" function spaces on T, L1 (R) is relatively small; in particular Lp (R) 6⊂ L1 (R) for p > 1. The definition of Fourier transforms in L1 (R) has now a much more special character and a new definition (i.e., an extension of the definition) is needed for Lp (R), p > 1. The situation turns out to be quite different for p ≤ 2 and for p > 2. If p < 2, Fourier transforms of functions in Lp (R) can be defined by continuity as functions in Lq (R), q = p/(p − 1); however, if p > 2, the only reasonable way to define the Fourier transform on Lp (R) is through duality and Fourier transforms are now defined as distributions. The plan of this chapter is as follows: in section 1 we define the Fourier transform in L1 (R) and discuss its elementary properties. We also mention the connection between Fourier transforms and Fourier coefficients and prove Poisson’s formula. In section 2 we define Fourier-Stieltjes transforms and obtain various characterizations of Fourier-Stieltjes transforms of arbitrary and positive measures. In section 3 we prove Plancherel’s theorem and the Hausdorff-Young inequality, thereby defining Fourier transforms in Lp (R), 1 < p < 2. In section 4 we use Parseval’s formula, that is duality, to define the Fourier transforms of tempered distributions, and study some of the properties of Fourier transforms of functions in Lp (R), p ≤ ∞. Sections 5 and 6 deal with spectral anal132

VI. F OURIER T RANSFORMS ON THE L INE

133

ysis and synthesis in L∞ (R). In section 5 we consider the problems relative to the norm topology and show that the class of functions for which we have satisfactory theory is precisely that of Bohr’s almost periodic functions. In section 6 we study the analogous problems for the weak-star topology. Sections 7 and 8 are devoted to relations between Fourier transforms and analytic functions. Finally, section 9 contains Kronecker’s theorem (which we have already used in chapter II) and some variations on the same theme. 1 FOURIER TRANSFORMS FOR L1 (R)

1.1 We denote by L1 (R) the space of Lebesgue integrable functions on the real line. For f ∈ L1 (R) we write Z ∞ |f (x)|dx, kf kL1 (R) = −∞

and when there is no risk of confusion, we write kf kL1 or simply kf k instead of kf kL1 (R) . The Fourier transform fˆ of f is defined by Z (1.1) fˆ(ξ) = f (x)e−iξx dx for all real ξ † This definition is analogous to I.(1.5), and the disappearance of the factor 1/2π is due to none other than our (arbitrary) choice to remove it. It was a natural normalizing factor for the Lebesgue measure on T; but, at this point, it seems arbitrary for R. The factor 1/2π will reappear in the inversion formula and p some authors, seeking more symmetry for the inversion formula, write 1/2π in front of the integral (1.1) so that the same factor appear in the Fourier transform and its inverse. The added symmetry, however, may increase the possibility of confusion between the domains of definition of a function and its transform. In L1 (T) the functions are defined on T whereas the Fourier transforms are defined on the integers; in L1 (R) the functions are defined on R and the domain of definition of the Fourier is again the real line. It may be helpful to consider two copies of the real line: one is R and the other, which will serve as the domain of definition of Fourier transforms of ˆ . This notation is in accordance with functions in L1 (R), we denote by R that of chapter VII. † Throughout this chapter, integrals with unspecified limits of integration are always to be taken over the entire real line.

134


Most of the elementary properties of Fourier coefficients are valid for Fourier transforms. Theorem. Let f, g ∈ L1 (R). Then (f[ + g)(ξ) = fˆ(ξ) + gˆ(ξ)

(a)

(b) For any complex number α d)(ξ) = αfˆ(ξ) (αf

(c) If f¯ is the complex conjugate of f , then

¯(ξ) = fˆ(−ξ) fˆ

(d) Denote fy (x) = f (x − y), y ∈ R. Then fˆy (ξ) = fˆ(ξ)e−iξy R |fˆ(ξ)| ≤ |f (x)|dx = kf k

(e)

(f) For positive λ denote ϕ(x) = λf (λx);

then

ξ ϕ(ξ) ˆ = fˆ( ). λ

P ROOF : The theorem follows immediately from (1.1). Parts (a) through (e) are analogous to the corresponding parts of I.1.4. Part (f) is obtained by a change of variable y = λx: Z Z ξ −i(ξ/λ)λx ϕ(ξ) ˆ = f (λx)e dλx = f (y)e−i(ξ/λy) dy = fˆ( ). J λ ˆ. 1.2 Theorem. Let f ∈ L1 (R). Then fˆ is uniformly continuous on R

P ROOF : fˆ(ξ + η) − fˆ(ξ) =

Z

f (x) e−i(ξ+η)x − e−iξx dx,

hence (1.2)

|fˆ(ξ + η) − fˆ(ξ)| ≤

Z

|f (x)||e−iηx − 1|dx.

The integral on the right of (1.2) is independent of ξ , the integrand is bounded by 2|f (x)| and tends to zero everywhere as η → 0. J


135

1.3 The following are immediate adaptations of the corresponding theorems in chapter I. Theorem. Let f, g ∈ L1 (R). For almost all x, f (x−y)g(y) is integrable (as a function of y ) and, if we write Z h(x) = f (x − y)g(y)dy,

then h ∈ L1 (R) and khk ≤ kf k kgk;

moreover, ˆ h(ξ) = fˆ(ξ)ˆ g (ξ)

for all ξ .

As in chapter I we denote h = f ∗g , call h the convolution of f and g , and notice that the convolution operation is commutative, associative, and distributive. 1.4 Theorem. Let f, h ∈ L1 (R) and Z 1 h(x) = H(ξ)eiξx dξ 2π

with integrable H(ξ). Then (h ∗ f )(x) =

(1.3)

1 2π

Z

H(ξ)fˆ(ξ)eiξx dξ.

P ROOF : The function H(ξ)f (y) is integrable in (ξ, y), hence, by Fubini’s theorem, Z ZZ 1 (h ∗ f )(x) = h(x − y)f (y)dy = H(ξ)eiξx e−iξy f (y)dξdy 2π Z Z Z 1 1 iξx −iξy = H(ξ)e e f (y)dydξ = H(ξ)fˆ(ξ)eiξx dξ. 2π 2π J 1.5 Theorem. Let f ∈ L1 (R) and define Z x f (y)dy. F (x) = −∞

1

Then, if F ∈ L (R) we have (1.4)

1 Fˆ (ξ) = fˆ(ξ) iξ

all real ξ 6= 0.

136


An equivalent statement of the theorem is: if F, F 0 ∈ L1 (R), then c0 (ξ) = iξ Fˆ (ξ). F

1.6 Theorem. Let f ∈ L1 (R) and xf (x) ∈ L1 (R). Then fˆ is differentiable and

(1.5)

d ˆ \)(ξ). f (ξ) = (−ixf dξ

P ROOF : (1.6)

Z e−ihx − 1 fˆ(ξ + h) − fˆ(ξ) = f (x)e−iξx dx. h h

The integrand in (1.6) is bounded by |xf (x)| (which is in L1 (R) by assumption) and tends to −ixf (x)e−iξx pointwise, hence (Lebesgue) it converges to −ixf (x)e−iξx in the L1 (R) norm. This implies that, as \)(ξ) and the h → 0, the right-hand side of (1.6) converges to (−ixf theorem follows. J 1.7 Theorem (Riemann-Lebesgue lemma). For f ∈ L1 (R) lim fˆ(ξ) = 0.

|ξ|→0

P ROOF : If g is continuously differentiable and with compact support we have, by 1.5 and 1.1, |ξˆ g (ξ)| ≤ kg 0 kL1 (R) hence lim|ξ|→∞ |ˆ g (ξ)| = 0. 1 For arbitrary f ∈ L (R), let ε > 0 and g be a continuously differentiable, compactly supported function such that kf − gkL1 (R) . We have both |fˆ(ξ) − gˆ(ξ)| < ε and limξ→∞ |ˆ g (ξ)| = 0; hence lim sup|ξ|→∞ ||fˆ(ξ)|| < ε. This being true for all ε > 0, we obtain limξ→∞ fˆ(ξ) = 0. J ˆ the space of all functions ϕ on R ˆ , which are 1.8 We denote by A(R) 1 the Fourier transforms of functions in L (R). By the results above, A(R) is an algebra of continuous functions vanishing at infinity, that ˆ is, a subalgebra of C0 (R), the algebra of all continuous functions on R ˆ which vanish at infinity. We introduce a norm to A(R) by transferring to it the norm of L1 (R), that is, we write kfˆkA(R) ˆ = kf kL1 (R) .

It follows from 1.3 that the norm k kA(R) ˆ , is multiplicative, that is, satisfies the inequality: kϕ1 ϕ2 kA(R) ˆ kϕ1 kA(R) ˆ kϕ2 kA(R) ˆ


137

The norm k kA(R) ˆ is not equivalent to the supremum norm; conseˆ is a proper subalgebra of C0 (R) ˆ . quently, A(R) 1.9 A summability kernel on the real line is a family of continuous functions {kλ } on R, with either discrete or continuous parameter‡ satisfying the following: Z kλ (x)dx = 1 kkλ k = O∗ (1) as λ → ∞ Z lim |kλ (x)|dx = 0, for all a > 0.

(1.7)

λ→∞

|x|>δ

A common way to produceR summability kernels on R is to take a function f ∈ L1 (R) such that f (x)dx = 1 and to write kλ (x) = λf (λx) for λ > 0. Condition (1.7) is satisfied since, introducing the change of variable y = λx, we obtain Z Z kλ (x)dx = f (y)dy = 1 kkλ k =

and

Z

Z

|kλ (x)|dx =

|kλ (x)|dx =

|x|>δ

Z

Z

|f (y)|dy = kf k

|f (y)|dy → 0 as λ → ∞.

|y|>λδ

The Fejér kernel on R is defined by Kλ (x) = λK(λx),

λ > 0,

where (1.8)

1 K(x) = 2π

sin x/2 x/2

2

1 = 2π

Z

1

−1

1 − |ξ| eiξx dξ .

The second equality in (1.8) is obtained directly by integration. By the previous remark it is clear that the only thing we need to Rcheck, in order to establish that {Kλ } is a summability kernel, is that K(x)dx = 1. This can be done directly, for example, by contour integration, or using ‡ The indexing parameter λ is often real valued; however, it should not be considered as an element of R so that no confusion with the notation of 1.1.d should arise.

138


the information that we have about the Fejér kernel of the circle, that is, that for all 0 < δ < π 1 lim n→∞ 2π

(1.9)

Z

δ

−δ

1 n+1

sin(n + 1)x/2 sin x/2

2

dx = 1.

R K(x)dx = Kλ (x)dx, we may take λ = n + 1, in which case 2 sin(n+1)x/2 1 Kλ (x) = 2π(n+1) , and notice that if δ > 0 is small enough, x/2 the ratio of 2πKλ (x) to the integrand in (1.9) is arbitrarily close to one in |x| < δ . More precisely, we obtain (λ = n + 1):

Since

R

sin δ δ

2

1 2π

2 Z δ 1 1 sin(n + 1)x/2 dx < Kλ (x) x/2 2π −δ −δ n + 1 2 Z π 1 1 sin(n + 1)x/2 dx < 2π −π n + 1 sin x/2

Z

δ

R Rδ Letting n → ∞ we see that K(x)dx = limλ→∞ −δ Kλ (x)dx is a numR ber between sin2 δ/δ 2 and 1; since δ > 0 is arbitrary K(x)dx = 1.

1.10 Theorem. Let f ∈ L1 (R) and let {kλ } be a summability kernel on R, then lim kf − kλ ∗ f kL1 (R) = 0.

λ→∞

P ROOF : Repeat the proof of theorem I.2.3 and lemma I.2.4.

J

1.11 Specifying theorem 1.10 to the Fejér kernel and using theorem 1.4, we obtain Theorem. Let f ∈ L1 (R), then

(1.10)

1 λ→∞ 2π

f = lim

λ

|ξ| ˆ 1− f (ξ)eiξx dξ λ −λ

Z

in the L1 (R) norm. Corollary (The uniqueness theorem). Let f ∈ L1 (R) and assume ˆ then f = 0. that fˆ(ξ) = 0 for all ξ ∈ R


139

1.12 If it happens that f is Lebesgue integrable, the integral on the R 1 right-hand side of (1.10) converges, uniformly in x, to 2π fˆ(ξ)eiξx dξ . We see that f is equivalent to a uniformly continuous function and obtain the so-called "inversion formula": Z 1 (1.11) f (x) = fˆ(ξ)eiξx dξ . 2π

An immediate consequence of (1.11) is cλ (ξ) = max 1 − |ξ| , 0 (1.12) K λ and, by theorem 1.3, (1.13)

\ (K λ ∗ f )(ξ) =

  1−

|ξ| λ

fˆ(ξ)

|ξ| ≤ λ |ξ| ≥ λ.

0,

Combining this with theorem 1.10, we obtain Theorem. The functions with compactly carried Fourier transforms form a dense subspace of L1 (R).

This theorem is analogous to the statement that trigonometric polynomials form a dense subspace of L1 (T). 1.13 Besides the Fejér kernel we mention the following: De la Vallée Poussin’s kernel

(1.14)

Vλ (x) = 2K2λ (x) − Kλ (x),

whose Fourier transform is given by    1, ˆ (1.15) Vλ (ξ) = 2 − |ξ| λ ,   0,

|ξ| ≤ 1 λ ≤ |ξ| ≤ 2λ 2λ ≤ |ξ|.

Poisson’s kernel Pλ (x) = λP(λx),

where (1.16)

P(x) =

1 π(1 + x2 )

140


and (1.17)

ˆ P(ξ) = e−|ξ| ,

and finally Gauss’ kernel Gλ (x) = λG(λx),

where (1.18)

1

G(x) = (2π)− 2 e−

x2 2

and 2

(1.19)

ξ ˆ G(ξ) = e− 2 .

To the inversion formula (1.11) and the summability in norm (theorems 1.10 and 1.11, one should add results about pointwise summability. Both the statements and the proofs of section I.3 can be adapted to L1 (R) almost verbatim and we avoid the repetition. 1.14 As in chapter I, we can replace the L1 (R) norm, in the statement of theorems 1.10 and 1.11, by the norm of any homogeneous Banach space B ⊂ L1 (R). As in chapter I, a homogeneous Banach space is a space of functions which is invariant under translation and such that for every f ∈ B, fy (defined by fy (x) = f (x − y)) depends continuously on y . The assumption B ⊂ L1 (R) is more restrictive than was the assumption B ⊂ L1 (T) in chapter I; it excludes such natural spaces as Lp (R), p > 1. We can obtain a reasonably general theory by considering homogeneous Banach space of locally summable functions, that is, functions which are Lebesgue integrable on every finite interval. We denote by L the space of all measurable functions f on R such that Z y+1 kf kL = supy |f (x)|dx < ∞ y

and by Lc the subspace of L consisting of all the functions f which satisfy kfy − f kL → 0 as y → 0. Theorem. If B is a homogeneous Banach space of locally summable functions on R and if convergence in B implies convergence in measure, then the L norm is majorized by the B norm and, in particular, B ⊂ Lc .


141

P ROOF : If the L norm is not majorized by k kB , we can choose a sequence fn ∈ B such that kfn kB < 2−n and kfnRkL > 3n . Replacing fn by fn (x − yn ) (if necessary) we may assume |fn (x)|dx > 3n Since kfn kB → 0, fn converges to zero in measure and it follows that if P nj → ∞ fast enough fnj , which belongs to B , is not integrable on (0, 1). J We can now extend Theorem 1.10 to homogeneous Banach spaces of locally summable functions (see exercises 11-14 at the end of this section); Theorem 1.11 can be generalized only after we extend the definition of the Fourier transformation. 1.15 We finish this section with a remark concerning the relation between Fourier coefficients and Fourier transforms. Let f ∈ L1 (R) and define ϕ by ϕ(t) = 2π

∞ X

f (t + 2πj).

j=−∞

t is a real number, but it is clear that ϕ(t) depends only on t (mod 2π) so that we can consider ϕ as defined on T. We clearly have ϕ ∈ L1 (T) and kϕkL1 (T) ≤ kf kL1 (R) .

For n ∈ Z, we have Z ∞ Z 2π X 1 −int ϕ(n) ˆ = ϕ(t)e dt = f (t + 2πj)e−int dt 2π 0 j=−∞ Z = f (x)e−inx dx = fˆ(n). so that ϕˆ is simply the restriction to the integers of fˆ. Similarly, if we write fλ (x) = λf (λx) and: (1.20)

ϕλ (t) = 2π

∞ X

fλ (t + 2πj),

j=−∞

we obtain, using 1.1, (1.21)

n ϕˆλ (n) = fˆ( ). λ

The preceding remarks, as simple as they sound, link the theory of Fourier integrals to that of Fourier series, and we can obtain a great

142


many facts about Fourier integrals from the corresponding facts about Fourier series. (For examples, see exercises 5 and 6 at the end of this section.) An application to the procedure above is the very important formula of Poisson: (1.22)

2πλ

∞ X

f (2πλn) =

n−=∞

∞ X

n fˆ( ). λ n=−∞

In order to establish Poisson’s formula, to understand its meaning and its domain of validity, all that we need to do is simply rewrite it as (1.23)

ϕλ (0) =

∞ X

ϕˆλ (n).

n=−∞

If ϕλ (0), as defined by (1.20), is well defined and if the Fourier series of ϕλ converges to ϕλ (0) for t = 0, then (1.23) and (1.22) are valid. One enhances the generality of (1.22) considerably by interpreting the sum on the right as N X |n| ˆ n lim 1− f ( ), N →∞ N λ −N that is, using C-1 summability instead of summation. Using Fejér’s theorem, for instance, one obtains that, with this interpretation, (1.22) is valid if t = 0 is a point of continuity of ϕλ . We remark that the continuity of f and f¯ is not sufficient to imply (1.22) even if both sides of (1.22) converge absolutely (see exercise 15). EXERCISES FOR SECTION I 1. Perform the integration R sin x/2 in (1.8). 1 2. Prove that 2π dx = 1 by contour integration. x/2 3. Prove (1.17). Hint: Use contour integration. 4. Prove (1.19). ˆ ˆ ˆ , (use 1.5 and Hint: Show that G(ξ) satisfies the equation d/dξ G(ξ) = −ξ G(ξ) 1.6). 5. Let f ∈ L1 (R) and ϕλ (t) defined by (1.20). Show that limλ→∞ kϕλ kL1 (T) = kf kL1 (R) ; hence deduce the uniqueness theorem from (1.21). 6. Prove Theorem 1.7 using (1.21), the uniform continuity of Fourier transforms and the Riemann-Lebesgue lemma for Fourier coefficients.


143

ˆ contains every twice continuously differentiable func7. Show that A(R) ˆ is uniformly dense in C0 (R) ˆ ; tion with compact support on R. Deduce that A(R) ˆ 6= C0 (R) ˆ . however, show that A(R) ˆ. 8. Let f ∈ L1 (R) be continuous Rat x = 0 and assume that fˆ(ξ) ≥ 0, ξ ∈ R 1 ˆ 1 ˆ ˆ Show that f ∈ L (R) and f (0) = 2π f (ξ)dξ . Hint: Use the analog to Fejér’s theorem and the fact that for positive functions C-1 summability is equivalent to convergence. 9. Show that C0 ∩ L1 (R), with the norm khk = supx |f (x)| + kf kL1 (R) is 1 a homogeneous Banach R λ space onR and conclude that if f ∈ C0 ∩ L (R) then 1 ˆ f (x) = limλ→∞ 2π −λ 1 − |ξ|/λ f (ξ)dξ uniformly. 10. Let f be bounded andRcontinuous on R and let {kλ } be a summability kernel. Show that kλ ∗ f = kλ (x − y)f (y)dy converges to f uniformly on compact sets on R. 11. Let f ∈ Lc and let ϕ be continuous with compact support; write ϕ∗f =

Z

ϕ(y)f (x − y)dy.

Interpreting the integral above as an Lc -valued integral, show that ϕ ∗ f ∈ Lc and kϕ ∗ f kL ≤ kϕkL1 (R) kf kL . Use this to define g ∗ f for g ∈ L1 (R) and f ∈ Lc . 12. Show that if f ∈ Lc then |f | ∈ Lc (notice, however, that eix log|x| 6∈ Lc ) and, using exercise 11, prove that if f ∈ Lc and g ∈ L1 (R) then for almost all 1 x R ∈ R, g(y)f (x − y) ∈ L (R) and g ∗ f , as defined in exercise 11, is equal to g(y)f (x − y)dy . 13. Let f ∈ L and let g R∈ L1 (R). Prove that g(y)f (x − y) ∈ L1 (R) for almost all x, and that h(x) = g(y)f (x − y)dy satisfies khkL ≤ kgkL1 (R) kf kL . 14. Let {kλ } be a summability kernel in L1 (R) and let B ⊂ Lc be a homogeneous Banach space. Show that for every R λ f eB , kkλ ∗ f − f kB → O, and conclude that if f ∈ B ∩ L1 (R), f = limλ→∞ −λ (1 − |ξ|/λ)fˆ(ξ)eiξx dξ in the B norm. ˆ , 15. Construct a continuous function f ∈ L1 (R) such that fˆ ∈ L1 (R) ˆ ˆ f (2πn) = 0 for all integers n, f (0) = 1 and f (n) = 0 for all integers n 6= 0. Hints: R 1 (a) We denote kf kA(R) = 2π |fˆ(ξ)|dξ . Let g be continuous with support in [0, 2π] and such that gˆ ∈ L1 (R). Write

gN (x) =

N |j| 1 X 1− N +1 N +1

g(x − 2πj).

N

Show that gˆN (ξ) = (N + 1)−1 KN (ξ)ˆ g (ξ) where KN is the 2π -periodic Fejér kernel, and deduce that kgN kA(R) → 0 as N → ∞.

144


1 ˆ c (j) (b) Let g (j) be nonnegative ( continuous functions such that g ∈ L (R) P∞ 1 0 < x < 1, and such that 1 g (j) (x) = Then, if Nj → ∞ fast 0 otherwise. (j) enough, kgN k ≤ 2−j and f = j A(R)

P∞

(j)

j=1

gNj has the desired properties.

2 FOURIER-STIELTJES TRANSFORMS.

2.1 We denote by M (R) the space of all finite Borel measures on R. M (R) is identified with the dual space of C0 (R)—the (sup-normed) space of all continuous functions on R which vanish at infinity—by means of the coupling Z (2.1) hf, µi = f dµ f ∈ C0 (R), µ ∈ M (R), R The total mass norm on M (R) is defined by kµkM (R) = |dµ| and is identical to the "dual space" norm defined by means of (2.1). The mapping f 7→ f (x)dx identifies L1 (R) with a closed subspace of M (R). The convolution of a measure µ ∈ M (R) and a function ϕ ∈ C0 (R) is defined by the integral Z (2.2) (µ ∗ f )(x) = ϕ(x − y)dµ(y).

and it is clear that µ∗ϕ ∈ C0 (R) and that kµ∗ϕk∞ ≤ kµkM (R) kϕk∞ . The convolution of two measures, µ, ν ∈ M (R), can be defined by means of duality and (2.2), analogously to what we have done in 1.7, or directly by defining Z (µ ∗ ν)(E) =

µ(E − y)dν(y)

for every Borel set E . Whichever way we do it, we obtain easily that kµ ∗ νkM (R) ≤ kµkM (R) kνkM (R) . 2.2 The Fourier-Stieltjes transform of a measure µ ∈ M (R) is defined by: Z Z ˆ (2.3) µ ˆ(ξ) = eiξx dµ(x) = e−iξx dµ(x) ξ ∈ R.

It is clear that if µ is absolutely continuous with respect to Lebesgue measure, say dµ = f (x)dx, then µ ˆ(ξ) = fˆ(ξ). Many of the properties 1 of L Fourier transforms are shared by Fourier-Stieltjes transforms: if µ, ν ∈ M (R) then |ˆ µ(ξ)| ≤ kµkM (R) , µ ˆ(ξ) is uniformly continuous, and


145

µd ∗ ν(ξ) = µ ˆ(ξ)ˆ ν (ξ). A departure from the theory of L1 Fourier transforms is the failing of the Riemann-Lebesgue lemma (the same way it fails for M (T)); the Fourier-Stieltjes transform of a measure µ need not vanish at infinity.

Theorem (Parseval’s formula). Let µ ∈ M (R) and let f be a conˆ . Then† tinuous function in L1 (R) such that fˆ ∈ L1 (R) Z Z 1 (2.4) f (x)dµ(x) = fˆ(ξ)ˆ µ(−ξ). 2π

P ROOF : By (1.11) f (x) =

hence Z

1 f (x)dµ(x) = 2π

ZZ

1 2π

Z

fˆ(ξ)eiξx dξ;

1 fˆ(ξ)eiξx dµ(x) dξ = 2π

Z

fˆ(ξ)ˆ µ(−ξ).

J

Corollary (uniqueness theorem). If µ ˆ(ξ) = 0 for all ξ , then µ = 0. ˆ justifies the change of order of integraThe assumption fˆ(ξ) ∈ L1 (R) tion (by Fubini’s theorem); however, it is not really needed. Formula ˆ , and is (2.4) is valid under the weaker assumption fˆ(ξ)ˆ µ(−ξ) ∈ L1 (R) 1 valid for all bounded continuous f ∈L (R) if we replace the integral on Rλ |ξ| 1 the right by limλ→∞ 2π 1 − fˆ(ξ)ˆ µ(−ξ)dξ (cf. exercise 1.10). λ −λ

2.3 The problem of characterizing Fourier-Stieltjes transforms among ˆ is very hard. As bounded and uniformly continuous functions on R far as local behavior is concerned this is equivalent to characterizing ˆ : every f ∈ A(R) ˆ is a Fourier-Stieltjes transform, and on the other A(R) hand, if µ ∈ M (R) and Vλ is de la Vallée Poussin’s kernel (1.14), then ∗ Vλ (ξ) = µ ˆ(ξ) for |ξ| ≤ λ. µ ∗ Vλ ∈ L1 (T) and µ\ The following theorem is analogous to I.7.3: ˆ , define Φλ by: Theorem. Let ϕ be continuous on R Z λ 1 |ξ| Φλ (x) = 1− ϕ(ξ)eiξx dξ . 2π −λ λ

Then ϕ is a Fourier-Stieltjes transform if, and only if, Φλ ∈ L1 (R) for all λ > 0, and kΦλ kL1 (R) is bounded as λ → ∞. R R † Notice that (2.4) is equivalent to

f (x)dµ(x) = 1/2π

fˆ(ξ)ˆ µ(ξ)dξ

146


P ROOF : If ϕ = µˆ with µ ∈ M (R), then Φλ = µ ∗ Kλ . It follows that for all λ > 0, Φλ ∈ L1 (R) and kΦλ kL1 (R) ≤ kµkM (R) . If we assume that Φλ ∈ L1 (R) with uniformly bounded norms, we consider the measures Φλ (x)dx and denote by µ a weak-star limit point of Φλ (x)dx as λ → ∞. We claim that ϕ = µ ˆ and since both functions are continuous, this will follow if we show that Z Z (2.5) ϕ(−ξ)g(ξ)dξ = µ ˆ(−ξ)g(ξ)dξ for every twice continuously differentiable g with compact support. For ˆ with G ∈ L1 ∩ C0 (R); by Parseval’s formula such g we have g = G Z Z λ |ξ| g(ξ)ϕ(−ξ) 1 − g(ξ)ϕ(−ξ)dξ = lim dξ λ→∞ −λ λ Z Z = lim 2π G(x)Φλ (x)dx = 2π G(x)dµ(x) λ→∞ Z = g(ξ)ˆ µ(−ξ)dξ and the proof is complete.

J

Remark: The application of Parseval’s formula above is typical and is the, more or less, standard way to check that weak-star limits in M (R) are what we expect them to be. Nothing like that was needed in the case of M (T) since weak-star convergence in M (T) implies pointwise convergence of the Fourier-Stieltjes coefficients (the exponentials be1ng to C(T) of which M (T) is the dual). The exponentials on R do not be1ong to C0 (R) and it is false that weak-star convergence in M (R) implies pointwise convergence of the Fourier-Stieltjes transforms (cf. exercise 1 at the end of this section.) However, the argument above gives: Lemma. Let µn ∈ M (R) and assume that µn → µ in the weak-star topology. Assume also that µ ˆn (ξ) → ϕ(ξ) pointwise, ϕ being continuous ˆ . Then µ on R ˆ = ϕ. 2.4 A similar application of Parseval’s formula gives the following useful criterion: ˆ , is a FourierTheorem. A function ϕ defined and continuous on R Stieltjes transform if, and only if, there exists a constant C such that Z 1 fˆ(ξ)ϕ(−ξ)dξ| ≤ C supx |f (x)| (2.6) | 2π


147

for every continuous f ∈ L1 (R) such that fˆ has compact support.

P ROOF : If ϕ = µˆ, (2.6) follows from (2.4) with C = kµkM (R) . if (2.6) R 1 holds, f 7→ 2π fˆ(ξ)ϕ(−ξ)dξ defines a bounded linear functional on a dense subspace of C0 (R), namely on the space of all the functions f ∈ C0 ∩L1 (R) such that fˆ has a compact support. This functional has a unique bounded extension to CR0 (R), which, by the Riesz representation theorem, has the form f 7→ f (x)dµ(x). Moreover, kµkM (R) ≤ C . Using (2.4) again we see that µ ˆ − ϕ is orthogonal to all the continuous, compactly supported functions fˆ with f ∈ L1 (R), and consequently ϕ=µ ˆ. J Remark: The family {f } of test functions for which (2.6) should be valid can be taken in many ways. The only properties that have been ˆ . Thus we used are that {f } is dense in C0 (R) and {fˆ} is dense in C0 (R) could require the validity of (2.6) only for (a) functions f such that fˆ is infinitely differentiable with compact support; or (b) functions f which are themselves infinitely differentiable with compact support, and so on. 2.5 With measures on R we can associate measures on T simply by integrating 2π -periodic functions. Formally: if E is a Borel set on T (T being identified with (−π, π]) we denote by En the set E + 2πn and write E˜ = ∪En ; if µ ∈ M (R) we define ˜ . µT (E) = µ(E)

It is clear that µT is a measure on T and that, identifying continuous functions on T with 2π -periodic functions on R Z Z (2.7) f (x)dx = f (t)dt. R

T

The mapping µ 7→ µT is an operator of norm one from M (R) onto M (T), and its restriction to L1 (R) is the mapping that we have discussed in section 1.15. It follows from (2.7) that µ ˆ(n) = µ ˆT (n) for all n; thus the restriction of a Fourier-Stieltjes transform to the integers gives a sequence of Fourier-Stieltjes coefficients. ˆ , is a FourierTheorem. A function ϕ defined and continuous on R Stieltjes transform if, and only if, there exists a constant C > 0, such that for all λ > 0, {ϕ(λn)}∞ n=−∞ are the Fourier-Stieltjes coefficients of a measure of norm ≤ C on T.

148


P ROOF : If ϕ = µˆ with µ ∈ M (R) we have ϕ(n) = µˆ(n) = µˆT (n) with kµT k ≤ kµk. Writing dµ(x/λ) for the measure satisfying Z x Z f (x) dµ = f (λx) dµx λ \ we have kµ(x/λ)kM (R) = kµkM (R) and µ(x/λ)(ξ) =µ ˆ(ξλ). This implies \ T (n) and the "only if" part is established. ϕ(λn) = µ(x/λ)

For the converse we use 2.4. Let f be continuous and integrable on R and assume that fˆ is infinitely differentiable and compactly supR 1 ported. We want to estimate the integral 2π fˆ(ξ)ϕ(−ξ)dξ and, since the integrand is continuous and compactly supported, we can approximate the integral by its Riemann sums. Thus, for arbitrary ε > 0, if λ is small enough: 1 Z λ X (2.8) fˆ(ξ)ϕ(−ξ)dξ < fˆ(λn)ϕ(−λn) + ε . 2π 2π Now, (λ/2π)fˆ(λn) are the Fourier coefficients of the function ψλ (t) = P∞ m=−∞ f ((t + 2πm)/λ) on T, and since the infinite differentiability of fˆ implies a very fast decrease of f (x) as |x| → ∞, we see that if λ is sufficiently small (2.9)

sup|ψλ (t)| ≤ sup|f (x)| + ε .

Assuming that ϕ(λn) = µˆλ (n), µλ ∈ M (T) and kµλ kM (T) ≤ C , we obtain from Parseval’s formula λ X X ˆ ˆλ (n)ˆ = f (λn)ϕ(−λn) ψ µ (−n) ≤ C sup|ψλ (t)|; λ 2π by (2.8) and (2.9) Z 1 ˆ(ξ)ϕ(−ξ) dξ ≤ C sup|f (x)| + (C + 1)ε f 2π and since ε > 0 is arbitrary, (2.6) is satisfied and the theorem follows from theorem 2.4. J 2.6 Parseval’s formula also offers an obvious criterion for determining when a function ϕ is the Fourier-Stieltjes transform of a positive measure. The analog to 2.4 is


149

ˆ , is the FourierTheorem. A function ϕ, bounded and continuous on R Stieltjes transform of a positive measure on R if, and only if, Z (2.10) fˆ(ξ)ϕ(−ξ) ≥ 0

for every nonnegative function f which is infinitely differentiable and compactly supported. PROOF : Parseval’s formula clearly implies the "only if" part and also the fact that if we assume ϕ = µ ˆ with µ ∈ M (R), then µ is a positive measure. To complete the proof we show that (2.10) implies (2.6), with C = ϕ(0), for every real-valued, compactly supported infinitely differentiable f (hence with C = 2ϕ(0) for complex-valued f ). As usual, we denote by Kλ (x) the Fejér kernel (1.8) and notice that 2 sin λx/2 1 λ−1 Kλ (x) = K(λx) = 2π is nonnegative and tends to 1/2π , λx/2 as λ → 0, uniformly on compact subsets of R. By (1.12) the Fourier transform of K(λx) is λ−1 max(1 − |ξ|/λ, 0) and, as ϕ(ξ) is continuous at ξ = 0, Z 1ˆ (2.11) lim Kλ (ξ)ϕ(−ξ) dξ = ϕ(0). λ→0 λ

If f is real-valued and compactly supported and ε > 0, then, for sufficiently small λ and all x, 2π(ε + sup|f |)K(λx) − f (x) ≥ 0; ˆ hence, by (2.10) and (2.11), if fˆ ∈ L1 (R) Z 1 (2.12) fˆ(ξ)ϕ(−ξ) dξ ≤ ϕ(0)(2ε + sup|f |), 2π

rewriting (2.12) for −f and letting ε → 0 we obtain: Z 1 ˆ(ξ)ϕ(−ξ) dξ ≤ ϕ(0) sup|f |. f 2π

J

2.7 The analog to 2.5 is: ˆ , is the FourierTheorem. A function ϕ, defined and continuous on R Stieltjes transform of a positive measure, if and only if, for all λ > 0, {ϕ(λn)}∞ n=−∞ are the Fourier-Stieltjes coefficients of a positive measure on T.

150


P ROOF : The "only if" part follows as in 2.5. For the "if" part we notice first that if ϕ(λn) = µ ˆλ (n) with µλ > 0 on T, then kµλ k = ϕ(0) and consequently, by 2.5, ϕ is a Fourier-Stieltjes transform. Using the continuity of ϕ we can now establish (2.10) by approximating the integral by its Riemann sums as in the proof of 2.5. J ˆ is said to be positive definite 2.8 Definition: A function ϕ defined on R ˆ and complex numbers z1 , . . . , zN , if, for every choice of ξ1 , . . . , ξN ∈ R we have N X

(2.13)

ϕ(ξj − ξk )zj zk ≥ 0 .

j,k=1

Immediate consequences of (2.13) are: (2.14)

ϕ(−ξ) = ϕ(ξ)

and |ϕ(ξ)| ≤ ϕ(0) .

(2.15)

In order to prove (2.14) and (2.15), we take N = 2, z1 = 1, z2 = z ; then (2.13) reads ϕ(0)(1 + |z|2 ) + ϕ(ξ)z + ϕ(−ξ)¯ z ≥ 0;

set z = 1, we get ϕ(ξ) + ϕ(−ξ) real; set z = i, we get i(ϕ(ξ) − ϕ(−ξ) real, hence (2.14). If we take z such that zϕ(ξ) = −|ϕ(ξ)| we obtain: 2ϕ(0) − 2|ϕ(ξ)| ≥ 0

which establishes (2.15). ˆ , is a Fourier-Stieltjes Theorem (Bochner). A function ϕ defined on R transform of a positive measure if, and only if, it is positive definite and continuous.

P ROOF : Assume first ϕ = µˆ with µ ≥ 0. Let ξ1 , . . . , ξN ∈ N and z1 , . . . , zN be complex numbers; then Z X X e−iξj x zj eiξk x zk dµ(x) ϕ(ξj − ξk )zj zk = (2.16)

j,k

Z X N 2 = zj e−iξj x dµ(x) ≥ 0 1


151

so that Fourier-Stieltjes transforms of positive measures are positive definite. If, on the other hand, we assume that ϕ is positive definite, it follows that for all λ > 0, {ϕ(λn)} is a positive definite sequence (cf. I.7.6). By Herglotz’ theorem I.7.6, ϕ(λn) = µ ˆλ (n) for some positive measure µλ on T, and by theorem 2.7, ϕ = µ ˆ for some positive µ ∈ M (R). J ?2.9 Some assumption of continuity of ϕ in Bochner’s theorem is essential, but one may assume only that it is continuous at ξ = 0 since a positive definite function which is continuous at ξ = 0 is uniformly continuous on the line. This can be obtained directly from condition (2.13) or as a consequence of Lemma I.7.7 applied to ϕ(λn), and letting λ → 0 (see exercise 9).

Lemma. Let ϕ = µ ˆ for some positive µ ∈ M (R). Assume that ϕ is twice Rdifferentiable at ξ = 0 or just that 2ϕ(0) − ϕ(h) − ϕ(−h) = O h2 . Then x2 dµ < ∞, and ϕ has a uniformly continuous second derivative ˆ. on R

P ROOF : The assumption is that for some constant C , Z −2 2h (ϕ(0) − ϕ(h) − ϕ(−h)) = 2h−2 (1 − cos hx)dµ(x) ≤ C. Since the integrand is nonnegative, for every a > o, Z Z a x2 dµ(x) ≤ lim inf 2h−2 (1 − cos hx)dµ(x) ≤ C. −a

h→0

Now, ν = x2 µ ∈ M (R), and ϕ00 = −ˆ ν.

J

Notice that if 2ϕ(0) − ϕ(h) − ϕ(−h) = o(h2 ), we have µ = ϕ(0)δ0 . By induction on m we obtain Proposition. Let ϕ = µ ˆ for some positiveRµ ∈ M (R). Assume that ϕ is 2m-times differentiable at ξ = 0, then x2m dµ < ∞, and ϕ has ˆ . If ϕ(2m) (0) = 0, a uniformly continuous derivative of order 2m on R then µ = ϕ(0)δ0 . ?2.10 Positive definite functions which are analytic at ξ = 0 are automatically analytic in a strip {ζ : ζ = ξ + iη, |η| < a}, with a > 0. By Bochner’s theorem (and the previous remark) such functions are Fourier-Stieltjes transforms of positive measures.

152


Lemma. Let µ be a positive measure on R. AssumeR that F (ξ) = µ ˆ(ξ) is analytic at ξ = 0. Then there exists b > 0 such that eb|x| dµ < ∞ and ˆ of the function µ ˆ is the restriction to R Z (2.17) F (ζ) = e−iζx dµ(x). P F (n) (0) n P ROOF : The assumption is: for some a > 0, F (ξ) = ∞ ξ 0 n! (n) −n in |ξ| ≤ a . This implies |F (0)| ≤ Cn!a , and in particular that R 2m x dµ ≤ C(2m)!a−2m . Since |x|2m+1 ≤ x2m + x2m+2 , we have Z |x|2m+1 dµ ≤ (2 + a2 )C(2m + 2)!a−2m+2

and (2.18)

Z

η|x|

e

R n X Z η n |x|n X n |x| dµ dµ = dµ = η λ |dµ| = 0. a. Prove that B ∈ U BS implies equi-continuity of {ˆ µ : µ ∈ B}. b. If µn are probability measures, and {ˆ µn } is equi-continuous at ξ = 0 then {µn } ∈ U BS . c. If {µn } ∈ U BS , and µn → µ in the weak-star topology, then µ ˆn (ξ) → µ ˆ(ξ) ˆ. uniformly on compact subsets of R 4. Let µn ∈ M (R) such that kµn k ≤ 1. Assume that µ ˆn converges pointwise to a continuous function ϕ. Show that ϕ = µ ˆ for some µ ∈ M (R) such that kµk ≤ 1; moreover, µn → µ weak-star. p Pn 5. Let λn be integers such that λn+1 /λn > 3. Write Xn = n2 1 cos λj t. (Xn are real-valued random variables on the probability space T endowed with the normalized Lebesgue measure). Prove that Xn converge in distribution to a normal variable. Hint: Exercise V.1.11. ˆ and (2.10) is valid, then ϕ is posi6. Show that if ϕ is continuous on R tive definite. Conclude from (2.15) that the boundedness assumption of 2.6 is superfluous. 7. Prove Theorem 2.12. 8. Express µ{[a, b]} and µ{(a, b)} in terms of µ ˆ. ([a, b] is the closed interval with endpoints a and b, and (a, b) is the open one.) 9. Let an be complex numbers, |an | ≤ 1. Set bn = an+1 − an and cn = √ bn − bn−1 = an+1 + an−1 − 2an . Prove that if |cn | ≤ ε, then |bn | ≤ 2 ε. a. How does this imply that a positive definite function which is continuous at ˆ? ξ = 0 is uniformly continuous on R b. Prove that if ϕ is positive definite and ϕ(0) − 0. Hint: Construct a continuous function with compact support which is not a Fourier transform of a summable function. 3 FOURIER TRANSFORMS IN Lp (R); 1 < p ≤ 2.

The definition of the Fourier transform (i.e., Fourier coefficients) for functions in various function spaces on T, was largely simplified by the fact that all these spaces were contained in L1 (T). The fact that the Lebesgue measure of R is infinite changes the situation radically. If p > 1 we no longer have Lp ⊂ L1 , and, if we want to have Fourier transforms for functions in Lp (R) (or other function spaces on R), we have to find a new way to define them. In this section we consider the case 1 < p ≤ 2 and obtain a reasonably satisfactory extension of the Fourier transformation for this case. 3.1 We start with L2 (R). Lemma. Let f be continuous and with compact support on R; then Z Z 1 |fˆ(ξ)|2 dξ = |f (x)|2 dx . 2π

We give two proofs.

P ROOF I: Assume first that the support of f is included in (−π, π). By theorem I.5.5, 1 2π

Z

|f (x)|2 dx =

∞ X 1 ˆ 2 f (n) 2π n=−∞

and replacing f by e−iαx f we have (3.1)

Z

∞ 1 X ˆ |f (x)| dx = |f (n + α)|2 ; 2π n=−∞ 2

integrating both sides of (3.1) with respect to α on 0 ≤ α ≤ 1, we obtain Z Z 1 2 |f (x)| dx = |fˆ(ξ)|2 dξ. 2π

156


If the support of f is not in (−π, π), we consider g(x) = λ1/2 f (λx). If λ is sufficiently big, the support of g is included in (−π, π) and, since gˆ(ξ) = λ−1/2 fˆ(ξ/λ), we obtain Z Z Z Z 1 1 2 2 2 |f (x)| dx = |g(x)| dx = |ˆ g (ξ)| dξ = |fˆ(ξ)|2 dξ. 2π 2π J R = f (x)f (x)dx = R RP ROOF 2II: Write g = f ∗ f2 (−x); we have g(0) |f (x)| dx and gˆ(ξ) = |fˆ(ξ)| . If we know that |fˆ(ξ)|2 dξ < ∞ (e.g., we assume that f is differentiable), it follows from the inversion formula (1.11) that Z Z 1 |fˆ(ξ)|2 dξ = g(0) = |f (x)|2 dx . 2π

In the general case we may apply Fejér’s theorem and obtain Z 1 |ξ| lim 1− |fˆ(ξ)|2 dξ = g(0) λ→∞ 2π λ and, since the integrand is nonnegative, its C-1 summability is equivalent to its convergence and the proof is complete. J ˆ we write D EFINITION : For g ∈ L2 (R) kgkL2 (R) ˆ =

1 2π

Z

|g(ξ)|2 dξ

1/2

.

Theorem (Plancherel). There exists a unique operator F from L2 (R) ˆ having the properties: onto L2 (R)

(3.2)

Ff = fˆ for f ∈ L1 ∩ L2 (R),

(3.3)

kFf kL2 (R) ˆ = kf kL2 (R) .

Remark: In view of (3.2) we shall often write fˆ instead of Ff .

P ROOF : We notice first that L1 ∩ L2 (R) is dense in L2 (R) and consequently any continuous operator defined on L2 (R) is determined by its values on L1 ∩ L2 (R). This shows that there exists at most one operator satisfying (3.2) and (3.3). By the lemma, (3.3) is satisfied if f is continuous with compact support, and since continuous functions with compact support are dense in L1 ∩ L2 (R) (with respect to the norm k kL1 (R) + k kL2 (R) ), (3.3) holds for all f ∈ L1 ∩ L2 (R). The mapping


157

f 7→ fˆ clearly can be extended by continuity to an isometry from L2 (R) ˆ . Finally, since every twice differentiable compactly supinto L2 (R) ˆ is the Fourier transform of a bounded integrable ported function on R function on R (1.5 and the inversion formula), it follows that the range ˆ and hence coincides with it. of f 7→ fˆ is dense in L2 (R) J

Remarks: (a) Given a function f ∈ L2 (R) we define fˆ as the limit (in ˆ ) of fˆn , where fn is any sequence in L1 ∩ L2 (R) which converges L (R) to f in L2 (R). As such a sequence we can take ( f (x) |x| < n fn = 0 |x| ≥ n 2

and obtain the following form of Plancherel’s theorem: the sequence Z n (3.4) f (x)e−iξx dx fc n (ξ) = n

ˆ , to a function which we denote by fˆ, and for which converges, in L2 (R) (3.2) and (3.3) are valid. ˆ , (b) The mapping f 7→ fˆ being an isometry of L2 (R) onto L2 (R) clearly has an inverse. Using theorem 1.11 and the fact that we have an isometry, we obtain the inverse map by f = lim f(n) in L2 (R) where Z n 1 (3.5) fˆ(ξ) dξ . f(n) (x) = 2π −n

(c) Parseval’s formula Z π Z 1 (3.6) fˆ(ξ)ˆ g (ξ)dξ f (x)g(x)dx = 2π −π for f. g ∈ L2 (R), follows immediately from (3.3) (and in fact is equivalent to it). 3.2 We turn now to define Fourier transforms for functions in Lp (R), 1 < p < 2. Using the Riesz-Thorin theorem and the fact that F : f 7→ fˆ ˆ and from L2 (R) onto has norm 1 as operator from L1 (R) into L∞ (R) ˆ , we obtain as in IV.2: L2 (R) Theorem (Hausdorff-Young). Let 1 < p < 2, q = p/(p − 1) and f ∈ L1 ∩ L2 (R). Then 1/p 1/q Z Z 1 . ≤ |f (x)|p dx |fˆ(ξ)|q dξ 2π

158


For f ∈ Lp (R), 1 < p < 2, we define fˆ by continuity; for R n now −iξx q ˆ example, as the limit in L (R) of −n e f (x)dx. The mapping F : ˆ ˆ ; f 7→ f so defined is an operator of norm 1 from Lp (R) into Lq (R) however, it is no longer an isometry and the range is not the whole of (see exercise 10 at the end of this section). 3.3 The fact that for p < 2, F is not an invertible operator from Lp (R) ˆ makes the inversion problem more delicate than it is for onto Lq (R) 2 L . The situation in the case of Lp (R) is similar to that which we encountered for Lp (T). We have inversion formulas both in terms of summability and in terms of convergence. The summability result can be stated in terms of general summability kernels without reference to the Fourier transform as we did in 1.10 for L1 (R); and in fact the statement of Theorem 1.10 remains valid if we replace in it L1 (R)by Lp (R), 1 ≤ p < ∞. For p ≤ 2 we can generalize theorem 1.11. We first check (see exercise 9 at the end of this section) that if g ∈ Lp (R) and ∗ g = fˆgˆ. f ∈ L1 (R) then f ∗ g is a well-defined element in Lp (R) and fd This is particularly simple if we take for f the Fejér kernel Kλ : we have |ξ| \ K ∗ g = 1 − gˆ λ λ

and, since Kλ ∗ g is clearly bounded (Kλ ∈ Lq (R), q = p/(p − 1)) and hence belongs to L1 ∩ L∞ (R) ⊂ L2 (R), it follows that Z λ 1 |ξ| Kλ ∗ g = 1− gˆ(ξ)eiξx dξ 2π −λ λ and from the general form of theorem 1.10 we obtain: Theorem. Let g ∈ Lp (R), 1 ≤ p ≤ 2; then 1 λ→∞ 2π

g = lim

λ

|ξ| 1− gˆ(ξ)eiξx dξ λ −λ

Z

in the Lp (R) norm. Corollary. The functions whose Fourier transforms have compact support form a dense subspace of Lp (R). 3.4 The analog to the inversion given by 3.1, remark (b) (i.e., convergence rather than summability) is valid for 1 < p < 2 but not as easy to prove as for p = 2; it corresponds to theorem II.1.5 and can be


159

proved either through the study of conjugate harmonic functions in the half-plane, analogous to that done for the disc in chapter III, or directly from II.1.5. The idea needed in order to obtain the norm inversion formula for Lp (R), 1 < p < 2, from II.1.5 is basically the one we have used in proof I of lemma 3.1. RN 1 For f ∈ ∪1≤p≤2 Lp (R) we write SN (f, x) = 2π fˆ(ξ)eiξx dξ. −N Lemma. For 1 0.

P ROOF : Inequality (3.7) is equivalent to the statement that, for M → ∞ !1/p Z M

|SN (f, x)|p dx

(3.8)

≤ Cp kf kLp (R) .

−M

Writing ϕM (x) = M 1/p f (M x) we see that kϕM kLp (R) = kf kLp (R) and check that SM N (ϕM , x) = M 1/p SN (f, M x).

(3.9)

In view of (3.9), (3.8) is equivalent to 1/p Z 1 (3.10) |SM N (ϕM , x)|p dx ≤ Cp kϕM kLp (R) . −1

As M → ∞, the support of ϕM shrinks to zero and consequently the lemma will be proved if we show that (3.8) is valid, with an appropriate Cp , for all f with support contained in (−π, π) for M = 1 (or any other fixed positive number) and for all integers N . We now write Z 1 NX −1 1 ˆ (3.11) SN (f, x) = f (n + α)ei(n+α)x dα 2π 0 −N P −1 1 ˆ inx and notice that N is a partial sum of the Fourier −N ( 2π )f (n + α)e −iαx series of f (x)e (it is carried by (−π, π) which we now identify with T). As an Lp (T)-valued function of α, the integrand in (3.11) is clearly continuous† and, by II.1.2 and II.1.5, it is bounded in Lp (T) by a constant multiple of kf eiαx klt[p] = (2π)−1/p kf kLp (R) . We therefore obtain 1/p 1/p Z π Z 1 p p ≤ Cp kf kLp (R) |SN (f, x)| dx |SN (f, x)| dx ≤ −1 † Note

−π

that f , having a compact a support, is in L1 (R) and f is therefore continuous.

160



J

Corollary. For 1 < p ≤ 2, inequality (3.7) is valid for all f ∈ Lp (R).

P ROOF : Write f = limn→∞ fn , with fn ∈ Lp (R), fn having compact supports and the limit being taken in the Lp (R) norm. By theorem 3.2, fˆ = lim fˆn in Lq (R) and consequently, we have for every fixed N > 0, SN (f, x) = limn SN (fn , x) uniformly in x. It follows that kSN (f )kLp (R) ≤ lim inf kSN (fn )kLp (R) ≤ n

Cp limkfn kLp (R) = Cp kf kLp (R) . n

J

Theorem. Let f ∈ Lp (R), 1 < p ≤ 2. Then lim kSN (f ) − f kLp (R) = 0.

N →∞

P ROOF : {SN } is a uniformly bounded family of operators which converge to the identity, as N → ∞, on all functions with compactly supported Fourier transform, and hence, by corollary 3.3, it converges to the identity in the strong topology. J

EXERCISES FOR SECTION 3 1. Let B ⊂ Lc be a homogeneous Banach space on R and let f ∈ B . Show that: (a) for every ϕ ∈ L1 (R), ϕ ∗ f can be approximated (in the B norm) by linear combinations of translates of f . In other words: for every ε > 0, there existPnumbers y1 , . . . , yn ∈ R and complex numbers A1 , . . . , An such that n kϕ ∗ f − j=1 Aj fyj k < ε, where fy (x) = f (x − y). (b) For every y ∈ R, fy can be approximated by functions of the form ϕ ∗ f with ϕ ∈ L1 (R). Deduce that a closed subspace H of B is translation invariant (i.e., f ∈ H implies fy ∈ H for all y ∈ R) if, and only if, f ∈ H implies ϕ ∗ f ∈ H for every ϕ ∈ L1 (R). ˆ and assume F (ξ) = 0 implies G(ξ) = 0 for almost all 2. Let F, G ∈ L2 (R) ˆ ξ ∈ R. Show that, given ε > 0, there exists a twice-differentiable compactly supported function Φ such that kΦF − GkL2 (R) ˆ < ε.

3. Let f, g ∈ L2 (R) and assume that fˆ(ξ) = 0 implies gˆ(ξ) = 0 for almost ˆ . Show that g can be approximated on L2 (R) by linear combinations all ξ ∈ R of translates of f . Hint: Use exercises I and 2 and Plancherel’s theorem.


161

ˆ denote HE = {f ∈ L2 (R) : fˆ = 11E fˆ}. 4. For measurable sets E ⊂ R Prove that HE is a closed translation invariant subspace of L2 (R), and that every closed translation invariant subspace of L2 (R) is obtained this way. 5. Show that every closed translation invariant subspace of L2 (R) is singly generated. 6. Let f ∈ L2 (R). Show that the translates of f generate L2 (R) if, and only if, fˆ 6= 0 almost everywhere. 7. The information obtained from exercises 2 through 6 can be obtained very easily by duality arguments (i.e., using the Hahn-Banach theorem). For instance, by Plancherel’s theorem, exercise 6 is equivalent to the statement that ˆ if, and only if, fˆ(ξ) 6= 0 almost everywhere. By the {fˆ(ξ)eiξx }x∈R spans L2 (R) ˆ if, and only if, there Hahn-Banach theorem {fˆ(ξ)eiξx }x∈R does not span L2R(R) ˆ , not identically zero, such that fˆ(ξ)ψ(ξ)eiξx dξ = 0 for is a function ψ ∈ L2 (R) all x ∈ R. By the uniqueness theorem this is equivalent to: fˆψ¯ = 0 identically, that is, f vanishes on the support of ψ . Use the same method to prove exercises 3 through 5. 8. Both the "if" and the "only if" parts of exercise 6 are based on Plancherel’s theorem and are both false for Lp (R), p < 2. Assuming the existence of a measure µ carried by a closed set of measure zero and such that µ ˆ ∈ Lq for all q > 2, 1 ∞ ˆ construct a function f ∈ L ∩ L (R) such that f (ξ) 6= 0 almost everywhere and such that the translates of f do not span Lp (R) for any p < 2. ˆ. Hint: Put µ on R 9. Show that if f ∈ L1 (R) and g ∈ Lp (R), then fd ∗ g = fˆgˆ. p ˆ We denote by F L the space of all functions f such that f ∈ Lp (R), (thus 1 ˆ ). By definition: F L = A(R) kfˆkF Lp = kf kLp (R) .

10. If µ ∈ M (R) and ϕ ∈ F Lp , 1 ≤ p ≤ 2, then µ ˆϕ ∈ F Lp . p 11. Show that if ϕ ∈ F L , 1 0

0

ξ ≤ 0,

then ψ ∈ F Lp , and kψk ≤ Cp kϕF Lp k. 12. Let α and β be real numbers, αβ 6= 0. Show that if ϕ ∈ F Lp , I 0 ψα,β (ξ) = ϕ(βξ) ξ ≤ 0, then ψα,β ∈ F Lp . R 13. Let f ∈ Lp (R), 1 < p ≤ 2. Show that h(x) = π −1 f (x − y) sin y/ydy is well defined and continuous on R, h ∈ Lp (R) and khkLp (R) ≤ Cp kf kLp (R) . 14. Show that, for 1 ≤ p < 2, the norms kϕkF Lp and kϕkLq (R) ˆ are not ˆ (q = p/(p − 1)). Hint: See IV.2. equivalent. Deduce that F Lp 6= Lq (R)

162


4 TEMPERED DISTRIBUTIONS AND PSEUDO-MEASURES

In the previous section we defined the Fourier transforms for functions in Lp (R), 1 < p ≤ 2, by showing that on dense subspace on which F : f 7→ fˆ is already well defined (e.g., on L1 ∩ L2 (R)), we have the norm inequality kfˆkLq (R) ˆ ≤ kf kLp (R) and consequently there exists a unique continuous extension of F , as ˆ . If p > 2 this procedure fails. It is an operator from Lp (R) into Lq (R) not hard to see that not only is it impossible to extend the validity of the Hausdorff-Young theorem for p > 2, but also there is no homogeneous Banach space B on R such that for some p > 2, some constant C and all f ∈ L1 ∩ L∞ (R), kfˆkB ≤ Ckf kLp (R) . So, a different procedure is needed if we want to extend the notion of Fourier transforms to Lp (R), p > 2. Clearly, we try to extend the notion, keeping as many of its properties as possible; in particular, we would like to keep some form of the inversion formula and the very useful Parseval’s formula. We realize immediately that, since the Fourier transforms of measures are bounded functions, if any reasonable form of inversion is to be valid, the Fourier transforms of some bounded functions will have to be measures; and once we accept the idea that Fourier transforms need not be functions but could be other objects, such as measures, the procedure that we look for is given to us by Parseval’s formula. So far we have established Parseval’s formula for various function spaces as a theorem following the definition of the Fourier transforms of functions in the corresponding spaces. In this section we consider Parseval’s formula as a definition of Fourier transform for a much larger class of objects. Having proved Parseval’s formula for Lp (R), 1 ≤ p ≤ 2, we are assured that our new definition is consistent with the previous ones. 4.1 We denote by S(R) the space of all infinitely differentiable functions on R which satisfy:

(4.1)

lim xn f (j) (x) = 0

|x|→∞

for all n ≥ 0, j ≥ 0.

S(R) is a topological vector space, the topology given by the family of seminorms

(4.2)

kf kj,n = sup|xn f (j) (x)|.


163

This topology on S(R) is clearly metrizable† and S(R) is complete, in other words, S(R) is a Frechet space.

D EFINITION : A tempered distribution on R is a continuous linear functional on S(R). We denote the space of tempered distributions on R that is, the dual of S(R), by S ∗ (R). S(R) is a natural space to study within the theory of Fourier transforms. By theorems 1.5 and 1.6 we see that if f ∈ ˆ ) and, as ξ n fˆ(j) (ξ) is ˆ (the analogous space on R S(R) then fˆ ∈ S(R) n j d (x f (x)) the Fourier transform of (−i)n+j dxn , we see that the mapping ˆ . By the inversion formula f 7→ fˆ is continuous from S(R) into S(R) ˆ and is bicontinuous. this mapping is onto S(R) ˆ We now define µ ˆ, for µ ∈ S ∗ (R), as the tempered distribution on R satisfying (4.3)

hfˆ, µ î = hf, µi

for all f ∈ S(R). The space of tempered distributions on R is quite large. Every function g which is measurable and locally summable, and which is bounded at infinity by a power of x can be identified with a tempered distribution by means of: Z hf, gi = f (x)g(x)dx f ∈ S(R) and so can every g ∈ Lp (R), for any p ≥ 1, and every measure µ ∈ M (R); thus our definition has a very satisfactory domain. However, the range of the definition is as large and this is clearly a disadvantage; it gives relatively little information about the Fourier transform. We thus have to supplement this definition with studies of the following general problem: knowing that a distribution µ ∈ S ∗ (R) has some special properties, what can we say about µ? Much of what we have done in the first three sections of this chapter falls into this category: if µ is (identified with) a summable function, then µ ˆ is (identified with) a function in C0 (R); if µ is a measure, then † A sequence of functions f m ∈ S(R) converges to f if limm→∞ kfm − f kj,n = 0 for all j ≥ 0 and n ≥ 0. The metric in S(R) can be defined by:

dist(f, g) =

X

j,n≥0

kf − gkj,n 1 . 2j+n 1 + kf − gkj,n

164


µ ˆ is a uniformly continuous bounded function; if µ ∈ Lp (R) with ˆ q = p/(p − 1). 1 < p ≤ 2, then µ ˆ ∈ Lq (R), We shall presently obtain some information about Fourier transforms of functions in Lp (R) with 2 < p ≤ ∞ but we should not leave the general setup without mentioning the notion of the support of a tempered distribution.

4.2 D EFINITION : A distribution ν ∈ S ∗ (R) vanishes on an open set O ⊂ R, if hϕ, νi = 0 for all ϕ ∈ S(R) with compact support contained in O. Lemma. Let O1 , O2 be open on R and let K be a compact set such that K ⊂ O1 ∪ O2 . Then there exist two compactly supported C ∞ functions ϕ1 and ϕ2 satisfying: support of ϕj ⊂ oj and ϕ1 + ϕ2 = 1 on K .

P ROOF : Let Uj ⊂ Oj have the following properties: Uj is open, U j is compact and included in Oj , and K ⊂ U1 ∪ U2 . Denote the indicator function of U1 by ψ1 and that of U2 \ U1 by ψ2 . Let ε > 0 be smaller than the distance of K to the boundary of U1 ∪ U2 and also smaller than the distance of Uj to the complement of Oj , j = 1, 2. Let δ(x) be an infinitely differentiable function carried by (−ε, ε) and whose integral is 1. Then we can take ϕj = ψj ∗ δ J Corollary. If ν ∈ S ∗ (R) vanishes on O1 and on O2 , it vanishes on O1 ∪ O2 .

P ROOF : Let f ∈ S(R) have a compact support included in O1 ∪ O2 . Denote the support of f by K and let ϕ1 , ϕ2 be the functions described in the lemma. Then ϕj ∈ S(R), f = f (ϕ1 + ϕ2 ) = f ϕ1 + f ϕ2 . Now hf ϕ1 , νi = 0, hf ϕ1 , νi = 0, and consequently hf, νi = 0. J Our corollary clearly implies that the union of any finite number of open sets on which ν vanishes has the same property, and since our test functions all have compact support, the same is valid for arbitrary unions. The union of all the open sets on which ν vanishes is clearly the largest such set. 4.3 D EFINITION : The support Σ(ν) of ν ∈ S ∗ (R) is the complement of the largest open set O ⊂ R on which ν vanishes.

Remarks: (a) If ν is (identified with) a continuous function g then Σ(ν) is the closure of {x : g(x) 6= 0}. If ν is a measurable function g then Σ(ν)


165

is the closure of the set of points of density of {x : g(x) 6= 0}. The set of points of density of {x : g(x) 6= 0} is a finer notion of support which may be useful (cf. exercise 3.4). (b) The definition of Σ(ν) implies that if ϕ ∈ S(R) and if the support of ϕ is compact and disjoint from Σ(ν) then hϕ, νi = 0. It may be useful to notice that if ψ ∈ S(R) and if δ is infinitely differentiable with compact support and δ(0) = 1, then ψ = limλ→0 δ(λx)ψ in S(R), and consequently if the support of ψ is disjoint from Σ(ν) (but not necessarily compact) we have hψ, νi = limλ→0 hδ(λx)ψ, νi = 0. In particular, if Σ(ν) = ∅ then ν = 0. (c) Let B ⊃ S(R) be a function space and assume that every f ∈ B with compact support can be approximated in the topology of B by functions ϕn ∈ S(R) such that the supports of ϕn tend to that of f . Let ν ∈ S ∗ (R) and assume that ν can be extended to a continuous linear functional on B . If f ∈ B has a compact support disjoint from Σ(ν), then hf, νi = 0. 4.4 S(R) is an algebra under pointwise multiplication. The product f ν of a function f ∈ S(R) and a distribution ν ∈ S ∗ (R) is defined by hg, f νi = hg f¯, νi,

g ∈ S(R).

i.e., the multiplication by f in S ∗ (R) is the adjoint of the multiplication by f in S(R). From the definitions above, it is clear that Σ(f ν) ⊂ Σ(f ) ∩ Σ(ν). ˆ the space of distributions on R ˆ 4.5 We denote by FLp = FLp (R)) p which are Fourier transforms of functions in L (R), 1 < p ≤ ∞ (we ˆ for FL1 (R) ˆ ). FLp inherits from Lp (R) its Bakeep the notation A(R) nach space structure; we simply put kfˆkF Lp = kf kLp (R); and we can identify FLp with the dual of FLq if q = p/(p − 1) < ∞. In parˆ . This identification may be considticular, FL∞ is the dual of A(R) ered as purely formal: writing hfˆ, gî = hf, gi carries the duality from (Lp (R), Lq (R)) to (FLp , FLq ); however, we have already made enough formal identifications to allow a somewhat clearer meaning to the one above. Having identified functions with the corresponding distribuˆ ⊂ FLp and, if p < ∞, S(R) ˆ is dense tions, we clearly have S(R) p in FL ; consequently, every continuous linear functional on FLp is canonically identified with a tempered distribution. The identification of FLq as the dual of FLp now becomes a theorem stating that a disˆ is continuous on S(R) ˆ with respect to the norm tribution ν ∈ S ∗ (R)

166


induced by FLp if, and only if, ν ∈ FLq . We leave the proof as an exercise to the reader. ˆ , We now confine our attention to FL∞ . If µ is a measure M (R) R in iξx it is the Fourier transform of the bounded function h(x) = e dµ(ξ); ˆ ⊂ FL∞ . The elements of FL∞ are commonly referred to thus M (R) ˆ is a relatively small part of as pseudo-measures. It is clear that M (R) ∞ ∞ FL ; for instance, if ϕ ∈ L is not uniformly continuous on R, ϕˆ cannot be a measure. ˆ1 ∗ h ˆ 2 of the pseudo-measures h ˆ 1 and D EFINITION : The convolution h ∞ ˆ 2 , (hj ∈ L (R)), is the Fourier transform of h1 h2 . h Again we reverse the roles; we take something which we have proved for measures, as a definition for the larger class of pseudo-measures. ˆ 1 and h ˆ 2 happen to be measures, h ˆ1 ∗ h ˆ 2 is their (measure Thus, if h theoretic) convolution. 4.6 Another case in which we can identify the convolution is given by Lemma. Let h1 ∈ L∞ (R) and h2 ∈ L1 ∩ L∞ (R); then

(4.4)

ˆ 1 (η)i. ˆ1 ∗ h ˆ 2 )(ξ) = hh ˆ 2 (ξ − η), h (h

We remark first that h1 h2 ∈ L1 ∩ L∞ (R) and consequently ˆ1 ∗ h ˆ 2 = h[ h 1 H2 ∈ A(R) so that we can talk about its value at ξ ∈ R. If h1 ∈ S(R) we have Z ZZ 1 −iξx ˆ ˆ ˆ 1 (η)h2 (x)ei(η−ξ)x dx dη (h1 ∗ h2 )(ξ) = h1 (x)h2 (x)e dx = h 2π Z 1 ˆ 1 (η)i. ˆ 2 (ξ − η)h ˆ 1 (η)dη = hh ˆ 2 (ξ − η), h = h 2π PROOF :

Since S(R) is dense in L∞ (R) in the weak-star topology (as dual of L1 (R)), and since both sides of (4.4) depend on h1 continuously with respect to the weak-star topology, (4.4) is valid for arbitrary h1 ∈ L∞ (R). Corollary. If h1 ∈ L∞ (R) and h2 ∈ L1 ∩ L∞ (R), then ˆ1 ∗ h ˆ 2 ) ⊂ Σ(h ˆ 1 ) + Σ(h ˆ 2 ). Σ(h

4.7 This corollary can be improved: Lemma. Assume h1 , h2 ∈ L∞ (R). Then ˆ1 ∗ h ˆ 2 ) ⊂ Σ(h ˆ 1 ) + Σ(h ˆ 2 ). Σ(h


167

ˆ with compact support P ROOF : Consider a smooth function fˆ ∈ A(R) ˆ ˆ disjoint from Σ(h1 ) + Σ(h2 ). We have to show that hfˆ, hd 1 h2 i = 0 which R d ¯ ¯ c is the same as f h1 h2 dx = hf h1 , h2 i = 0. c) = {ξ : −ξ ∈ Σ(b c ). Now Σ(h h1 )} and, by 4.6, Σ(fd h1 ) ⊂ Σ(fˆ)+Σ(h 1 1 d ˆ b b If ξ ∈ Σ(f h ) ∩ Σ(h ), then there exist η ∈ Σ(f ) and η ∈ Σ(h ) such 0

1

2

0

1

1

that ξ0 = η0 − η1 , that is, η0 = ξ0 + η1 . This would contradict the assumption Σ(fb) ∩ Σ(b h1 ) + Σ(b h2 ) = ∅. It follows that Σ(fd h1 ) is d c b d c disjoint from Σ(h2 ), hence hf , h1 h2 i = hf h1 , h2 i = 0 and the lemma is proved. J

4.8 The reader might have noticed that we were using not only the duality between L1 (R) and L∞ (R) but also the fact that a multiplication by a bounded function is a bounded operator on L1 (R). Another operation between L1 (R) and L∞ (R) which we have used is the convolution that takes L1 × L∞ into L∞ (R). Passing to Fourier transforms we see ˆ the multiplication of a pseudomeasure that FL∞ is a module over A(R) ˆ ˆ . by a function in A(R) being the adjoint of the multiplication in A(R) This extends the notion of multiplication introduced in 4.4. ˆ 4.9 Let k be an R infinitely differentiable function on R, carried by [−1, 1] ˆ we set and such that k(ξ)dξ = 1. For fˆ ∈ A(R) Z fˆλ = λk(λξ) ∗ fˆ = λ k(λη)fˆ(ξ − η)dη. fˆλ is infinitely differentiable, Σ(fˆλ ) ⊂ Σ(fˆ) + [−1/λ, 1/λ], and as ˆ . λ → ∞, fˆλ → fˆ in A(R) ˆ has By 4.3, remark (c) it follows that if ν ∈ FL∞ and if fˆ ∈ A(R) ˆ a compact support disjoint from Σ(ν), we have hf , νi = 0. Further, if ˆ and Σ(fˆ) ∩ Σ(ν) = ∅, it follows that h(1 − |ξ|/λ)fˆ, νi = 0 fˆ ∈ A(R) for all λ > 0 and letting λ → ∞, we obtain hfˆ, νi = 0. For convenient reference we state this as: ˆ . If Σ(fˆ) ∩ Σ(ν) = ∅ then Lemma. Let ν ∈ FL∞ and fˆ ∈ A(R) ˆ hf , νi = 0.

4.10 We leave the proof of the following lemma as an exercise to the reader. ˆ : then Lemma. Let ν ∈ FL∞ and fˆ ∈ A(R) Σ(fˆν) ⊂ Σ(fˆ) ∩ Σ(ν).

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4.11 We show now that a pseudo-measure with finite support is a meaˆ we see that a pseusure. Using the multiplication by elements of A(R) domeasure with finite support is a linear combination of pseudomeasures carried by one point each; thus it would be sufficient to prove: Theorem. A pseudo-measure carried by one point is a measure. ˆ = {0}. If ϕ1 , ϕ2 ∈ AR ˆ P ROOF : Let h ∈ L∞ (R) and assume Σ(h) ˆ ˆ and ϕ1 (ξ) = ϕ2 (ξ) in a neighborhood of ξ = 0, then hϕ1 , hi = hϕ2 , hi. ˆ where ϕ is any function in A(R) ˆ such that ϕ(ξ) = 1 Put c = hϕ, hi near ξ = 0. As usual we denote by K the Fejér kernel and recall that ˆ K(ξ) = sup(1 − |ξ|, 0). By lemma 4.6 we have

(4.5)

c ˆ − η), h(η)i. ˆ hK(ξ) = hK(ξ

c For |ξ| ≥ 1 we clearly have hK(ξ) = 0. If −1 < ξ1 < ξ2 < 0 we have ˆ ˆ K(ξ2 − η) = K(ξ1 − η) = ξ2 − ξ1 for η near zero. By (4.5) and the c 2 − hK(ξ c 2 = c(ξ2 − ξ1 ), and since definition of c we conclude that hK(ξ c c hK(ξ) is continuous, upon letting ξ1 → −1 we obtain hK(ξ) = c(1 + ξ) for −1 < ξ < 0. Repeating the argument for 0 ≤ ξ ≤ 1 we obtain c ˆ hK(ξ) = cK(ξ) and by the uniqueness theorem h(x) = c a.e. It follows ˆ is the measure of mass c concentrated at the origin. that h J

4.12 We add a few remarks about distributions in FLp , 2 < p < ∞. There is clearly no inclusion relation between Lp (R) and L∞ (R) but 0 it might be useful to notice that locally FLp ⊂ F Lp if p ≤ p0 and in particular all distributions in FLp are locally pseudomeasures. (We recall that a tempered distribution ν belongs locally to a set G ⊂ S ∗ (R) ˆ there exists µ ∈ G such that Σ(µ − ν) does not if for every ξ ∈ R ˆ we may take λ > |ξ| and concontain ξ ). If ν ∈ FLp and ξ ∈ R b b λ (ξ) = 1 sider µ = Vλ ν where Vλ is de la Vallée Poussin’s kernel (V −1 for |ξ| ≤ λ, = 2 − |ξ|λ for λ < |ξ| < 2λ, and = 0 for |ξ| ≥ λ). It is clear that ν = µ on (−λ, λ), that is, Σ(µ − ν) ∩ (−λ, λ) = ∅ and if p ∞ ν = fˆ with f ∈ Lp (R), then µ = V\ λ ∗ f and Vλ ∗ f ∈ L ∩ L (R) since 1 q Vλ ∈ L ∩ L (R), q = p/(p − 1). In particular, if Σ(ν) is compact, say Σ(ν) ⊂ (−λ, λ), then µ = ν ; we have thus proved: Theorem. If ν ∈ FLp and Σ(ν) is compact, then ν ∈ FL∞ . 4.13 If ν ∈ FLp ∩ FL∞ we can consider the repeated convolution of ν with itself; writing ν = fˆ with f ∈ Lp ∩ L∞ (R), the convolution of ν with itself m times is the Fourier transform of f m , and if m ≥ p,


169

ˆ . In particular, assuming ν 6= 0 f m ∈ L1 (R) so that ν ∗ · · · ∗ ν ∈ A(R) Σ(ν ∗ · · · ∗ ν) ⊂ Σ(ν) + · · · + Σ(ν) contains an interval. As an immediate consequence we obtain: ˆ, Theorem. Let ν ∈ FLp , p < ∞, and let J be an open interval on R ˆ. such that J ∩ Σ(ν) 6= ∅. Then J ∩ Σ(ν) is a basis for R

Theorems 4.11 and 4.13 are equivalent to the following approximation theorems: ˆ and denote 4.11’ Theorem. Let ξ ∈ R ˆ f (ξ) = 0} I(ξ) = {f : f ∈ A(R), ˆ ξ 6∈ Σ(f )} I0 (ξ) = {f : f ∈ S(R), ˆ topology. Then I0 (ξ) is dense in I(ξ) in the A(R) ˆ be closed, and denote 4.13’ Theorem. Let E ⊂ R ˆ Σ(f ) ∩ E = ∅}. I0 (E) = {f : f ∈ S(R),

Assume that E + E + · · · + E (m times) has no interior. Let 1 < p ≤ m and q = p/(p − 1). Then I0 (E) is (norm) dense in FLq . The proofs of 4.11’ and 4.13’ are essentially the same and follow immediately from the Hahn-Banach theorem (and 4.11, 4.13, respecˆ which annihilates I0 (ξ) is a pseutively). A linear functional on A(R) domeasure supported by {ξ}, hence is constant multiple of the Dirac measure at ξ , and hence annihilates I(ξ). A linear functional on FLq which annihilates I0 (E) is an element of FLp supported by E ; hence it must be zero. J EXERCISES FOR SECTION 4 1. 2. 3. 4.

Deduce 4.11 from 4.11’. Deduce 4.13 from 4.13’. ˆ is finite? What is a function h ∈ L∞ (R) such that Σ(h) ˆ and ν ∈ F L∞ , then If f ∈ A(R) kf νkF L∞ ≤ kf kA(R) ˆ kνkF L∞

d)) ⊂ Σ(fˆ) ∪ (−Σ(fˆ)). 5. Let f ∈ L∞ (R). Show that Σ( 0. An ε-almost-period of f is a number τ such that

supx |f (x − τ ) − f (x)| < ε. Examples: t = 0 is a trivial ε-almost-period for all ε > 0; if f is periodic then its period, or any integral multiple thereof, is an ε-almost-period for all ε > 0; if f is uniformly continuous, every sufficiently small t is an ε-almost-period. 5.2 D EFINITION : A function f is (uniformly) almost-periodic on R if it is continuous and if for every ε > 0 there exists a number Λ = Λ(ε, f ) such that every interval of length Λ on R contains an ε-almost-period of f . We denote by AP (R) the set of all almost-periodic functions on R. Examples: (a) Continuous periodic functions are almost-periodic. (b) We shall show (see 5.7) that the sum of two almost-periodic functions is almost-periodic; hence f = cos x+cos πx is almost periodic (see also exercise 1 at the end of this section); noticing, however, that f (x) = 2 only for x = 0, we see that f is not periodic.


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(c) If f is almost-periodic, so are |f |, fˆ, af for any complex number a, and f (λx) for any real λ. 5.3 Lemma. Almost-periodic functions are bounded.

P ROOF : Let f be almost-periodic. Take ε = 1 and let Λ = Λ(1, f ). For arbitrary x ∈ R let τ be a 1-almost-period in the interval [x − Λ, x]. We have 0 ≤ x − r ≤ Λ and |f (x) − f (x − τ )| < 1, consequently |f (x)| ≤ sup0≤y≤Λ |f (y)| + 1. J Corollary. If f is almost-periodic, so is f 2 .

P ROOF : Without loss of generality we may assume |f (x)| ≤ 1/2 for all x ∈ R. We have f 2 (x − τ ) − f 2 (x) = (f (x − τ ) + f (x))(f (x − τ ) − f (x)) which implies that, for every ε > 0, ε-almost-periods of f are also εalmost-periods of f 2 . J 5.4 Lemma. Almost-periodic functions are uniformly continuous.

P ROOF : Let f be almost-periodic, ε > 0, Λ = Λ(ε/3, f ). Since f is uniformly continuous on [0, Λ], there exists η0 > 0 such that for all |η| < η0 sup0<x 0, it can be covered by a union of a finite number of balls of radius ε. Assume first that f is almost-periodic and let us show that W0 (f ) is totally bounded in L∞ (R). Let ε > 0 be given and let Λ = Λ(ε/2, f ); by the uniform continuity of f we can find numbers η1 , . . . , ηM in [0, Λ] such that if 0 ≤ y0 ≤ Λ, inf 1≤j≤M kfy0 − fηj k < ε/2. For arbitrary y ∈ R let τ be an ε/2-almost-period of f in [y − Λ, y]; writing y0 = y − τ we obtain 0 ≤ y0 ≤ Λ and kfy − fy0 k∞ < ε/2; consequently, inf 1≤j≤M kfy − fηj k∞ < ε and W0 (f ) is covered by the union of balls of radius ε, centered at fηj , j = 1, . . . , M . Assume now that W0 (f ) is precompact. Let ε > 0 and let O1 , . . . , OM be balls of radius ε/2 such that W0 (f ) ⊂ ∪M 1 Oj . We may clearly assume that Oj ∩ W0 (f ) 6= ∅ and hence pick fyj ∈ Oj , j = 1, . . . , M . The balls of radius ε centered at yj cover W0 (f ). We claim that every interval J of length Λ = 2 max1≤j≤M |yj | contains an ε-almost-period of f . If J is such an interval, denote by y its midpoint. There exists a j0 such that kfy − fyj0 k∞ < ε; writing τ = y − yj0 it is clear that τ ∈ J and, on the other hand, kfτ − f k∞ = kfτ +yj0 − fyj0 k∞ < ε.

All that we have to do in order to complete the proof is show that, under the assumption that W0 (f ) is precompact, f is continuous.‡ We show that it is uniformly continuous, that is, limη→0 kfη − f k∞ = 0. Given ε > 0, let O1 , . . . , OM be balls of radius ε/2 covering W0 (f ), as above, and write Ej = {τ : fτ ∈ Oj }. Since ∪Ej = R, at least one of these, say Ej0 , has positive mesure. But then Ej0 − Ej0 is a neighborhood of 0 in R, and for y ∈ Ej − Ej we have kfy − f k∞ ≤ ε. J 5.6 D EFINITION : The translation convex hull, W (f ), of a function S f ∈ L∞ (R) is the closed convex hull of |a|≤1 W0 (af ). Equivalently, it is the set of uniform limits of functions of the form X X (5.1) ak fxk , xk ∈ R, |ak | ≤ 1.

Remark: If f is uniformly continuous we can define W (f ) as the closure of the set of all functions of the form (5.1’) ‡ That

ϕ∗f

with

ϕ ∈ L1 (R), kϕkL1 (R) ≤ 1.

is: f is equal a.e. to a continuous function.


173

Another observation that will be useful later is: (5.2)

W (eiξx f ) = {eiξx g : g ∈ W (f )}.

By its very definition W (f ) is convex and closed in L∞ (R). Since W (f ) ⊃ W0 (f ), it is clear that if W (f ) is compact then W0 (f ) is precompact; the converse is also true: if W0 (f ) is precompact, there exist for every ε > 0, a finite number of translates {fyj }M j=1 such that every translate of f lies within less than ε from fyj for some 1 ≤ j ≤ M . Thus, every function of the form (5.1) lies within e of a function having P P the form M |bj | ≤ 1. In the unit disc |b| ≤ 1 we can j=1 bj fyj with pick a finite number of points {ck }N k=1 such that every b in the unit disc lies within εM −1 kf k−1 from one of the ck ’s; thus every combination L∞ (R) P P bj fyj |bj | ≤ 1 lies within ε of some (5.3)

M X

b0j fyj ,

b0j ∈ {ck }N k=1 .

1

It follows that W (f ) is covered by the union of M N balls of radius 3ε centered at the functions of the form (5.3); hence W (f ) is precompact and being closed it is compact. We have proved: Lemma. W (f ) is compact if, and only if, W0 (f ) is precompact, that is, if, and only if, f ∈ AP (R). 5.7 Theorem. AP (R) is a closed subalgebra of L∞ (R).

P ROOF : In order to show that AP (R)is a subspace, we have to show that if f, g ∈ AP (R) so does f +g . We clearly have W (f +g) ⊂ W (f )+W (g) and since, by 5.6, W (f ) and W (g) are both compact, W (f ) + W (g) is compact and hence W (f + g) is precompact. Since W (f + g) is closed, it is compact, and by 5.6, f + g ∈ AP (R). It follows from the corollary 5.3 that f 2 , g 2 , (f + g)2 ∈ AP (R) and consequently f g = 1/2((f + g)2 − f 2 − g 2 ) is almost-periodic and we have proved that AP (R)is a subalgebra of L∞ (R). In order to show that it is closed, we consider a function f in its closure. Since f is the uniform limit of continuous functions, it is continuous. Given ε > 0 we can find a g ∈ AP (R) such that kf − gk∞ < ε/3, and if τ , is an ε/3 almost-period of g we have fτ − f = (fτ − gτ ) + (gτ − g) + (g − f ),

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hence kfτ − f k∞ < ε/3 + ε/3 + ε/3 = ε and τ is an ε-almost-period of f . Thus every interval of length Λ(ε/3, g) contains an ε-almost-period of f , and f is almost-periodic. J 5.8 D EFINITION : the form

A trigonometric polynomial on R is a function of f (x) =

n X

aj eiξj x ,

ˆ ξj ∈ R.

1

The numbers ξj are called the frequencies of f . By theorem 5.7, all trigonometric polynomials and all uniform limits of trigonometric polynomials are almost-periodic. The main theorem in the theory of almost-periodic functions states that every almostperiodic function is the uniform limit of trigonometric polynomials, and actually gives a recipe, analogous to Fejér’s theorem for periodic functions, for finding the approximating polynomials (see 5.20. 5.9 D EFINITION : The norm spectrum of a function h ∈ L∞ (R) is the set ˆ aeiξx ∈ W (h) for sufficiently small a 6= 0}. σ(h) = {ξ : ξ ∈ R, σ(h) may well be empty even if h 6= 0; for instance, if h ∈ C0 (R) we have W (h) ⊂ C0 (R) and consequently σ(h) = ∅. We notice that from (5.2) and our definition above it follows immediately that

(5.4)

σ(eiξx h) = ξ + σ(h) = {ξ + η : η ∈ σ(h)}.

ˆ . Lemma. If h ∈ L∞ (R) then σ(h) ⊂ Σ(h) ˆ y = eiξy h ˆ it is clear that Σ(h ˆ y ) = Σ(h) ˆ and conseP ROOF : Since h iξx ˆ ˆ quently Σ(f ) ⊂ Σ(h) for any f ∈ W (h). If f = ae , then fˆ = aδξ (δξ is the measure of mass one concentrated at ξ ) and Σ(fˆ) = {ξ}; thus ˆ . if ξ ∈ σ(h) then ξ ∈ Σ(h) J

5.10 Lemma. Let h be bounded and uniformly continuous. Assume that ηK(ηx) ∗ h converges uniformly as η → 0 to a limit which is not identically zero. Then 0 ∈ σ(h). ˆ ˆ , so that P ROOF : Writing gη = ηK(ηx) ∗ h we have gˆη = K(ξ/η) h \ Σ(ˆ gη ) ⊂ [−η, η] and hence Σ(limη→0 gη ) = {0}. By 4.11, limη→0 gη is a constant, and by the remark following definition 5.6, gη ∈ W (h) and hence limη→0 gη ∈ W (h) ; now, as lim gη is a constant different from zero, we obtain 0 ∈ σ(h). J


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ˆ Corollary. R iξx Let µ be a measure on R and assume µ({0}) 6= 0. Let ˆ h(x) = e dµ(ξ) (so that µ = h); then 0 ∈ σ(h). ˆ P ROOF : Keeping the notations above, we have gˆη = K(ξ/η)µ and conˆ sequently gˆη tends to µ({0})δ0 in M (R) which implies gη → µ({0}) uniformly. J ˆ plays no specific role in 5.9; 5.11 Remarks: It is clear that 0 ∈ R if µ({ξ}) 6= 0 we have ξ ∈ σ(h) (h as above). Also, it is not essential to use Fejér’s kernel: if F ∈ L1 (R), and if we assume that Fη ∗ h converges uniformly to a nonvanishing limit, where Fη = ηF (ηx), it follows that 0 ∈ σ(h). This can be seen as follows: given a sequence ˆn εn → 0, we can write F = Gn + Hn such that Gn , Hn ∈ L1 (R), G has compact support, say included in (−cn , cn ), and kHn kL1 (R) < εn . Writing Gn,η (x) = ηGn (ηx), Hn,η (x) = ηHn (ηx) and noticing that kHn,η ∗ hkL∞ (R) < εn khk, we obtain limn→∞ Gn,η ∗ h = Fη ∗ h. Remembering that Σ(G\ n,η ∗ h) ⊂ (−ηcn , ηcn ) we obtain, letting η → 0 faster than cn → ∞, Σ(lim\ Fη ∗ h) = 0 as before. The condition of existence of a uniform limit of Fη ∗ h as η → 0 can clearly be replaced by the less stringent condition of the existence of a nonvanishing limit point, that is. a limit of some sequence Fηn ∗ h with ηn→ 0. We restate these remarks as:

Lemma. Let f ∈ AP (R) and assume 0 6∈ σ(f ); then for all F ∈ L1 (R) limη→0 kηF (ηx) ∗ f kL∞ (R) = 0.

P ROOF : Let F ∈ L1 (R); with no loss of generality we may assume that kF kL1 (R) ≤ 1. It follows that ηF (ηx)∗f ∈ W (f ) and, if it did not tend to zero as η → 0, it would have, W (f ) being compact, other limit points. By the preceding remarks this would imply 0 ∈ σ(f ). J 5.12 Lemma 5.11 has the following converse: R Lemma. Let f ∈ AP (R), F ∈ L1 (R) and F (x)dx 6= 0. If for some sequence ηn → 0, limn→∞ kηn F (ηn x) ∗ f k = 0, then 0 6∈ σ(f ).

P ROOF : We notice first that for any translate of f , hence for any linear combination of translates, and hence for any g ∈ W (f ), we have limn→∞ kηn F (ηn x) ∗ gkL∞ (R) = 0. If g = const , ηn F (ηn x) ∗ g = Fˆ (0)g and consequently the only constant in W(f) is zero, that is, 0 6∈ σ(f ). J

176


5.13 Theorem. To every f ∈ AP (R) there corresponds a unique number M (f ), called the mean value of f , having the property that 0 6∈ σ(f − M (f )).

P ROOF : We have seen before that uniform limit points of ηK(ηx) ∗ f as η → 0 are necessarily constants. Since ηK(ηx) ∗ f ∈ W (f ) and since W (f ) is compact, there exists a number α such that for an appropriate sequence ηn → 0, ηn K(ηn x) ∗ f converges uniformly to α. ˆ Since K(0) = 1, ηn K(ηn x) ∗ (f − α) → 0 uniformly; hence, by 5.12, 0 6∈ σ(f − α). If β is another number such that 0 6∈ σ(f − β) we obtain, using 5.11, that as η → 0 ηK(ηx) ∗ [(f − α) − (f − β)] = ηK(ηx) ∗ (f − α) − ηK(ηx) ∗ (f − β)

converges to zero uniformly. But ηK(ηx) ∗ [(f − α) − (f − β)] = β − α identically and consequently β = α. Thus the property 0 6∈ σ(f − α) determines α uniquely and we set M (f ) = α. J Corollary. If f ∈ AP (R) and F ∈ L1 (R), then ηF (ηx) ∗ f converges uniformly as η → 0 to Fˆ (0)M (f ). ( 1/2 |x| < 1 In particular, taking F (x) = writing T = η −1 , and 0 |x| ≥ 1 evaluating the convolution at the origin, we obtain: Corollary. For f ∈ AP (R),

(5.5)

1 T →∞ 2T

M (f ) = lim

Z

T

f (x)dx.

−T

Using the mean value we can determine the norm spectrum of f completely. By (5.4) it is clear that ξ ∈ σ(f ) if, and only if, 0 ∈ σ(f e−iξx ) and consequently (5.6)

ξ ∈ σ(f ) ⇔ M (f e−iξx ) 6= 0

By our definition of M (f ) and by corollary 5.9 it is clear that if fˆ is a measure then fˆ({0}) = M (f ) and similarly (5.6’)

fˆ(ξ) = M (f e−iξx );

thus we can recover the discrete part of fˆ. We shall soon see that f has no continuous part when f ∈ AP (R).


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5.14 The mean value clearly has the basic properties of a translation invariant integral, namely:

(5.7)

M (f + g) = M (f ) + M (g),

(5.8)

M (af ) = aM (f ),

(5.9)

M (fy ) = M (f )

(where fy (x) = f (x − y)).

It is also positive: Lemma. Assume f ∈ AP (R), f (x) ≥ 0 on R, and f not identically zero. Then M (f ) > 0.

P ROOF : By (5.7) we may assume f (0) > 0 and consequently, if α > 0 is small enough, f (x) > α on −α < x < α. Let Λ = Λ(α/2, f ); every interval of length Λ contains an α/2-almost-period of f , say τ , and f (x) > α/2 in (τ − α, τ + α). It follows that the integral of f over any interval of length Λ is at least α2 ; hence M (f ) > α2 /Λ. J 5.15 We define the inner product of almost-periodic functions by: hf, giM = M (f g¯)

(5.10)

and claim that with the inner product so defined, AP (R) is a preHilbert space, that is, satisfies all the axioms of a Hilbert space except for completeness. The bilinearity of hf, giM is obvious and the fact that hf, giM > 0 unless f = 0 has been established in 5.14. In this preHilbert space, the exponentials {eiξx }ξ∈Rˆ form an orthonormal family, since ( Z T 1 if ξ = η 1 i(ξ−η)x iξx iηx e dx = he , e iM = lim T →∞ 2T −T if ξ 6= η. 0 We now introduce the notation§ (5.11)

fˆ({ξ}) = hf, eiξx iM = M (f e−iξx ).

that is, fˆ({ξ}) are the Fourier coefficients of f relative to the orthonormal family {eiξx }ξ∈Rˆ . Bessel’s inequality now reads X (5.12) |fˆ({ξ})|2 ≤ hf, f iM = M (|f |2 ) ˆ ξ∈R

§ If f is a measure on R ˆ , (5.11) agrees with (5.6’). By abuse of language we shall sometimes refer to fˆ({ξ}) for arbitrary f ∈ AP (R), as the mass of the pseudomeasure fˆ at ξ .

178


and it follows that fˆ({ξ}) = 0 except possibly for a countable set of ξ ’s. Combining this with (5.6) we obtain that for all f ∈ AP (R), σ(f ) is countable. 5.16 We now introduce the mean convolution f ∗ g of two almostM

periodic functions. Let f, g ∈ AP (R); then for every x ∈ R, f (x − y)g(y), as a function of y , is almost periodic and My (f (x − y)g(y)) is well defined. Write: (f ∗ g)(x) = My (f (x − y)g(y)) = lim

(5.13)

T →∞

M

Z

T

f (x − y)g(y)dy.

−T

Lemma. f ∗ g is almost-periodic. If M (|g|) ≤ 1, then f ∗ g ∈ W (f ). M

M

P ROOF : Without loss of generality we assume that M (|g|) < 1. It follows that for all sufficiently large T 1 2T

Z

T

f (x − y)g(y)dy ∈ W (f )

−T

and, combining the compactness of W (f ) with the fact that the pointwise limit in (5.13) is well defined, we obtain f ∗ g as the uniform limit M RT 1 of 2T f (x − y)g(y)dy . J −T The convolution f ∗ g has all the properties of convolutions on T M and R; in particular (f ∗ g)b({ξ}) = Mx My f (x − y)g(y) e−iξx M (5.14) = Mx My f¯(x − y)e−iξ(x−y) g(y)e−iξ(y) = fˆ({ξ})ˆ g ({ξ}).

Also, f ∗ eiξx = My (f (x − y)eiξy ) = Mt (f (t)eiξ(x−t) ) = fˆ({ξ})eiξx , M

so that if g(x) =

P

gˆ({ξ})eiξx (finite sum) then f ∗g= M

X

gˆ({ξ})fˆ({ξ})eiξx .


179

5.17 For f ∈ AP (R), write f ∗ (x) = f¯(−x), and

(5.15)

h = f ∗ f ∗ = My f (y)f¯(x + y) . M

Since fc∗ ({ξ}) = fˆ({ξ}) we have by (5.14),

ˆ h({ξ}) = |fˆ({ξ})|2 .

If kf k∞ ≤ 1, which we assume for convenience, then h ∈ W (f ). Lemma. h, defined by (5.15), is positive definite.

P ROOF : Let xj ∈ R and zj are complex numbers, j = 1, . . . , N , then Z T X X 1 h(xj − xk )zj z¯k = lim f (xj + y)f (xk + y)zj z¯k dy n→∞ 2T −T Z T X 2 1 zj f (xj + y) dy ≥ 0 . = lim n→∞ 2T −T

J

Since h is continuous, Bochner’s theorem 2.8 says that h is the Fourier ˆ is a positive measure. transform of a positive measure or, equivalently, h ˆ , then fˆ = P fˆ({ξ})δξ , 5.18 Proposition. If f ∈ AP (R) and fˆ ∈ M (R) Pˆ P ˆ f ({ξ})eiξx . kfˆkM (R) |f ({ξ})|, and f (x) = ˆ = Pˆ P ROOF : By (5.6’), the discrete part of fˆ is f ({ξ})δξ , and we have P ˆ ˆ |f ({ξ})| ≤ kfˆkM (R) ˆ . We claim that the continuous part of f is zero. Denote the continuous part of fˆ by µ; it is the Fourier transform Pˆ of the almost-periodic function g = f − f ({ξ})eiξx . By Wiener’s R 2 −1 T theorem 2.12, lim(2T ) |g(x)| dx = 0 and, by 5.14, µ = 0. J −T 5.19 Theorem (Parseval’s identity). Let f ∈ AP (R), then X (5.16) |fˆ({ξ})|2 = M (|f |2 ) .

P ROOF : Define h by (5.15). By Proposition 5.18 we have X X ˆ |fˆ({ξ})|2 = h({ξ}) = h(0) = M (|f |2 ). J

Corollary (Completeness). {eiξx }ξ∈Rˆ is a complete orthonormal basis for AP (R). Corollary (Uniqueness). Let f ∈ AP (R), f 6= 0. Then σ(f ) 6= ∅.

180


Pˆ f ({ξ})eiξx , to which we 5.20 For arbitrary f ∈ AP (R), the series refer as the Fourier series of f , converges to f in the norm induced by the bilinear form h·, ·iM . Our next goal is to show that, as in the case of periodic functions, the Fourier series of any f ∈ AP (R) is summable to f in the uniform norm. ˆ and an 5.21 Lemma. Given a finite number of points ξ1 , . . . , ξN ∈ R ε > 0, there exists a trigonometric polynomial B having the following properties:

(a)

B(x) ≥ 0

(b)

M (B) = 1 ˆ j }) > 1 − ε B({ξ

(c)

for j = 1, . . . , N.

P ROOF : We notice first that if ξ1 , . . . , ξN happen to be integers and if m is an integer larger than ε−1 max|ξj |, then the Fejér kernel of order m, P |k| ikx namely Km = m has all the properties mentioned. In −m 1 − m+1 e the general case let λ1 , . . . , λq be a basis for ξ1 , . . . , ξN ; that is, λ1 , . . . , λq are linearly independent over the rationals and every ξj can be written P in the form ξj = q1 Aj,k λk with integral Aj,k . Let ε1 > 0 be such that (1 − ε1 )q > 1 − ε, and let m > ε−1 1 maxj,k |Aj,k |; we contend that Qq B = 1 Km (λk x) has all the required properties. Property (a) is obvious since B is a product of nonnegative functions. In order to check (b) and (c) we rewrite B as (5.17)

B(x) =

X

1−

|k1 | |kq | i(k1 λ1 +···+kq λq )x ... 1 − e , m+1 m+1

the summation extending over |k1 | ≤ m, . . . , |kq | ≤ m. Because of the independence of the λj ’s there is no regrouping of terms having the ˆ same frequency and we conclude from (5.17) that B(0) = the constant term in (5.17) = M (B) = 1, which establishes (b), and q q X Y |Aj,k | ˆ ˆ B({ξj }) = B { Aj,k λk } = > (1 − ε1 )q > 1 − ε, 1− m+1 1 k=1

which establishes (c).

J

Theorem. Let f ∈ AP (R). Then f can be approximated uniformly by trigonometric polynomials Pn ∈ W (f ).


181

P ROOF : Since σ(f ) is countable we can write it as {ξj }∞ j=1 . For each n let Bn be the polynomial described in the lemma for ξ1 , . . . , ξn and ε = 1/n. Write Pn = f ∗ Bn . By 5.17, Pn ∈ W (f ) and taking account M of (c) above, lim Pˆn ({ξj }) = fˆ({ξ}) for every ξj ∈ σ(f ). If ξ 6∈ σ(f ) we have Pˆn ({ξ}) = fˆ({ξ}) = 0 for all n. It follows that if g is a limit point of Pn in W (f ), then gˆ({ξ}) = fˆ({ξ}) for all ξ and by the uniqueness theorem g = f . Thus, f is the only limit point of the sequence Pn in the compact space W (f ) and it follows that Pn converge to f (in norm, i.e., uniformly.) J Corollary. Every closed translation invariant subspace of AP (R) is spanned by exponentials. 5.22 We finish this section with two theorems providing sufficient conditions for functions to be almost-periodic. Though apparently different they are essentially equivalent and both are derived from the same principle. We start with some preliminary definitions and lemmas. ˆ is an almostFor h ∈ AP (R), we say, by abuse of language, that h periodic pseudo-measure. D EFINITION : A pseudo-measure ν is almost-periodic at a point ˆ , if there exists a function ϕ ∈ A(R), ˆ ϕ(ξ) = 1 in some neighborξ0 ∈ R hood of ξ0 , such that ϕν is almost-periodic. It is clear that ν is almost-periodic at ξ0 if, and only if, ψν is almostˆ whose support is sufficiently close to ξ0 periodic for every ψ ∈ A(R) (e.g., within the neighborhood of ξ0 on which the function ϕ above is equal to one). In particular, ν is almost-periodic at every ξ 6∈ Σ(ν). Lemma. Let ν ∈ FL∞ and assume that Σ(ν) is compact and that ν is almost-periodic at every point of Σ(ν). Then ν is almost-periodic.

P ROOF : By a standard compactness argument we see that there exists ˆ which is an η > 0 such that ν is almost-periodic for every ψ ∈ A(R) ˆ have their supsupported by an interval of length η . Let ψj ∈ A(R) ports contained in intervals of length η , j = 1, 2, . . . , N , and such that PN 1 ψj = 1 on a neighborhood of Σ(ν). By the assumption concerning the supports of ψj , ψj ν is almost-periodic for all j , and consequently N X 1

is almost-periodic.

(ψj ν) =

N X

ψj ν = ν

1

J

182


ˆ is compact and 5.23 Theorem. Let h ∈ L∞ (R) and assume that Σ(h) ˆ that νˆ is almost-periodic at every ξ ∈ R except, possibly, at ξ = 0. Then h ∈ AP (R).

5.24 Theorem (Bohr). Let h ∈ L∞ (R) and assume that it is differentiable and that h0 ∈ AP (R). Then h ∈ AP (R).

These two theorems are very closely related. We shall first show how theorem 5.23 follows from 5.24, and then prove 5.24. ˆ is compact, then 5.23: We begin by showing that if Σ(h) 0 ˆ ˆ h is differentiable and h = iξ h (see exercise 4.6). Let f ∈ S(R) be ˆ . We have h = fˆh and such that fˆ(ξ) = 1 in a neighborhood of Σ(h) R consequently h = f ∗ h or h(x) = f (x − y)h(y)dy . Since h is bounded and f ∈ S(R) we can differentiate under the integral sign and obtain that h is (infinitely) differentiable and that h0 = f 0 ∗ h. Remembering ˆ , we obtain hb0 = fb0 h = iξh. that fˆ0 (ξ) = iξ in a neighborhood of Σ(h) By theorem 4.11’ there exists a sequence {ϕn } in L1 (R) such that b ϕˆn (ξ) = 0 in a neighborhood of ξ = 0, and such that kϕˆn − fˆ0 kA(R) → 0. ˆ − hb0 k \ This implies (exercise 4.4) that kϕˆn h → 0, that is, h0 is the F L∞ uniform limit of ϕn ∗ h. Now, since ϕˆn , vanishes in a neighborhood of ξ = 0, it follows from 5.22 that ϕn ∗ h ∈ AP (R); by 5.7, h0 ∈ AP (R), and by 5.24 h ∈ AP (R). PROOF OF

PROOF OF 5.24: Since h is clearly continuous we only have to show that for every ε > 0 there exists a constant Λ(ε, h) such that every interval of length Λ(ε, h) contains an ε-almost period of h. In view of 5.7 we may consider the real and the imaginary parts of h separately, so that we may assume that h is real-valued. Denote

(5.18)

M = supx h(x),

m = inf h(x). x

Let ε > 0. Let x0 and x1 be real numbers such that (5.19) we put e1 =

ε h(x0 ) < m + , 8 ε 4|x1 −x0 |

ε h(x1 ) > M − ; 8

and claim that if τ is an ε1 -almost period of h0 then


183

h(x0 − τ ) < m + ε/2. In order to see this we write Z x1 h(x1 − τ ) − h(x0 τ ) = h0 (x − τ )dx x0 Z x1 Z x1 = h0 (x)dx + h0 (x − τ ) − h0 (x) dx (5.20) x0 x0 Z x1 = h(x1 ) − h(x0 ) + h0 (x − τ ) − h0 (x) dx, x0

and, since the last integral is bounded by |x1 − x0 |ε1 = ε/4 it follows from (5.19) and (5.20) that ε h(x1 − τ ) − h(x0 − τ ) > M − m − 2 and, since h(x1 − τ ) ≤ M , we obtain h(x0 − τ ) < m + ε/2. We now use the points {x0 − τ }, where τ is an ε1 /2-almost-period of h0 as reference points. Let Λ1 = Λ(ε1 /2, h0 ) and define ε2 by ε2 = min(q/2, ε1 /2, ε/Λ1 ). We claim that every ε2 -almost-period of h0 is an ε-almost-period of h. In order to prove it let x ∈ R and let τ1 be an ε2 almost-period of h0 ; we take τ0 to be an ε1 /2-almost-period of h0 such that x ≤ x0 − τ0 ≤ x + Λ1 , and write h(x − τ1 ) − h(x) = h(x − τ1 ) − h(x0 − τ0 − τ1 ) + h(x0 − τ0 − τ1 ) − h(x0 − τ0 ) + h(x0 − τ0 ) − h(x) Z x0 −τ0 = h(x0 − τ0 − τ1 ) − h(x0 − τ0 ) + h0 (y) − h0 (y − τ1 ) dy .

(5.19)

x

Since τ0 and τ0 + τ1 are both ε1 -almost-periods we have m ≤ h(x0 − τ0 − τ1 ) ≤ m + ε/2 and m ≤ h(x0 − τ0 ) ≤ m + ε/2,

hence |h(x0 − τ0 − τ1 ) − h(x0 − τ0 )| ≤ ε/2. The integral in (5.21) is bounded by e2 Λ1 ≤ ε/2 and it follows that |h(x − τ1 ) − h(x)| < ε. Thus, every interval of length Λ(ε2 , h0 ) contains an ε-almost-period of h and the proof is complete. J 5.25 Theorem. Let h ∈ L∞ (R) and assume that Σ(h) is compact and countable. Then h ∈ AP (R). ˆ is not P ROOF : This is a corollary of 5.20. The set of points ξ such that h almost-periodic at ξ is a subset of Σ(h) and, by 5.23, has no isolated points. Since a countable set contains no nonempty perfect sets, h is ˆ and, by 5.22, h ∈ AP (R). almost-periodic at every ξ ∈ R J

184


EXERCISES FOR SECTION 5 1. Show that f = cos 2πx + cos x is almost-periodic by showing directly that given ε > 0, there exists an integer M such that at least one of any M consecutive integers lies within ε from an integral multiple of 2π . 2. Let h ∈ L∞ (R). Show that, if h is uniformly continuous, σ(h) contains ˆ . every isolated point of Σ(h) Pˆ 3. Let f, g ∈ AP (R). Show that f ∗ g = f ({ξ})ˆ g ({ξ})eiξx . M

4. Let f ∈ AP (R) and assume that W (f ) is minimal in the sense that if h ∈ W (f ) and h 6= 0 then af ∈ W (h) for sufficiently small a. Show that f is a constant multiple of an exponential. 5. Let f ∈ AP (R) and assume that f 0 is uniformly continuous. Show that 0 f ∈ AP (R). ˆ is compact is essential in the state6. Show that the assumption that Σ(h) ment of theorem 5.25. Hint: Consider discontinuous periodic functions. ˆ is 7. Show that in the statement of theorem 5.25, the assumption that Σ(h) compact can be replaced by the weaker condition that h be uniformly continuous. 8. Deduce 5.24 from 5.23. 9. Let P be a trigonometric polynomial on R, and let ε > 0. Show that there exists a positive η = η(P, ε) such that if Q ∈ L∞ (R), kQk < 1 and ˆ ⊂ (−η, η), then Σ(Q) range(P + Q) + (−e, e) ⊃ range(P ) + range(Q).

Hint: The conditions on Q imply that kQ0 k ≤ η ; see exercise 4.6. ˆ ∈ FL∞ , ξ0 ∈ R ˆ and {ηn } a sequence tending to zero. Show that 10. Let h −1 ˆ ˆ if K(ηn (ξ − ξ0 ))h tends to a limit (in the weak-star topology), then the limit ˆ has that P ˆthe form aδξ0 . 2Introducing the notation a = h({ξ0 }, K, {ηn }), show ˆ such |h({ξ0 }, K, {ηn })| < ∞ where the summation extends over all ξ0 ∈ R ˆ exists. ˆ n−1 (ξ − ξ0 ))h that weak-star-limn→∞ K(η ∞ ˆ ˆ , except possibly countably 11. Let h ∈ FL . Show that for all ξ0 ∈ R −1 ˆ ˆ many, weak-star-limn→∞ K(ηn (ξ − ξ0 ))h exists and is equal to zero. 12. Show that if h ∈ L∞ (R), σ(h) is countable. 13. Let B be a homogeneous Banach space on R such that AP (R) ⊂ B ⊂ Lc (see 1.14). Describe the closure in B of AP (R). 6 THE WEAK-STAR SPECTRUM OF BOUNDED FUNCTIONS

6.1 Given a function h ∈ L∞ (R), we denote by [h] the smallest translation invariant subspace of L∞ (R) that contains h; that is, the span of {hy }y∈R . We denote by [h] the norm closure of [h] in L∞ (R), and


185

by [h]w∗ , the weak-star closure [h] in L∞ (R). Our definition 5.9 of the norm spectrum of h is clearly equivalent to σ(h) = {ξ : eiξx ∈ [h]}

and we define the weak-star spectrum by σw∗ (h) = {ξ : eiξx ∈ [h]w∗ } .

Let h ∈ L∞ (R). The problem of weak-star spectral analysis is: find σw∗ (h). The problem of weak-star spectral synthesis is: does h belongs to the weak-star closure of span {eiξx }ξ∈σw∗ (h) ? The corresponding problems for the uniform topology were studied in section 5. We have obtained some information about σ(h) for arbitrary h and complete information in the case that h was almost-periodic (see (5.6)); we proved that the norm spectral synthesis is valid for h if, and only if, h ∈ AP (R). The problem of weak-star spectral analysis admits the following answer: ˆ . Theorem. For h ∈ L∞ (R), σw∗ (h) = Σ(h)

P ROOF : The subspace of L1 (R) orthogonal to [h] is composed of all the functions f ∈ L1 (R) satisfying Z f (x)h(x − y)dx = 0 for all y ∈ R which is equivalent to (6.1)

ˆ f ∗ h(−x) = 0.

We denote this subspace of L1 (R) by [h]⊥ . By the Hahn-Banach theorem, eiξx ∈ [h]w∗ if, and only if, Z f (x)eiξx dx = fˆ(ξ) = 0 for all f ∈ [h]⊥ . We thus have an equivalent definition of σw∗ (h) as the set of all common zeros of {f : f ∈ [h]⊥ }. ˆ ; if ε > 0 is small enough (ξ0 −ε, ξ0 +ε)∩Σ(h) ˆ =∅ Assume ξ0 6∈ Σ(h) 1 ˆ so that if f ∈ L (R) and the support of f is contained in (ξ0 − ε, ξ0 + ε) we have Z ˆ ˆ hf , hi = f (x)h(x)dx = 0.

186


We claim that f is orthogonal not only to h, but also to all the translates of h, hence to [h]. This follows from Z Z (6.2) f (x)h(x − y)dx = f (x + y)h(y)dx, and since the Fourier transform of f (x + y) is eiξy fˆ, hence supported by (ξ0 −ε, ξ0 +ε), both sides of (6.2) must vanish. There are many functions fˆ in A(R) supported by (ξ0 − ε, ξ0 + ε) such that fˆ(ξ0 ) 6= 0; it follows that ξ0 is not a common zero of {f : f ∈ [h]⊥ } hence ξ0 6∈ σw∗ (h); this ˆ. proves σw∗ (h) ⊂ Σ[h] In the course of the proof of the converse inclusion we shall need the following lemma, due to Wiener. The proof of the lemma will come in chapter VIII (see VIII.6.2). Lemma. Assume fˆ, fˆ1 ∈ A(R) and assume that the support of fˆ1 is contained in a bounded interval U on which fˆ is bounded away from zero. Then fˆ1 = gˆfˆ for some g ∈ L1 (R). ˆ ⊂ σw∗ (h), we have to show that if ξ0 6∈ σw∗ (h), then To prove Σ[h] ˆ vanishes in some neighborhood of ξ0 . Now, since ξ0 6∈ σw∗ (h), there h exists a function f ∈ L1 (R) satisfying (6.1) and such that fˆ(ξ0 ) 6= 0 and consequently fˆ is bounded away from zero on some neighborhood U of ˆ vanishes in U , a contention that will be proved if ξ0 . We contend that h we show that if f1 ∈ L1 (R) and the support of fˆ1 is contained in U then f1 ∗ h(−x) = 0. By Wiener’s lemma there exists a function g ∈ L1 (R) such that fˆ1 = gˆfˆ or equivalently f1 = g ∗ f . Now f1 ∗ h(−x) = (g ∗ f ) ∗ h(−x) = g ∗ (f ∗ h(−x)) = 0


J

6.2 The Hahn-Banach theorem, used as in the foregoing proof, gives a convenient restatement of the problem of spectral synthesis. We inˆ write troduce first the following notations: if E is a closed set on R

(6.3)

ˆ I(E) = {f : f ∈ L! (R), fˆ(ξ) = 0 on E}

and (6.4)

ˆ Ω(E) = {g : g ∈ L∞ (R) and hf, gi = 0 for all f ∈ I(E)}.


187

ˆ I(E) is clearly the orthogonal complement in L1 (R) to the span of iξx ˆ . {e }ξ∈E and Ω(E) is the orthogonal complement in L∞ (R) of I(E) By the Hahn-Banach theorem Ω(E) is precisely the weak-star closure of span {eiξx }ξ∈E and the problem of (weak-star) spectral synthesis for h ∈ L∞ (R) can be formulated as: is it true that h ∈ Ω(σw∗ (h))? Equivˆ alently, is it true that for fˆ ∈ A(R)

(6.5)

fˆ(ξ) = 0 on σw∗ (h) ⇒ hf, hi = 0?

ˆ or: is it true that, (f ∈ A(R))

(6.6)

ˆ = 0? fˆ(ξ) = 0 on ⇒ fˆh

(The equivalence of (6.5) and (6.6) follows from (6.2)). ˆ ∈ FL∞ and assume that fˆ(ξ) = 0 on ˆ and h Theorem. Let f ∈ A(R) ˆ . Then Σ(fˆh) ˆ is a perfect subset of Σ(fˆ) ∩ bdry(Σ(h)) ˆ . Σ(h) ˆ ⊂ Σ(fˆ) ∩ Σ(h) ˆ and since f vanishes on Σ(h) ˆ , P ROOF : By 4.10, Σ(fˆh) ˆ ˆ no interior point of Σ(h) is in Σ(f ). Let ξ0 be an isolated point of ˆ ; with no loss of generality we may assume ξ0 = 0 and that Σ(fˆh) ˆ . (−η, η) contains no other point of Σ(fˆh) −1 cη fˆh) ˆ = {0} cη (ξ) = K(η ˆ Write K ξ) = sup(0, 1−|η −1 ξ|). We have Σ(K c ˆ ˆ and consequently (see 4.11) Kη f h = aδ , with a 6= 0 a constant, and δ the unit mass concentrated at ξ = 0. By 4.11’ there exists a function g ∈ L1 (R) such that gˆ vanishes in a neighborhood of ξ = 0, say in (−η1 , η1 ), and such that kg − f kL1 (R) < (|a|/2)khk−1 L∞ (R) , (remember b η k = 1, we have kK cη (fˆ − gˆ)hk ˆ < |a|/2 and, that fˆ(0) = 0). Since kK b b η gˆ = 0), multiplying everything by Kη1 we obtain, (remember that K 1 ˆ has no |a| = kaδkF L∞ < |a|/2 which is a contradiction. Thus Σ(fˆh) isolated points and the proof is complete. J ˆ has countable boundary then h admits weak-star Corollary. If Σ(h) spectral synthesis; that is, h ∈ Ω(σw∗ (h)). ˆ itself, and not just its boundary, is countable, We recall that if Σ(h) and if h is uniformly continuous, then h ∈ AP (R) (theorem 5.25), that is, admits norm spectral synthesis. Weak-star spectral synthesis is closely related to the structure of closed ideals in A(R), and we shall discuss it further in chapter VIII. In particular, we shall show that weak-star spectral synthesis in FL∞ is not always possible.

188


7 THE PALEY–WIENER THEOREMS

7.1 Our purpose in this section is to study the relationship between properties of analyticity and growth of a function on R, and the growth ˆ . The situation is similar to, though not as of its Fourier transform on R simple as, the case of functions on the circle. We have seen in chapter I (see exercise I.4.4) that a function f , defined on T, is analytic if, and only if, fˆ(n) tends to zero exponentially as |n| → ∞. The simplicity of this characterization of analytic functions on T is due to the compactness of T. If we consider the canonical identification of T with the unit circle in the complex plane (i.e. t ↔ eit ), then a function f is analytic on T (i.e., is locally the sum of a convergent power series) if, and only if, f is the restriction to T of a function F , holomorphic in some annulus, concentric and containing the unit circle. This function F is automatically bounded in an annulus containing the unit circle, and the Fourier series of f is simply the restriction to T of the Laurent expansion of F . Considering R as the real axis in the complex plane, it is clear that a function f is analytic on R if, and only if, it is the restriction to R of a function F , holomorphic in some domain containing R; however, this domain need not contain a whole strip {z : z = x + iy, |y| < a}, nor need F be bounded in strips around R or on R itself (cf. exercises 1 through 3 at the end of this section). If we assume exponential decrease of fˆ at infinity we can deduce more than just the analyticity of f on R; in fact, writing Z 1 F (z) = fˆ(ξ)eiξz dξ, 2π we see that if fˆ(ξ) = O e−a|ξ| for some a > 0, then F is well defined and holomorphic in the strip {z : |y| < a}, and is bounded in every strip {z : |y| < a1 }, a1 < a; by the inversion formula† , F R = f . Under the same assumption we obtain also that, since fˆ ∈ L2 (R), f ∈ L2 (R); and since for |y| < a, F (x + iy) is the inverse Fourier transforms of e−ξy fˆ, we see that, as a function of x, F (x + iy) ∈ L2 (R) for all |y| < a. Even with all this added information about the analytic function extending f to a strip, we cannot obtain exponential decrease of f ; we can only ˆ for all |y| < a. obtain that e−ξy fˆ ∈ L2 (R) Theorem (Paley-Wiener). For f ∈ L2 (R), the following two conditions are equivalent: †F

R denotes the restriction of F to R.


189

(1) f is the restriction to R of a function F holomorphic in the strip {z : |y| < a} and satisfying Z (7.1) |F (x + iy)2 |dx ≤ const |y| < a. ˆ ea|ξ| fˆ ∈ L2 (R).

(2)

P ROOF : (2) ⇒ (1): write (7.2)

F (z) =

1 2π

Z

fˆ(ξ)eiξz dξ;

then by the inversion formula F R = f ; the function F is well defined and holomorphic in {z : |y| < a}, and, by Plancherel’s theorem: Z Z 1 2 |F (x + iy) |dx = |fˆ(ξ)|2 e2ξy dξ ≤ kfêa|ξ| k2L2 (R) ˆ . 2π (1) ⇒ (2); write fy (x) = F (x + iy) (thus f = f0 ), and consider the Fourier transforms fˆy . We want to show that fˆy (ξ) = fRˆ(ξ)e−ξy since by Plancherel’s theorem and (7.1), we would then have |fˆ(ξ)|2 e2ξy dx uniformly bounded in |y| < a, which clearly implies (2). Notice that if we assume (2) then, by the first part of the proof, we do have fˆy (ξ) = fˆ(ξ)e−ξy . For λ > 0 and z in the strip {z : |y| < a} we put: Z ∞ (7.3) F (z − u)Kλ (u)du, Gλ (z) = Kλ ∗ F = −∞

where K denotes Fejér’s kernel. Gλ is clearly holomorphic in the strip {z : |y| < a} and we notice that gλ,y (x) = Gλ (x+iy) = Kλ ∗fy and hence cˆ gd d λ,y (ξ) = Kλ fy (ξ). Now since g λ,y (ξ) has a compact support (contained −ξy in [−λ, λ]) we have gd d and consequently if |ξ| < λ, λ,y (ξ) = g λ,0 (ξ)e −ξy ˆ ˆ fy (ξ) = f (ξ)e . Since λ > 0 is arbitrary, the above holds for all ξ and the proof is complete. J We may clearly replace the "symmetric" conditions of 7.1 by nonsymmetric ones. The assumption (7.1) for −a1 < y < a, with a, a1 > 0, ˆ . is equivalent to: (ea1 ξ + e−aξ )fˆ(ξ) ∈ L2 (R) 7.2 Theorem (Paley-Wiener). For f ∈ L2 (R) the following two conditions are equivalent:

190


(1) There exists a function F , holomorphic in the upper half-plane {z : y > 0}, and satisfying: Z |F (x + iy)|2 dx < const , y>0 (7.4) and (7.5) (2)

lim y↓0

Z

|F (x + iy) − f (x)|2 dx = 0. fˆ(ξ) = 0

for ξ < 0.

P ROOF : (2) ⇒ (1): Define F (z), for y > 0, by (7.2). F is clearly holomorphic, F (x + iy) is the inverse Fourier transform of e−ξy fˆ and, by Plancherel’s theorem, ˆ kF (x + iy)kL2 (R) = kfê−ξy kL2 (R) ˆ ≤ kf kL2 (R) ˆ

which establishes (7.4), and also kF (x + iy) − f kL2 (R) = kfˆ(e−ξy − 1)kL2 (R) ˆ →0

as y ↓ 0. (1) ⇒ (2); write f1 (x) = F (x + i). By 7.1: kfê−ξy kL2 (R) ˆ = kF (x + i + iy)kL2 (R)

for −1 < y < ∞

and, in particular, by (7.4): Z (7.6) |fˆ1 (ξ)|2 e−2ξy dξ ≤ const . Letting y → ∞, (7.6) clearly implies that fˆ(ξ) = 0 for ξ < 0. By 7.1, the Fourier transform of F (x + iy) is fˆ(ξ)eξ(1−y) ; hence, by (7.5), fˆ(ξ) = fˆ1 (ξ)eξ , and fˆ(ξ) = 0 for ξ < 0. J ?7.3 The foregoing proofs yield more information than that stated explicitly. The proof of the implication (2) ⇒ (1) also shows that F is R bounded for y ≥ ε > 0 since |fˆ(ξ)e−ξy |dξ is then bounded. In the proof (1) ⇒ (2) no mention of f is needed nor is the assumption (7.5); if we simply assume that F is holomorphic in the upper half-plane and satisfies (7.4), we obtain, keeping the notations of the proof above, that ˆ and, denoting by f the function in L2 (R) of which fˆ1 eξ fˆ1 eξ ∈ L2 (R) is the Fourier transform, we obtain (7.5) as a consequence (rather than as an assumption). The Phragmén-Lindelöf theorem allows a further improvement:


191

Lemma. Let F be holomorphic in a neighborhood of the closed upper half-plane {z : y ≥ 0} and assume that Z (7.7) |F (x)|2 dx < ∞

and (7.8)

lim r−1 log+ |F (reiϑ )| = 0

r→∞

for all 0 < ϑ < π . Then (7.4) is valid.

P ROOF : Let ϕ be continuous with compact support on R, kϕkL2 ≤ 1. R∞ Write G(z) = ϕ ∗ F = −∞ F (z − u)ϕ(u)du; then G is holomorphic in {z : y ≥ 0}, satisfies the condition (7.8), and, on R,‡ |G(x)| ≤ kF RkL2 kϕkL2 ≤ kF RkL2 .

By the Phragmén-Lindelöf theorem we Rhave |G(z)| ≤ kF RkL2 throughout the upper half-plane, which means | F (x+iy)ϕ(−x)dx| ≤ kF RkL2 for y > 0. Since this is true for every ϕ (continuous and with compact support) such that kϕkL2 ≤ 1, it follows that Z Z 2 |F (x + iy)| dx ≤ |F (x)|2 dx J

7.4 Theorem. Let F be an entire function and a > 0. The following two conditions on F are equivalent: (1) F R ∈ L2 (R) and

(7.9)

|F (z)| = o(ea|z| )

ˆ fˆ(ξ) = 0 for |ξ| > a, such (2) There exists a function fˆ ∈ L2 (R), that Z a 1 (7.10) F (z) = fˆ(ξ)eiξz dξ. 2π −a

P ROOF : (2) ⇒ (1); if (7.10) is valid we have ˆ |F (z)| ≤ kfˆ(ξ)e−ξy kL1 (R) ˆ ≤ kf kL2 (R) ˆ ‡F

R denotes the restriction of F to R.

12 1 Z a e2ξy dξ . 2π −a

192


Now 1 2π

Z

a

−a

e2ξy dξ =

1 e2ay e2ay − e−2ay ≤ 4πy 2π|y|

and consequently ea|y| |F (z)| ≤ p kfˆkL2 (R) ˆ 2π|y|

which is clearly stricter than (7.9). The square summability of F R follows from Plancherel’s theorem. (1) ⇒ (2); assume first that F R is bounded. The function G(z) = iaz e F (z) is entire, satisfies (7.9) in the upper half-plane, and G(iy) → 0 as y → ∞. By the Phragmén-Lindelöf theorem, G is bounded in the upper half-plane and, writing g = G R, it follows from Lemma 7.3 and Theorem 7.2 that gˆ is carried by (0, ∞). Writing f = F R we clearly have fˆ(ξ) = gˆ(ξ + a) which implies fˆ(ξ) = 0 for ξ < −a. Similarly, considering G1 (z) =R e−iaz F (z), we obtain f (ξ) = 0 for ξ > a and, writing H(z) = 1/2π fˆ(ξ)eiξx dξ , we obtain, by the inversion theorem, H R = F R so that H = F and (7.10) is established. In the general case, that is without R assuming that F is bounded on R, we consider Fϕ (z) = ϕ ∗ F = F (z − u)ϕ(u)du where ϕ is an arbitrary continuous function with compact support. Fϕ satisfies the conditions in (1) and is bounded on R. Writing fϕ = Fϕ R we have fˆϕ (ξ) = fˆ(ξ)ϕ(ξ) ˆ and fˆϕ (ξ) = 0 if |ξ| > a. Since ϕ is arbitrary this implies fˆ(ξ) = 0 for |ξ| > a and the proof is completed as before. J EXERCISES FOR SECTION 7 P∞ −n

1. Show that F (z) = 2 [(z + n)2 + n−1 ]−1 is analytic on R and n=1 1 ∞ F R ∈ L ∩ L (R); however, F is not holomorphic in any strip {z : |y| < a}, a > 0. 2. Show that for a proper choice of the constants {an } and {bn } the function G(z) =

X

an e−bn (z−n)

2

is entire, G R ∈ L1 (R), but G is unbounded on R. z2 3. Show that H(z) = e−e is entire, H R ∈ L1 ∩ L∞ (R); however, H is unbounded on any line y = const 6= 0. 4. Let R F be holomorphic in a neighborhood of the strip {z : |y| ≤ a} and assume |F (x + iy)|2 dx < const for |y| ≤ a. Show that for z in the interior of the strip: Z ∞ F (u − ia) F (u + ia) 1 − du. F (z) = 2πi ∞ u − ia − z u + ia − z


193

ˆ , supported by [−a, a]. Define: 5. Let µ be a measure on R F (z) =

Z

e−iξz dµ(ξ).

Show that F is entire and satisfies F (z) = O ea|y| . Give an example to show that F need not satisfy (7.9). Hint: F (z) = cos az . ˆ 6. Let ν be a distribution R −iξzon R, supported by [−a, a]. Show that the function F , defined by F (z) = e dν , is entire and that there exists an integer N such that F (z) = O z N ea|y| as |z| → ∞. 7. Titchmarsh’s convolution theorem: (a) Let F be an entire function of exponential type (i.e., F (z) = O ea|z| for some a > 0) and assume that |F (x)| ≤ 1 for all real x and that F (iy) is real valued. Assuming that F is unbounded in the upper half-plane, show that the domain D = {z : y > 0, |F (z)| > 2} is symmetric with respect to the imaginary axis, is connected, and its intersection with the imaginary axis is unbounded. Hint: Phragmén-Lindelöf. (b) Let F1 and F2 both have the properties of F in part (a) and denote the corresponding domains by D1 , D2 , respectively. Show that D1 ∩ D2 6= ∅ and deduce that F1 F2 is unbounded in the upper half-plane. ˆ j = 1, 2, and assume that fj are both real-valued and (c) Let fj ∈ L2 (R), carried by [−a, 0]. Show that if f1 ∗ f2 vanishes in a neighborhood of ξ = 0, so does at least one of the functions fj . Remark: : Titchmarsh’s theorem is essentially statement (c) above. The assumption that fj are real-valued is introduced to ensure that the corresponding Fj , defined by an integral analogous to (7.10), is real-valued on the imaginary axis. This assumption is not essential; in fact, part (c) is an immediate consequence of the Paley-Wiener theorems in the case f1 = f2 (in which case part (b) is trivial), and the full part (c) can be obtained from it quite simply (see [18]). ?8 THE FOURIER-CARLEMAN TRANSFORM

We sketch briefly another way to extend the domain of the Fourier transformation. There is no aim here at maximum generality and we describe the main ideas using L∞ (R) as an example, although only minor modifications are needed in order to extend the theory to functions of polynomial growth at infinity or, more generally, to functions whose growth at infinity is slower than exponential. For more details we refer the reader to [3].

194


8.1 For h ∈ L∞ (R) we write Z 0 e−iζx h(x)dx F1 (h, ζ) = −∞ (8.1) Z ∞ e−iζx h(x)dx F2 (h, ζ) = −

ζ = ξ + iη, η > 0 ζ = ξ + iη, η < 0.

0

F1 (h, ζ) and F2 (h, ζ) are clearly holomorphic in their respective domains of definition, and it is apparent from (8.1) that if η > 0, then F1 (h, ξ + iη) − F2 (h, ξ − iη) is the Fourier transform of e−η|x| h. Hence if h ∈ L1 (R) we obtain

(8.2)

ˆ lim (F1 (h, ξ + iη) − F2 (h, ξ − iη)) = h(ξ)

η→0+

uniformly. Since e−η|x| h tends to h in the weak-star topology for any ˆ is allowed to h ∈ L∞ (R), (8.2) is valid for every h ∈ L∞ (R) provided h be a pseudo-measure and the limit is in the weak-star topology of FL∞ ˆ . as dual of A(R) ˆ disjoint Let us consider the case h ∈ L1 (R). If I is an interval on R ˆ from the support of h, and D is the disc of which I is a diameter, and if we define the function F in D by ( F1 (h, ζ) η ≥ 0 (8.3) F (h, ζ) = F2 (h, ζ) η ≤ 0, then it follows from (8.2) that F (h, ζ) is well defined and continuous in D and it is holomorphic in D \ I . It is well known that this implies (e.g., by Morera’s theorem) that F (h, ζ) is holomorphic in D. We see that in the case h ∈ L1 ∩L∞ (R), F1 (h, ζ) and F2 (h, ζ) are analytic continuations ˆ on R ˆ . On the other hand, of each other through the complement of Σ(h) if F1 (h, ζ) and F2 (h, ζ) are analytic continuations of each other through ˆ ˆ = ∅. an open interval I , h(ξ) = F1 (h, ξ)−F2 (h, ξ) = 0 on I , and I ∩Σ(h) Denoting by c(h) the set of concordance of (F1 (h, ζ), F2 (h, ζ)), that is, ˆ in the neighborhood of which F1 (h, ζ) and F2 (h, ζ) the set of points on R are analytic continuations of each other, we can state our result as ˆ is the complement of Lemma. Assume h ∈ L1 ∩ L∞ (R); then Σ(h) c(h).

8.2 We now show that the same is true without assuming h ∈ L1 (R). ˆ is the complement of Theorem. For every bounded function h, Σ(h) c(h).


195

P ROOF : Let E be a compact subset of c(h); then, as η → 0+, ˆ F1 (h, ξ + iη) − F2 (h, ξ − iη) → 0 uniformly for ξ ∈ E . If fˆ ∈ A(R) ˆ and the support of f is contained in E , then Z 1 ˆ ˆ (8.4) hf , hi = lim fˆ(ξ)F1 (h, ξ + iη) − F2 (h, ξ − iη)dξ = 0 η→0+ 2π ˆ ∩ c(h) = ∅. which proves Σ(h) ˆ implies ξ0 ∈ c(h) is obtained from Lemma The fact that ξ0 6∈ Σ(h) 8.1 and the following simple lemma about removable singularities:

8.3 Lemma. Let I be an interval on the real line, D the disc in the ζ plane of which I is a diameter, F a holomorphic function defined in D \ I , satisfying the growth condition |F (ξ + iη)| < const |η|−n .

(8.5)

Assume that there exist functions Φj which are holomorphic in D, satisfy (8.5) (with a constant independent of j ) and Φj (ζ) → F (ζ) in D \ I . Then F can be extended to a function holomorphic in D.

P ROOF : Let D1 be a concentric disc properly included in D and D2 a concentric disc properly included in D1 . Denote by ζ1 , ζ2 the points of intersection of the boundary of D1 with I . The functions (ζ − ζ1 )n (ζ − ζ2 )n Φj (ζ) are uniformly bounded on the boundary of D1 , hence in D1 , and consequently Φj are uniformly bounded in D2 . The Cauchy integral formula now shows that Φj converge uniformly in D2 to a holomorphic function which agrees with F on D2 \ I . Since D2 is an arbitrary concentric disc in D, the lemma follows. J 8.4 Lemma. Let h ∈ L∞ (R); then |F1 (h, ζ)| ≤ khk∞ η −1 |F2 (h, ζ)| ≤ khk∞ |η|

−1

ζ = ξ + iη, η > 0 ζ = ξ + iη, η > 0.

P ROOF : |F1 (h, ζ)| ≤

Z

0

−∞

and similarly for F2 .

eηx |h(x)|dx ≤ khk∞

Z

0

eηx dx = khk∞ η −1

−∞

J

196


We can now finish the proof of Theorem 8.2. We have to show that ˆ , then ξ0 ∈ c(h). Assume ξ0 6∈ Σ(h) ˆ ; by Lemma 4.7, ξ0 if ξ0 6∈ Σ(h) dλ ) provided λ has an interval I about it which does not intersect Σ(hK ˆ λ ) = [−1/λ, 1/λ]). If D is large enough (Kλ is the Fejér kernel and Σ(K is the disc for which I is a diameter, it follows from Lemma 8.1 that the pair (F1 (hKλ , ζ), F2 (hKλ , ζ)) defines holomorphic functions Φλ in D, which clearly converge, as λ → ∞ to (F1 (h, ζ), F2 (h, ζ)) on D \ l. By Lemma 8.4 we can apply Lemma 8.3 and the theorem follows. J The Fourier-Carleman transform thus gives an alternative definition of the weak-star spectrum of a bounded function. As an illustration we indicate briefly how Theorem 4.11 can be obtained by Carleman’s ˆ = {0}. The pair method. We assume again h ∈ L∞ (R) and Σ(h) (F1 (h, ζ), F2 (h, ζ)) defines an analytic function whose only singularity in the finite ζ plane is at the point ζ = 0. By Lemma 8.4 and the Phragmén-Lindelöf theorem, Φ tends to zero at infinity and has a simple pole at ζ = 0. Hence, for some constant c, Φ(ζ) = c/iζ , which is the Fourier-Carleman transform of the constant c. 9 KRONECKER’S THEOREM

9.1 Theorem (Kronecker). Let λ1 , λ2 , . . . , λn be real numbers, independent over the rationals. Let α1 , . . . , αn be real numbers and ε > 0. Then there exists a real number x such that

(9.1)

|eiλj x − eiαj | < ε,

j = 1, 2, . . . , n.

Kronecker’s theorem is equivalent to 9.2 Theorem. λ1 , λ2 , . . . , λn be real numbers, independent over the rationals, λ0 = 0, and let a0 , a1 , . . . , an be any complex numbers. Then

(9.2)

n n X X supx aj eiλj x = |aj |. j=0

j=0

We first establish the equivalence of Theorems 9.1 and 9.2 and then obtain 9.2 as a limit theorem.

P ROOF THAT 9.1 ⇒ 9.2: Write aj = rj eiαj , rj ≥ 0. By 9.1, there exist values of x for which |eiλj x − ei(α0 −αj ) | is small, j = 1, . . . , n. For these P P values of x, nj=0 aj eiλj x is close to eiα0 rj . J


197

P P ROOF THAT 9.2 ⇒ 9.1: Consider the polynomial 1+ n1 e−iαj eiλj x and notice that its absolute value can be close to n + 1 only if all the summands are close to 1, that is, only if (9.1) is satisfied. J

Remark: If λ1 , . . . , λn , π are linearly independent over the rationals, we can add the condition |e2πix − 1| < ε which essentially means that we can pick x in (9.1) to be an integer. Theorem 9.2 is a limiting case of Theorem 9.3N below. The idea in the proof is that used in the proof of Lemma V.1.3, that is, the application of Riesz products and of the inequality M (f g) ≤ kf k∞ M (|g|)

(9.3)

which is clearly valid for f, g ∈ AP (R) (see (5.5)). Actually, we use (9.3) for polynomials only, in which case the existence of the limit (5.5) and the fact that it equals the constant term are obvious, and this section is essentially independent of section 5. For the sake of clarity we state 9.3N first for N = 1, as 9.3 Theorem. Let λ1 , . . . , λn be real numbers having the following properties: (a) (b)

n X

1 n X

ε j λj = 0

εj = −1, 0, 1,

ε j λj = λk

εj = −1, 0, 1,

⇒ εj = 0 for all j . ⇒ εj = 0 for j 6= k .

1

Then, for any complex numbers a1 , . . . , an X 1X |aj |. (9.4) supx aj eiλj x ≥ 2

P ROOF : Write aj = rj eiαj , rj ≥ 0 and g(x) =

n Y

f (x) =

X

1

1 + cos(λj x + αj ) aj eiλj x .

g is a nonnegative trigonometric polynomial whose frequencies all have P the form εj λl , εj = −1, 0, 1. By (a), the constant term in g is 1, hence

198


M (g) = M (|g|) = 1. By (b), the constant term (which is the same as the P mean value) of f g is 12 n1 rj and, by (9.3), n

1X rj ≤ sup|f |. 2 1

J

9.3N Theorem. Let λ1 , . . . , λn be real numbers having the following properties: (a) (b)

n X

1 n X

ε j λj = 0

εj integers, |εj | ≤ N ⇒ εj = 0 for all j .

ε j λj = λk

εj integers, |εj | ≤ N ⇒ εj = 0 for j 6= k .

1

Then, for any complex numbers a1 , . . . , an X 1 X |aj |. (9.4N ) supx aj eiλj x ≥ 1 − N +1

P ROOF : Virtually identical to that of 9.3; we only have to replace g as defined there by n Y g(x) = KN (λj x + αj ) 1

where KN (x) =

PN −N

ijx e 1 − N|j| . We leave the details to the reader, +1 J

It is clear that if λ1 , . . . , λn are linearly independent, the conditions of 9.3N are satisfied for all N and consequently we obtain (9.4N ) for all N , hence (9.2). This completes the proof of theorem 9.2 and hence of Kronecker’s theorem. J For a different approach see VII.3. 9.4 The extension of theorem 9.1 to infinite, linearly independent sets presents a certain number of problems, not all of which are solved. We restrict our attention to compact linearly independent sets E and ask under what conditions is it possible to approximate uniformly on E every function of modulus 1, by an exponential. The obvious answer is that this is possible if, and only if, E is finite; this follows from Kronecker’s theorem ("if") and the fact that uniform limits of exponentials


199

must be continuous on E , and if E is infinite (and compact) not all functions of modulus 1 are continuous ("only if"). We therefore modify our questions and ask under what condition is it possible to approximate uniformly on E every continuous function of modulus 1 by an exponential. We do not have a satisfactory answer to this question; for some sets E the approximation is possible, for others it is not, and we introduce the following:

D EFINITION : A compact set E ⊂ R is a Kronecker set if every continuous function of modulus 1 on E can be approximated on E uniformly be exponentials. The existence of an infinite perfect Kronecker set is not hard to establish by a direct construction. We choose, however, to prove it by a less direct method which also may be used to obtain finer results (see [14]). Theorem. Let E be a perfect totally disconnected set on R. Denote by CR (E) the space of continuous, real-valued functions on E . Then there exists a set G of the first category† in CR (E) such that every ϕ ∈ CR (E) \ G maps E homeomorphically onto a Kronecker set.

P ROOF : A function ϕ ∈ CR (E) maps E homeomorphically onto a Kronecker set if, and only if, for every continuous function h of modulus 1 on E and for every ε > 0, there exists a real number λ such that (9.5) supx∈E eiλϕ(x) − h(x) < ε

We show first that if we fix h and ε, the set of functions ϕ for which (9.5) holds for an appropriate λ is everywhere dense in CR (E). For this, let ψ ∈ CR (E) and let η > 0. We take λ = 10η −1 and write E as a union of disjoint closed subsets Ej , j = 1, . . . , N, the Ej ’s being small enough so that the variation of either h or eiλψ on Ej does not exceed ε/3. Let eiαj be a value assumed by h on Ej and eiβj a value assumed by eiλψ on Ej ; we may clearly assume |αj | ≤ π and |βj | ≤ π for all j . We now define (9.6)

ϕ(x) = ψ(x) +

αj − βj λ

for x ∈ Ej .

We have ϕ ∈ CR (E) and kϕ − ψk∞ ≤ 2π/λ < η ; also, checking on each Ej , it is clear that (9.5) holds. † C (E), with the metric given by the norm kϕk ∞ = supx∈E |ϕ(x)|, is a complete R metric space.

200


It follows that the set G(h, ε) of all ϕ ∈ CR (E) for which (9.5) holds for no λ ∈ R, a set which is clearly closed, is nondense. Taking a sequence of continuous functions of modulus 1, say {hn }, which is dense in the set of all such functions, and taking a sequence of positive numbers {εm } such that εm → 0, it is clear that G = ∪n,m G(hn , em ) is of the first category. Also, if ϕ 6∈ G, then every hn can be approximated uniformly on E by eiλϕ with appropriate λ’s hence so can every continuous function of modulus 1 and the theorem follows. J

EXERCISES FOR SECTION 9 1. Let λ1 , . . . , λn be linearly independent over the rationals and let f1 , . . . , fn be continuous and periodic on R, having periods λ−1 respectively. Show that j the closure of the range of f = f1 + · · · + fn is precisely range (f0 ) + · · · +range (fn ). Deduce that if 0 ∈ range (fj ) for all j , then

X

fj

∞

≥

1X kfj k∞ . 6

Hint: Show that if c1 . . . , cn are complex numbers, P P one can find ε1 , . . . , εn , the εj ’s being zero or one, such that | εj cj | ≥ 16 |cj |. 2. Let f ∈ AP (R) and assume that σ(f ) is independent over the rationals. Show that fˆ is a measure and that kf k∞ = kfˆkM (R) ˆ . ˆ 3. Let f ∈ AP (R) and assume that σ(f ) ⊂ {3−j }∞ j=1 . Show that f is a ˆ measure and that kf kM (R) ˆ ≤ 2kf k∞ . 4. Let λ1 , . . . , λn be real numbers. Set λ0 = 0 and assume that for any choice of complex numbers a0 , . . . , an , (9.2) is valid. Show that λ1 , . . . , λn are linearly independent over the rationals. 5. Construct a sequence {λj } of linearly independent numbers such that λj → 0, and such that {λj } ∪ {0} is not a Kronecker set. 6. Show that every convergent sequence of linearly independent numbers contains an (infinite) subsequence which is a Kronecker set.

Chapter VII

Fourier Analysis on Locally Compact Abelian Groups

We have been dealing so far with spaces of functions defined on ˆ ). the circle group T, the group of integers Z, or the real line R (or R Most of the theory can be carried, without too much effort, to spaces of functions defined on any locally compact abelian group. The interest in such a generalization lies not only in the fact that we have a more general theory, but also in the light it sheds on the "classical" situations. We give only a brief sketch of the theory: proofs, many more facts, and other references can be found in [5], [9], [15] and [24]. 1 LOCALLY COMPACT ABELIAN GROUPS

A locally compact abelian (LCA) group is an abelian group, say G, which is at the same time a locally compact Hausdorff space and such that the group operations are continuous. To be precise: if we write the group operation as addition, the continuity requirement is that both mappings x 7→ −x of G onto G and (x, y) 7→ x + y of GxG onto G are continuous. For a fixed x ∈ G, the mappings y 7→ x + y is a homeomorphism of G onto itself which takes 0 into x. Thus the topological nature of G at any x ∈ G is the same as it is at 0. Examples: (a) Any abelian group G is trivially an LCA group with the discrete topology. (b) The circle group T and the real line R with the usual topology. (c) Let G be an LCA group and H a closed subgroup, then H with the induced structure is an LCA group. The same is true for the quotient group G/H if we put on it the canonical quotient topology 201

202


that is, if we agree that a set U in G/H is open if, and only if, its preimage in G is open. (d) The direct sum of a finite number of LCA groups is defined as the algebraic direct sum endowed with the product topology; it is again an LCA group. (e) The complete direct sum of a family {Gα }, α ∈ I , of abelian groups is the group of all "vectors" {xα }α∈I , xα ∈ Gα , where the addition is performed coordinatewise: {xα } + {yα } = {xα + yα }. If for all α ∈ I, Gα is a compact abelian group, the product topology on the complete direct sums make it a compact abelian group. This follows easily from Tychonoff’s theorem. If for every positive integer n, Gn is the group of order two, then the complete direct sum of {Gn } is the group of all sequences {εn }, εn = 0, 1 with coordinatewise addition modulo 2, and with the topology P that makes the mapping {εn } 7→ 2 εn 3−n a homeomorphism of the group onto the classical cantor set on the line. We denote this particular group by D. 2 THE HAAR MEASURE

Let G be a locally compact abelian group. A Haar measure on G is a positive regular Borel measure µ having the following two properties: (1) µ(E) < ∞ if E is compact; (2) µ(E + x) = µ(E) for all measurableE ⊂ G and all x ∈ G. One proves that a Haar measure always exists and that it is unique up to multiplication by a positive constant; by abuse of language one may therefore talk about the Haar measure. The Haar measure of G is finite if, and only if, G is compact and it is then usually† normalized to have total mass one. If G = T or G = Tn the Haar measure is simply the normalized Lebesgue measure. If G = R the Haar measure is again a multiple of the Lebesgue measure. If G is discrete, the Haar measure is usually† normalized to have mass one at each point. If G is the direct sum af G1 and G2 , the Haar measure of G is the product measure of the Haar measures of G1 and G2 . The Haar measure on the complete † Except when G is finite; it is as usual to introduce the "compact" normalization as it is the "discrete."

VII. F OURIER A NALYSIS ON L OCALLY C OMPACT A BELIAN G ROUPS 203

direct sum of a family of compact groups is the product of the corresponding normalized Haar measures. In particular, the Haar measure on the group D defined above corresponds to the well-known Lebesgue measure on the Cantor set, the homeomorphism defined above being also measure preserving. Let G be an LCA group; we denote the Haar measure Ron G by dx, and the Rintegral of f with respect to the Haar measure by G f (x)dx or simply f (x)dx. For 1 ≤ p ≤ ∞ we denote by Lp (G) the Lp space on G correspondingR to the Haar measure. One defines convolution on G by (f ∗ g)(y) = G f (y − x)g(x)dx and proves that if f, g ∈ L1 (G) then f ∗ g ∈ L1 (G) and kf ∗ gkL1 (G) ≤ kf kL1 (G) kgkL1 (G) so that L1 (G) is a Banach algebra under convolution. We may define homogeneous Banach spaces on any LCA group G as we did for T or R, that is, as Banach spaces B of locally integrable functions, norm invariant under translation and such that the mappings y 7→ fy are continuous from G to B for all f ∈ B . Remembering that for 1 < p < ∞ the continuous functions with compact support are norm dense in Lp (G), it is clear that Lp (G) is a homogeneous Banach space on G. Let B be a homogeneous Banach space on an LCA group G. Using vector-valued integration we can extend the definition of convolution so that f ∗ g is defined and belongs to B for all f ∈ L1 (G) and g ∈ B and show that kf ∗ gkB ≤ kf kL1 (G) kgkB . D EFINITION : A summability kernel on the LCA group G is a directed family {kα } in L1 (G) satisfying the following conditions: (a) kkα kL1 (G) < const; (b)

R

kα (x)dx = 1;

(c) if V is an neighborhood of 0 in G, limα

R

G\V

|kα (x)|dx = 0.

If {kα } is a summability kernel on G and if B is a homogeneous Banach space on G, then limα kkα ∗ g − gkB = 0 for all g ∈ B . 3 CHARACTERS AND THE DUAL GROUP

A character on an LCA group G is a continuous homomorphism of G into the multiplicative group of complex numbers of modulus 1, that is, a continuous complex-valued function ξ(x) on G satisfying: |ξ(x)| = 1

and

ξ(x + y) = ξ(x)ξ(y).

204


The trivial character is ξ(x) = 1 identically. If G is non-trivial there are non-trivial characters on it. ˆ of all the characters on G is clearly a commutative multiThe set G plicative group (under pointwise multiplication). We change the notaˆ as addition and replace ξ(x) by tion and write the group operation of G iξx hx, ξi or sometimes by e . ˆ by stipulating that convergence in We introduce a topology to G ˆ G is equivalent to uniform convergence on compact subsets of G (the ˆ being functions on G). Thus, a basis of neighborhoods elements of G ˆ of 0 in G is given by sets of the form {ξ : |hx, ξi − 1| < ε for all x ∈ K} where K is a compact subset of G and ε > 0. Neighborhoods of other ˆ are translates of neighborhhods of 0. It is not hard to see points in G ˆ is an LCA group; we call it the dual group of that with this topology G G. ˆ. For each x ∈ G, the mapping ξ 7→ hx, ξi defines a character on G ˆ has this The Pontryagin duality theorem states that every character on G form and that the topology of uniform convergence on compact subsets ˆ coincides with the original topology on G. In other words, if G ˆ is of G ˆ. the dual group of G, then G is the dual of G Examples: (a) For G = T with the usual topology every character has the form t 7→ e−int for some integer n, the topology of uniform ˆ = Z. Similarly, convergence on T is clearly the discrete topology and T ˆ = T; this illustrates the Pontryagin duality theorem. we check Z The example G = T hints the following general theorem: The dual group of any compact group is discrete (see exercise 5 at the end of this section). Also: The dual group of every discrete group is compact. (b) Characters on R all have the form x 7→ eiξx for some real ξ . ˆ is The dual group topology is the usual topology of the reals and R isomorphic to R. (c) If H is a closed subgroup of an LCA group G, the annihilator of H , denoted H ⊥ , is the set of all characters of G which are equal to ˆ . If ξ ∈ H ⊥ , ξ defines 1 on H . H ⊥ is clearly a closed subgroup of G canonically a character on G/H ; on the other hand, every character on G/H defines canonically (by composition with the mapping G 7→ G/H ) a character on G. This establishes an algebraic isomorphism between the dual group of G/H and H ⊥ . One checks that this is also a homeomorphism and the dual of G/H can be identified with H ⊥ . If H is a proper closed subgroup, then H ⊥ is non-trivial. ˆ ⊥ is the (d) By (c) above and the Pontryagin duality theorem: G/H


dual group of H . (e) If G1 and G2 are LCA groups, then G\ 1 ⊕ G2 can be identified ˆ1 ⊕ G ˆ 2 through with G < (x1 , x2 ), (ξ1 , ξ2 ) >=< x1 , ξ1 >< x2 , ξ2 > . n n In particular, the dual P group of T is Z , the characters have the form (t1 , . . . , tn ) 7→ e−i aj tj with aj ∈ Z. (f) If Gα , is a compact abelian group for every α belonging to some ˆ can be index set I , and if G is the complete direct sum of {Gα }, then G ˆ α } (with the discrete topology). The identified with the direct sum of {G ˆ α } of groups is the subgroup of the complete direct sum of a family {G ˆ α , such that direct sum consisting of those vectors {ξα }α∈I , ξα ∈ G ˆ α for all but a finite number of indices. ξα = 0 in G The dual group of the group of order two is again the group of order two. Consequently, the dual group of the group D introduced above is the direct sum of a sequence of groups of order two. If we identify the ˆ is the discrete group elements of D as sequences {εn }, εn = 0, 1, then D of sequences {ζn }, ζn = 0, 1 with only a finite number of ones, and P < {εn }, {ζn } >= (−1) εn ζn .

Remark: A natural way to look at Kronecker’s theorem VI.9.3 is: Assume that λ1 , . . . , λn ’s are rationally independent mod 2π and consider λ = (λ1 . . . λn ) ∈ Tn . The set Λ = {jλ : j ∈ Z} is a subgroup, and Kronecker’s theorem states that it is dense in Tn . If it weren’t, its closure Λ would be a closed proper subgroup and there would be a P non-trivial (a1 , . . . , an ) ∈ Zn which is trivial on Λ, i.e. aj λj ∈ 2πZ. 4 FOURIER TRANSFORMS

Let G be an LCA group; the Fourier transform of f ∈ L1 (G) is defined by Z ¯ fˆ(ξ) = < x, ξ >f (x) dx, ξ ∈ G. G

¯ the space of all Fourier transforms of functions in We denote by A(G) ˆ is an alL1 (G). Since we have (f\ + g) = fˆ + gˆ and fd ∗ g = fˆgˆ, A(G) ˆ gebra of functions on G under the pointwise operations. The functions ˆ are continuous on G; in fact, an equivalent way to define the in A(G) ˆ is as the weak topology determined by A(G) ˆ , that is, as topology on G ˆ the weakest topology for which all the functions in A(G) are continuous.

206


ˆ properly normalized one proves With the Haar measures on G and G inversion formulas stating essentially that f (−x) is the Fourier transform of fˆ in some appropriate sense, and literally if f is continuous ˆ . One deduces the uniqueness theorem stating that if and fˆ ∈ L1 (G) f ∈ L1 (G) and fˆ = 0 then f = 0. From the inversion formulas one can also prove Plancherel’s theorem. This states that the Fourier transformation is an isometry of ˆ and can therefore be exL1 ∩ L2 (G) onto a dense subspace of L2 (G) ˆ . One can now define the tended to an isometry of L2 (G) onto L2 (G) Fourier transform of functions in Lp (G), 1 < p < 2, by interpolation, and obtain inequalities generalizing the Hausdorff-Young theorem (as we did in VI.3 for the case G = R). We denote by M (G) the space of (finite) regular Borel measures on G. M (G) is a Banach space canonically identified with the dual of C 0 (G). The fact that the underlying space G is a group permits the definition of convolution in M (G) (analogous to that which we introduced in I.7 for the case G = T). With the convolution as multiplication, M (G) is a Banach algebra. We keep the notation µ ∗ ν for the convolution of the measures µ and ν . L1 (G) is identified as a closed subalgebra of M (G) through the correspondence f 7→ f dx. The Fourier (Fourier-Stieltjes) transform of µ ∈ M (G) is defined by Z ˆ µ ˆ(ξ) = < x, ξ >dµ(x), ξ ∈ G. ˆ . If µ = f dx with For all µ ∈ M (G), µ ˆ(ξ) is uniformly continuous on G 1 f ∈ L (G), then µ ˆ(ξ) = fˆ(ξ). The mapping µ 7→ µ ˆ is clearly linear ˆ d and we have µ ∗ ν = µ µ : µ ∈ M (G)} of ˆνˆ so that the family B(G) = {ˆ all Fourier-Stieltjes transforms is an algebra of uniformly continuous ˆ under pointwise addition and multiplication. functions on G ˆ is called positive definite if, for every A function ϕ defined on G ˆ and complex numbers z1 , . . . , zN we have choice of ξ1 , . . . , ξN ∈ G PN j,k=1 ϕ(ξj − ξk )zj zk ≥ 0. Weil’s generalization of Herglotz-Bochner’s ˆ is the Fourier transform of a theorem states that a function ϕ(ξ) on G positive measure on G if, and only if, it is continuous and positive definite. 5 ALMOST-PERIODIC FUNCTIONS AND THE BOHR COMPACTIFICATION

Let G be an LCA group. A function f ∈ L∞ (G) is, by definition, almost-periodic if the set of all translates of f , {fy }y∈G is precom-


pact in the norm topology of L∞ (G) (compare with VI.5.5). We denote the space of all almost-periodic functions on G by AP (G). One proves that almost-periodic functions are uniformly continuous and are uniform limits of trigonometric polynomials on G (i.e., of finite linear combinations of characters). Since trigonometric polynomials are clearly almost-periodic, one obtains that AP (G) is precisely the closure in L∞ (G) of the space of trigonometric polynomials. If G is compact we have AP (G) = C(G). In the general case we ˆ d , the dual group of G with its topology replaced consider the groups (G) ˆ d. G ˜ , the dual group of (G) ˜ is the group by the discrete topology, and G ˆ of all homomorphisms of G into T, and it therefore contains G (which ˆ into is identified with the group of all continuous homomorphisms of G ˜ T). One proves that the natural imbedding of G into G is a continuous ˜ . Being the dual of a discrete isomorphism and that G is dense in G ˜ group, G is compact; we call it the Bohr compactification of G. The Bohr compactification of the real line is the dual group of the discrete real line and is usually called the Bohr group. Assume f ∈ AP (G); let {Pj } be a sequence of trigonometric polynomials which converges to f uniformly. Then, since G is dense in ˜ , {Pj } converges uniformly on G ˜ (every character on G extends by G ˜ continuity to a character on G. It follows that f is the restriction to G ˜ . Conversely, since every continuous function F of lim Pj = F ∈ C(G) ˜ on G can be approximated uniformly by trigonometric polynomials, it ˜ . follows that AP (G) is simply the restriction to G of C(G) EXERCISES 1. Let G be an LCA group and µ the Haar measure on G. Show that if U is a nonempty open set in G then µ(U ) > 0. Hint: Every compact set E ⊂ G can be covered by a finite number of translates of U . 2. Let G be an LCA group and µ the Haar measure on G. Let H be a compact subgroup. Describe the Haar measure on G/H . 3. Let G1 and G2 be compact abelian groups and let G = G1 ⊕ G2 . Denote by µ, µ1 , µ2 the normalized Haar measures on G, G1 , G2 , respectively. Considering µj , j = 1, 2, as measures on G (carried by the closed subgroups Gj ), prove that µ = µ1 ∗ µ2 . 4. Let G be a compact group and {Hn } an increasing sequence of compact subgroups such that ∪∞ 1 Hn is dense in G. Denote by µ, µn , respectively, the

208


normalized Haar measure of G, Hn , respectively. Considering the µn ’s as measures on G, show that µn → µ in the weakstar topology of measures. 5. Let G be a group and let ξ1 and ξ2 be distinct characters on G. Show that √ supx∈G |< x, ξ1 > − < x, ξ2 >| ≥ 3. ˆ is discrete. Deduce that if G is a compact abelian group, then G 6. Let G be a compact abelian group with normalized Haar measure and let ˆ . Show that ξ∈G ( Z 1 if ξ = 0 < x, ξ > dx = 0 if ξ 6= 0. G

7. Let G be a compact abelian group. Show that the characters on G form a complete orthonormal family in L2 (G).

Chapter VIII

Commutative Banach Algebras

Many of the spaces we have been dealing with are algebras. We used this fact, implicitly or semi-explicitly, but only on the most elementary level. Our purpose in this chapter is to introduce the reader to the theory of commutative Banach algebras and to show, by means of examples, how natural and useful the Banach algebra setting can be in harmonic analysis. There is no claim, of course, that every problem in harmonic analysis has to be considered in this setting; however, if a space under study happens to be either a Banach algebra, or the dual space of one, keeping this fact in mind usually pays dividends. The introduction that we offer here is by no means unbiased. The topics discussed are those that seem to be the most pertinent to harmonic analysis and some very important aspects of the theory of commutative Banach algebras (as well as the entire realm of the noncommutative case) are omitted. As further reading on the theory of Banach algebras we mention [5], [15], [19] and [21]. 1 DEFINITION, EXAMPLES, AND ELEMENTARY PROPERTIES

1.1 D EFINITION : A complex Banach algebra is an algebra B over the field C of complex numbers, endowed with a norm k k under which it is a Banach space and such that

(1.1)

kxyk ≤ kxkkyk

for any x, y ∈ B . Examples: (1) The field C of complex numbers, with the absolute value as norm. (2) Let X be a compact Hausdorff space and C(X) the algebra of all continuous complex-valued functions on X with pointwise addition and multiplication. C(X) is a Banach algebra under the supremum 209

210


norm (also referred to as the sup-norm) (1.2)

kf k∞ = supx∈X |f (x)|.

(3) Similarly, if X is a locally compact Hausdorff space, we denote by C0 (X) the sup-normed algebra (with pointwise addition and multiplication) of all continuous functions on X which vanish at infinity (i.e., the functions f for which {x : |f (x)| ≥ ε} is compact for all ε > 0). (4 )C n (T)–the algebra of all n-times continuously differentiable functions on T with pointwise addition and multiplication and with the norm kf kC n =

n X 1 max|f (j) (t)|. t j! 0

(5) HC(D)—the algebra of all functions holomorphic in D (the unit ¯ , with pointwise addition and disc {z : |z| < 1}) and continuous in D multiplication and with the sup-norm. (6) L1 (T)–with pointwise addition and the convolution (I.1.8) as multiplication, and with the norm k kL1 . Condition (1.1) is proved in Theorem I.1.7. Similarly–L1 (R). (7) M (T )–the space of (Borel) measures on T with convolution as multiplication and with the norm k kM (T) (see I.1.7). Similarly–M (R). (8) The algebra of linear operators on a Banach space with the standard multiplication and the operator norm. (9) Let B be a Banach space; we introduce to B the trivial multiplication xy = 0 for all x, y ∈ B . With this multiplication B is a Banach algebra. All the foregoing examples, except (8), have the additional property that the multiplication is commutative. In all that follows we shall deal mainly with commutative Banach algebras. 1.2 In all the examples except for (3), (6), and (9), the algebras have a unit element for the multiplication: the number 1 in (1); the function f (x) = 1 in (2), (4), and (5); the unit mass at the origin in (7); and the identity operator in (8). It is clear from I.1.7 that if f ∈ L1 (T) were a unit element, we would have fˆ(n) = 1 for all n which, by the RiemannLebesgue lemma, is impossible; thus L1 (T) does not have a unit. Let B be a Banach algebra. We consider the direct sum B1 = B ⊕ C, that is, the set of pairs (x, λ), x ∈ B, λ ∈ C; and define addition,

VIII.

C OMMUTATIVE BANACH A LGEBRAS

211

multiplication, scalar multiplication and norm in B1 by: (x1 , λ1 ) + (x2 , λ2 ) = (x1 + x2 , λ1 + λ2 ) (x1 , λ1 )(x2 , λ2 ) = (x1 x2 + λ1 x2 + λ2 x1 , λ1 λ2 ) λ(x1 , λ1 ) = (λx1 , λλ1 ) k(x, λ)kB1 = kxkB + |λ|.

It is clear, by direct verification, that B 1 is a Banach algebra with a unit element (namely (0, 1)). We now identify B with the set of pairs of the form (x, 0), which is clearly an ideal of codimension 1 in B1 . We say that B1 is obtained from B by a formal adjoining of a unit element; this simple operation allows the reduction of many problems concerning Banach algebras without a unit to the corresponding problems for Banach algebras with unit. If B is an algebra with a unit element we often denote the unit by 1 and identify its scalar multiples with the corresponding complex numbers. Thus we write "1 ∈ B " instead of "B has a unit element," and so on. This notation will be used when convenient and may be dropped when the unit element has been identified differently. 1.3 Every normed algebra, that is, complex algebra with a norm satisfying (1.1) but under which it is not necessarily complete, can be completed into a Banach algebra. This is done in the same way a normed space is completed into a Banach space. If B0 is a normed algebra, we denote by B the space of equivalence classes of Cauchy sequences in B0 , determined by the equivalence relation: {xn } ∼ {yn }

if, and only if,

limkxn − yn k = 0.

One checks immediately, and we leave it to the reader, that if {xn } ∼ {x0n } and {yn } ∼ {yn0 } then {xn + yn } ∼ {x0n + yn0 }, {λxn } ∼ {λx0n }, {xn yn } ∼ {x0n yn0 } and limn→∞ kxn k = limn→∞ kx0n k; hence we can define addition, scalar multiplication, multiplication, and norm in B as follows: for x, y ∈ B , let {xn } (resp. {yn }) be a Cauchy sequence in the equivalence class x (resp. y ), then x + y (resp. λx, xy ) will be the equivalence class containing {xn + yn } (resp. {λxn }, {xn yn }) and kxk is, by definition, limn→∞ kxn k. With these definitions, B is a Banach algebra and the mapping which associates with an element a ∈ B0 the equivalence class of the "constant" sequence {xn }, xn = a for all n, is an isometric embedding of B0 in B as a dense subalgebra.

212


1.4 The condition (1.1) on the norm in a Banach algebra implies the continuity of the multiplication in both factors simultaneously. Conversely: Theorem. Let B be an algebra with unit and with a norm k k under which it is a Banach space. Assume that the multiplication is continuous in each factor separately. Then there exists a norm k k0 equivalent to k k, for which (1.1)is valid. PROOF : By the continuity assumption, every x ∈ B defines a continuous linear operator Ax : y 7→ xy on B . If x 6= 0, Ax (1) = x, and consequently Ax 6= 0; also Ax1 x2 (y) = x1 x2 y = Ax1 (x2 y) = Ax1 Ax2 y hence the mapping x 7→ Ax is an isomorphism of the algebra B into the algebra of all continuous linear operators on B . Let k k0 be the induced norm, that is,

(1.3)

kxk0 = kAx k = supkyk≤1 kxyk

then k k0 is clearly a norm on B and it clearly satisfies (1.1). We also remark that (1.4)

kxk0 ≥ k1k−1 kxk

(take y = k1k−1 in (1.3)) and, consequently, if xn , is a Cauchy sequence in k k0 , it is also a Cauchy sequence in k k, and so converges to some x0 ∈ B . We contend now that limkxn − x0 k0 = 0 which is the same as limkAxn − Ax0 k = 0. This follows from: (a) {Axn } is a Cauchy sequence in the algebra of linear operators on B hence converges in norm to some operator A0 ; (b) Axn y = xn y → x0 y = Ax0 y for all y ∈ B (here we use the continuity of xy in x). It follows that A0 = Ax0 and the contention is proved. We have proved that B is complete under the norm k k0 , and since the two norms, k k and k k0 are comparable, (1.4), they are in fact equivalent (closed graph theorem). J Remark: The norm k k0 has the additional property that k1k0 = 1; hence there is no loss of generality in assuming as we shall henceforth do implicitly, that whenever 1 ∈ B , k1k = 1. 1.5 Theorem. Let B be a commutative Banach algebra and let I be a closed ideal in B . The quotient algebra B/I endowed with the canonical quotient norm is a Banach algebra.

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PROOF : The only thing to verify is the validity of (1.1). Let ε > 0, let x˜, y˜ ∈ B/I and let x, y ∈ B be representatives of the cosets x˜, y˜ respectively, such that kxk ≤ k˜ xk + ε, kyk ≤ k˜ y k + ε. We have xy ∈ x ˜y˜ and consequently

k˜ xy˜k ≤ kxyk ≤ kxkkyk ≤ k˜ xkk˜ y k + ε(k˜ xk + k˜ y k) + ε2

and since ε > 0 is arbitrary, k˜ xy˜k ≤ k˜ xkk˜ y k.

J

EXERCISES FOR SECTION 1 1. Verify condition (1.1) in the case of C n (T) (example 4 above). 2. Let B be a homogeneous Banach space on T, define multiplication in B as convolution (inherited from L1 (T)). Show that with this multiplication B is a Banach algebra. 3. Let X be a locally compact, noncompact, Hausdorff space and denote by X∞ its one-point compactification. Show that C(X∞ ) is isomorphic (though not isometric) to the algebra obtained by formally adjoining a unit to C0 (X). 4. Let B be an algebra with two consistent norms (see IV.1.1, k k0 and k k1 . Assume that both these norms are multiplicative (i.e., satisfy condition (1.1)). Show that all the interpolating norms k kα , 0 < α < 1 (see IV.1.2), are multiplicative. Hint: B is a normed algebra and Bα are ideals in B. 2 MAXIMAL IDEALS AND MULTIPLICATIVE LINEAR FUNCTIONALS

2.1 Let B be a commutative Banach algebra with a unit 1. An element x ∈ B is invertible if there exists an element x−1 ∈ B such that xx−1 = 1. Lemma. Consider a Banach algebra B with a unit 1. Let x ∈ B and assume kx − 1k < 1. Then x is invertible and

(2.1)

x−1 =

∞ X

(1 − x)j .

j=0

By (1.1), k(1 − x)j k ≤ k(1 − x)kj ; hence the series on the right of (2.1) converges in B . Writing x = 1 − (1 − x) and multiplying term P j be term we obtain x ∞ J j=0 (1 − x) = 1 PROOF :

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2.2 Lemma. Let x ∈ B be invertible and y ∈ B satisfying ky − xk < kx−1 k−1 . Then y is invertible and

(2.2)

y −1 = x−1

∞ X

(1 − x−1 y)j .

j=0

PROOF : k1 − x−1 yk ≤ kx−1 kkx − yk.

Apply Lemma 2.1 to x−1 y .

J

Corollary. The set U of invertible elements in B is open and the function x 7→ x−1 is continuous on U . PROOF : We only need to check the continuity, Let x ∈ U, y → x; by P −1 j (2.2) we have y −1 − x−1 = x−1 ∞ y) ; hence j=1 (1 − x

ky −1 − x−1 k ≤

∞ X

kx−1 kj+1 kx − ykj ≤ 2kx−1 k2 kx − yk

j=1

provided kx − yk ≤ 12 kx−1 k−1 .

J

2.3 D EFINITION : The resolvent set R(x) = RB (x) of an element x in a Banach algebra B with a unit is the set of complex numbers λ such that x − λ is invertible. Lemma. For x ∈ B , R(x) is open and F (λ) = (x − λ)−1 is a holomorphic B -valued function on R(x). PROOF :

This is again an immediate consequence of Lemma 2.2. If λ0 ∈ R(x) and λ is close to λ0 , it follows from (2.2) that X (x − λ)−1 = (x − λ0 )−1 (1−(x − λ0 )−1 (x − λ0 + λ0 − λ)j (2.3) X =− (x − λ0 )−j−1 (λ0 − λ)j . (2.3) is the expansion of (x−λ)−1 to a convergent power series in λ−λ0 with coefficients in B. J 2.4 Lemma. R(x) can never be the entire complex plane.

Assume R(x) = C. The function (x − λ)−1 is an entire B valued function and as |λ| → ∞ −1

x

∼ |λ|−1 → 0. k(x − λ)−1 k = |λ|−1 −1 λ PROOF :

It follows from Liouville’s theorem (see appendix A) that (x−λ)−1 ≡ 0, which is impossible. J

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Theorem (Gelfand–Mazur). A complex commutative Banach algebra which is a field is isomorphic to C.

Let x ∈ B , λ a complex number λ 6∈ R(x); then x − λ is not invertible and, since the only noninvertible element in a field is zero, x = λ. Thus, having identified the unit of B with the number l, B is canonically identified with C. PROOF :

2.5 We now turn to establish some basic facts about ideals in a Banach algebra. Lemma. Let I be an ideal in an algebra B with a unit. Then I is contained in a maximal ideal. PROOF : Consider the family I of all the ideals in B which contain I . I is partially ordered by inclusion and, by Zorn’s lemma, contains a maximal linearly ordered subfamily I0 . The union of all the ideals in I0 is a proper ideal, since it does not contain the unit element of B , and it is clearly maximal by the maximality of I0 . J

Remark: The condition 1 ∈ B in the statement of the lemma can be relaxed somewhat. For instance, if I ⊂ B is an ideal and if u ∈ B is such that (u, I)–the ideal generated by u and I –is the whole algebra, then u belongs to no proper ideal containing I , and the union of all the ideals in I0 (in the proof above) is again a proper ideal since it does not contain u. 2.6 D EFINITION : The ideal I ⊂ B is regular if B has a unit mod I ; that is, if there exists an element u ∈ B such thatx − ux ∈ I for all x ∈ B . If B has a unit element, every I ⊂ B is regular. If I is regular in B and u is a unit mod I then, since for every x ∈ B , x = ux + (x − ux), we see that (u, I) = B . Using Remark 2.5 we obtain: Lemma. Let I be a regular ideal in an algebra B . Then I is contained in a (regular) maximal ideal. 2.7 Lemmas 2.5 and 2.6 did not depend on the topological structure of B . If B is a Banach algebra with a unit it follows from Lemma 2.1 that the distance of 1 to any proper ideal is one, and consequently the closure of any proper ideal is again a proper ideal; in particular, maximal ideals in B are closed. Our next lemma shows that the same is true even if B does not have a unit element provided we restrict our attention to regular maximal ideals.

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Lemma. Let I be a regular ideal in a Banach algebra B . Let u be a unit mod I . Then dist(()u, I) ≥ 1.

P ROOF : We show that if v ∈ B and ku − vk < 1, then (I, v) = B ; hence v 6∈ I . For x ∈ B we have ! ∞ ∞ ∞ X X X j j (2.4) x= (u − v) x − u (u − v) + v (u − v)j x. 0

0

0

P P∞ j j The difference ( ∞ 0 (u − v) x − u 0 (u − v) ) belongs to I since u is a unit mod I , and the third term is a multiple of v ; hence (I, v) = B and the lemma is proved. J

Corollary. Regular maximal ideals in a Banach algebra are closed.

2.8 D EFINITION : A multiplicative linear functional on a Banach algebra B is a nontrivial† linear functional w(x) satisfying (2.5)

w(xy) = w(x)w(y),

x, y ∈ B.

Equivalently, it is a homomorphism of B onto the complex numbers. We do not require in the definition that w be continuous—we can prove the continuity: Lemma. Multiplicative linear functionals are continuous and have norms bounded by 1.

P ROOF : Let w be a multiplicative linear functional; denote its kernel by M . M is clearly a regular maximal ideal and is consequently closed. The mapping x 7→ w(x) identifies canonically the quotient algebra B/M with C, and if we denote by k k0 the norm induced on C by B/M , we clearly have kλk0 = k1k0 |λ| for all λ ∈ C. By (1.1), k1k0 ≥ 1 and hence |λ| ≤ kλk0 for all λ ∈ C; it follows that for any x ∈ B , |w(x)| ≤ J kw(x)k0 ≤ kxk. Theorem. The mapping w 7→ ker(w) defines a one-one correspondence between the multiplicative linear functionals on B and its regular maximal ideals. †A

linear functional which is not identically zero.

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PROOF : A multiplicative linear functional w is completely determined by its kernel M : if x ∈ M then w(x) = 0: if x 6∈ M , w(x) is the unique complex number for which x2 − w(x)x ∈ M . On the other hand, if M is a regular maximal ideal in a commutative Banach algebra B , the quotient algebra B/M is a field. Since M is closed B/M is itself a complex Banach algebra (Theorem 1.5), and by Theorem 2.4, B/M is canonically identified with C. It follows that the mapping B 7→ B/M is a multiplicative linear functional on B . J

Corollary. Let B be a commutative Banach algebra with a unit element. An element x ∈ B is invertible if, and only if w(x) 6= 0 for every multiplicative linear functional w on B .

If x is invertible w(x)w(x−1 ) = 1 for every multiplicative linear functional w, hence w(x) 6= 0. If x is not invertible then xB is a proper ideal which by Lemma 2.5, is contained in a maximal ideal M . Since x = x · 1 ∈ xB ⊂ M it follows that w(x) = 0 where w is the multiplicative linear functional whose kernel is M . J PROOF :

2.9 At this point we can already give one of the nicest applications of the theory of Banach algebras to harmonic analysis. Theorem (Wiener). Let f ∈ A(T) and assume that f vanishes nowhere on T; then f −1 ∈ A(T).

P ROOF : We have seen in I.6.1 that A(T) is an algebra under pointwise multiplication and that the norm kf kA(T) =

∞ X

|fˆ(n)|

−∞

is multiplicative. Since A(T) is clearly a Banach space (isometric to `1 ), it follows that it is a Banach algebra. Let w be a multiplicative linear functional on A(T); denote λ = w(eit ) (the value of w at the function eit ∈ A(T)). Since keit kA(T) = 1 it follows from Lemma 2.8 that λ ≤ 1; similarly we obtain that λ−1 = w((eit )−1 ) = w(e−it ) satisfies |λ−1 | ≤ 1, and consequently |λ| = l, that is λ = eit0 for some t0 . By the multiplicativity of w, w(eint ) = eint0 for all n; by the linearity, w(P ) = P (t0 ) for every trigonometric polynomial P ; and by the continuity, w(f ) = f (t0 ) for all f ∈ A(T). It follows that every multiplicative linear functional on A(T) is an evaluation at some

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to t0 ∈ T; every to t0 ∈ T clearly gives rise to such a functional and we have thus identified all multiplicative linear functionals on A(T). Let f ∈ A(T) such that f (t) 6= 0 for all t ∈ T. By Corollary 2.8, f is invertible in A(T), that is, there exists a function g ∈ A(T) such that g(t)f (t) = 1 or equivalently f −1 ∈ A(T). J ˆ and Theorem 2.9 can 2.10 The algebra A(T) is closely related to A(R) ˆ (or, equivalently, be used in determining the maximal ideals of A(R) 1 L (R)); this can also be done directly and our proof below has the advantage of applying for many convolution algebras (see also exercise 4 at the end of this section).

Theorem. Every multiplicative linear functional on L1 (R) has the ˆ. form f 7→ fˆ(ξ0 ) for some ξ0 ∈ R 1 P ROOF : Let w be a multiplicative linear functional on R L (R). As any 1 linear functional on L (R) w has the form w(f ) = f (x)h(x)dx for some h ∈ L∞ Z Z(R). We have ZZ w(f ∗ g) = f (x − y)g(y)h(x)dy dx = f (x)g(y)h(x + y)dx dy Z Z ZZ w(f )w(g) = f (x)h(x)dx g(y)h(y)dy = f (x)g(y)h(x)h(y)dx dy

By the multiplicativity of w and the fact that the linear combinations of P the form fj (x)gj (y), fj , gj ∈ L1 (R) are dense in L1 (RxR), it follows that h(x + y) = h(x)h(y) almost everywhere in RxR. Thus (see exercise ˆ , and w(f ) = fˆ(ξ0 ). VI.4.7) h(x) = eiξ0 x for some ξ0 ∈ R J 2.11 We shall use the term “function algebra” for algebras of continuous functions on a compact or locally compact Hausdorff space with pointwise addition and multiplication. It is clear that if B is a function algebra on a space X and if x ∈ X , then f 7→ f (x) is either a multiplicative linear functional on B , or zero, and consequently (Lemma 2.8) if B is a Banach algebra under a norm k k, we have |f (x)| ≤ kf k for all x ∈ X and f ∈ B . 2.12 Let B be a function algebra on a locally compact Hausdorff space X and assume that for all x ∈ X there exists a function f ∈ B such that f (x) 6= 0. Denote by wx the multiplicative linear functional f 7→ f (x). Recall that B is separating on X if for any x1 , x2 ∈ X, x1 6= x2 , there exists an f B such that f (x1 ) 6= f (x2 ); this amounts to saying that if x1 6= x2 then wx1 6= wx2 . Thus, if B is separating on X and not all the

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functions in B vanish at any x ∈ X , the mapping x 7→ wx identifies X as a set of multiplicative linear functionals on B . In general we obtain only part of the set of multiplicative linear functionals as wx , x ∈ X (see exercise 6 at the end of this section); however, in some important cases, every multiplicative linear functional on B has the form wx for some x ∈ X . We give one typical illustration.

D EFINITION : A function algebra B on a space X is self-adjoint on X if whenever f ∈ B then also f¯ ∈ B (where f¯(x) = f (x)). D EFINITION : A function algebra B on a space X is inverse closed if 1 ∈ B and whenever f ∈ B and f (x) 6= 0 for all x ∈ X , then f −1 ∈ B . Thus we can restate Theorem 2.9 as: "A(T) is inverse closed." Theorem. Let B be a separating, self-adjoint, inverse-closed function algebra on a compact Hausdorff space X . Then every multiplicative linear functional on B has the form wx (i.e., f 7→ f (x)) for some x ∈ X .

If we denote Mx = {f : f (x) = 0}, or equivalently Mx = ker(wx ), then, by theorem 2.8, the assertion that we want to prove is equivalent to the assertion that every maximal ideal in B has the form Mx for some x ∈ X . We prove this by showing that every proper ideal is contained in at least one Mx . Let I be an ideal in B and assume I 6⊂ Mx for all xX . This means that for every x ∈ X there exists a function f ∈ I such that f (x) 6= 0. Since f is continuous, f (y) 6= 0 for all y in some neighborhood Ox of x. By the compactness of X we can find a finite number of points x1 , . . . , xn with corresponding fj ∈ l and neighborhoods Oj , j = 1, . . . , n, such that X = ∪n1 Oj and such that P fj (y) 6= 0 for y ∈ Oj . The function ϕ = j f¯j fj belongs to I , is positive on X , and since B is assumed to be inverse closed, ϕ is invertible and 1 ∈ I , that is, I = B . J PROOF :

Corollary. Let X be a compact Hausdorff space. Then every multiplicative linear functional on C(X) has the form wx (i.e., f 7→ f (x)) for some x ∈ X .

EXERCISES FOR SECTION 2 1. Use the method of the proof of 2.9 to determine all the multiplicative linear functionals on C(T). 2. The same for C n (T). Hint: keimt kC n = O (mn ).

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3. Check the results of exercises 1 and 2 using 2.12. 4. Let G be an LCA group, and let B denote the convolution algebra L1 (G). Show that every multiplicative linear functional on B has the form f 7→ fˆ(γ) ˆ . Hint: Repeat the proof of 2.10. for some γ ∈ G 5. Determine the multiplicative linear functionals on HC(D) (see section 1, example 5). 6. Let B be the sup-norm algebra of all the continuous functions f on T such that fˆ(n) = 0 for all negative integers n. Show that B is a separating function algebra on T; however, not every multiplicative linear functional on B has the form wt for some t ∈ T. Hint: What is the relationship between B and HC(D)? 7. Show that a commutative Banach algebra B may have no multiplicative linear functionals (hint: example 9 of section 1); however, if 1 ∈ B , B has at least one such functional. 8. Determine the multiplicative linear functionals on C0 (X), X being a locally compact Hausdorff space. 3 THE MAXIMAL-IDEAL SPACE AND THE GELFAND REPRESENTATION

3.1 Consider a commutative Banach algebra B and denote by M the set of all of its regular maximal ideals. By Theorem 2.8 we have canonical identification of every M ∈ M with a multiplicative linear functional, and hence, by Lemma 2.8, we can identify M with a subset of the unit ball U ∗ of B ∗ –the dual space of B . This identification induces on M whatever topological structure we have on U ∗ , and two important topologies come immediately to mind: the norm topology and the weak-star topology. We limit our discussion of the metric induced on M by the norm in B ∗ to exercises 1-3 at the end of this section and refer to [6] for a more complete discussion. The topology induced on M by the weak-star topology on B ∗ is more closely related to the algebraic properties of B ; we shall refer to it as the weak-star topology on M. Lemma. M ∪ {0} is closed in U ∗ in the weak-star topology. If 1 ∈ B then M is closed.

P ROOF : In order to prove the first statement we have to show that if u0 ∈ M, then u0 (xy) = u0 (x)u0 (y) for all x, y ∈ B . From this it would follow that either u0 ∈ M or u0 = 0. Let ε > 0, x, y ∈ B and consider the neighborhood of u0 in U ∗ defined by (3.1) {u : |u(x) − u0 (x)| < ε, |u(y) − u0 (y)| < ε, |u(xy) − u0 (xu)| < ε};

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Since u0 ∈ M there exists a w ∈ M in (3.1), and remembering that w(xy) = w(x)w(y) we obtain |u0 (xy) − u0 (x)u0 (y)| ≤ ε(1 + kxk + kyk).

Since ε > 0 is arbitrary, u0 (xy) = u0 (x)u0 (y). In order to prove the second statement we have to show that if 1 ∈ B , then 0 6∈ M. Since w(1) = 1 for all w ∈ M, it follows that {u : |u(1)| < 1 2 } is a neighborhood of 0 disjoint from M and the proof is complete. J Since U ∗ with the weak-star topology is a compact Hausdorff space, it follows that the same is true for M ∪ {0} or, if 1 ∈ B , for M. This is sufficiently important to be stated as: Corollary. M, with the weak-star topology, is a locally compact Hausdorff space. If 1 ∈ B then M is compact.

We shall see later (see Theorem 3.5) that in some cases the compactness of M implies 1 ∈ B . However, considering example (9) of section 1, we realize that M may be compact (as a matter of fact empty !) for algebras without unit. The reader who feels unconvinced by an example consisting of the empty set should refer to exercise 4 at the end of this section. 3.2 For x ∈ B and M ∈ M we now write xˆ(M ) = x mod M (i.e., the image of x under the multiplicative linear functional corresponding to M ). By its definition, the weak-star topology on M is the weakest topology such that all the functions {ˆ x(M ) : x ∈ B} are continuous. Lemma. If 1 ∈ B , the mapping x 7→ xˆ is a homomorphism of norm one of B into C(M).

P ROOF : The algebraic properties of the mapping are obvious. For every M ∈ M and x ∈ B , |ˆ x(M )| ≤ kxk (Lemma 2.8) and hence supM |ˆ 1(M ) = 1 and the norm of x(M )| ≤ kxk. On the other hand ˆ not smaller than one. J If we do not assume 1 ∈ B , the set {M : |ˆ x(M )| ≥ ε} is compact in M for every x ∈ B and ε > 0; consequently x 7→ xˆ is a homomorphism of norm at most one, of B into C0 (M). The subalgebra Bˆ of C(M) (resp. C0 (M)) obtained as the image of B under the homomorphism x 7→ x ˆ is called the Gelfand representation of B . The function x ˆ(M ) is sometimes referred to as the Fourier-Gelfand transform of x.

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3.3 In many cases we can identify the weak-star topology on concretely in virtue of the following simple fact (cf. [15], p.6): let τ, τ1 be Hausdorff topologies on a space M and assume that M is compact in both topologies and that the two topologies are comparable; then τ = τ1 . In our case this means that if 1 ∈ B and if τ is a Hausdorff topology on M in which M is compact, and such that all the functions xˆ(M ) in Bˆ are τ -continuous, then, since the weak-star topology is weaker than or equal to τ , the two are equal. By a formal adjoining of a unit we obtain, similarly, that if I 6∈ B and τ is a Hausdorff topology on M such that M is locally compact and Bˆ ⊂ C0 (M, τ ) then τ is the weak-star topology on M. 3.4 D EFINITION : The radical, Rad (B), of a commutative Banach algebra B is the intersection of all the regular maximal ideals in B . Rad (B) is clearly a closed ideal in B and is the kernel of the homomorphism x 7→ xˆ of B onto Bˆ . The radical of B may coincide with B (example 9 of section 1) in which case we say that B is a radical algebra; it may be a nontrivial proper ideal, or it may be reduced to zero. 3.5 D EFINITION : A commutative Banach algebra B is semisimple if Rad (B) = 0. Equivalently: B is semisimple if the mapping x 7→ x ˆ is an isomorphism. 3.6 D EFINITION : The spectral norm† of an element x ∈ B , denoted kxksp , is supM ∈M |ˆ x(M )|. The spectral norm can be computed from the B norm by: kxksp = limn→∞ kxn k1/n .

Lemma.

P ROOF : The claim is that the limit on the right exists and is equal to kxksp . This follows from the two inequalities: (a) (b) † The

kxksp ≤ lim infkxn k1/n ; kxksp ≥ lim supkxn k1/n .

origin of the term is in the fact that the set of complex numbers λ for which x − λ is not invertible (assuming 1 ∈ B ) is commonly called "the spectrum of x" and the spectral norm of x is defined as sup|λ| for λ in the spectrum of x. By Corollary 2.8, the spectrum of x coincides with the range of x ˆ(M ), which justifies our definition; we prefer to avoid using the much abused word "spectrum" in any sense other than that of chapter VI.

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We notice first that for every n, kxknsp = kxn ksp ≤ kxn k, that is, kxksp ≤ kxn k1/n , which proves (a). In the proof of (b) we assume first that 1 ∈ B , and consider (x − λ)−1 as a function of λ. By Lemma 2.3, (x − λ)−1 is holomorphic for |λ| > kxksp . If |λ| > kxk we have X (x − λ)−1 = −λ−1 (1 − x/λ)−1 = −λ xn λ−n ; and if F is any linear functional on B , h(x − λ)−1 , F i = −λ

X

hxn , F iλ−n

is holomorphic, hence convergent for |λ| > kxksp . Pick any λ > kxksp then hλ−n xn , F i is bounded (in fact, it tends to zero) for all F ∈ B ∗ , and it follows from the Uniform Boundedness Principle that λ−n kxn k is bounded, hence lim supkxn k1/n < λ. Since λ is any number of modulus greater than kxksp , (b) follows. If 1 6∈ B we may adjoin a unit formally. Both the norm and the spectral norm of an element x ∈ B are the same in the extended algebra and since (b) is valid in that algebra, the proof is complete. J Corollary. x ∈ Rad (B) ⇐⇒ limkxn k1/n = 0

P ROOF : x ∈ Rad (B) ⇐⇒ kxksp = 0.

J

3.7 Lemma 3.6 allows a simple characterization of the Banach algebras for which the spectral norm is equivalent to the original norm. Such an algebra is clearly semisimple, and the Gelfand representation identifies it with a (uniformly) closed subalgebra of the algebra of all continuous functions on its maximal ideal space. Theorem. A necessary and sufficient condition for the equivalence of k ksp and the original norm k k of a Banach algebra B is the existence of a constant K such that kxk2 ≤ Kkx2 k for all x ∈ B .

P ROOF : If k k ≤ K1 k ksp , then kxk2 ε and identically zero for |z| < ε/2 (where ε = 21 inf|ˆ x(M )|), we see that Fd (x) = 1 on M. If we assume that B is semisimple it follows that F (x) is a unit element in B . The assumption that xˆ is bounded away from zero for some xeB implies directly that M is compact (see the proof of 3.1); if we assume, on the other hand, that M is compact, then 0 is not a limit point of M in U ∗ and consequently there exists a neighborhood of zero in U ∗ , disjoint from M. By the definition of the weak-star topology this is equivalent to the existence of a finite number of elements x1 , . . . , xn in B such that |ˆ x1 |+· · ·+|ˆ xn | is bounded away from zero on M. Operating with functions of several complex variables one can prove again that 1 ∈ B . We refer the reader to ([5], §13) for a discussion of operation by functions of several complex variables on elements in B and state the following theorem without proof: Theorem. If B is semisimple and M is weak-star compact, then 1 ∈ B.

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3.12 Another very important theorem which is proved by application of holomorphic functions of several complex variables, deals with the existence of idempotents: Theorem. Assume that M is disconnected and let U ⊂ M be both open and compact in the weak-star topology. Then there exists an element u ∈ B such that (a) u2 = u (b) u ˆ(M ) =

(

0

M 6∈ U

1

M ∈U

Remarks: (i) If B is semisimple then (a) is a consequence of (b). (ii) The idempotent u allows a decomposition of B into a direct sum of ideals. Write I1 = uB = {x ∈ B : x = ux} and I2 = {x ∈ B : ux = 0}; it is clear that I1 and I2 are disjoint closed ideals and for any x ∈ B , x = ux + (x − ux); thus B = I1 ⊕ I2 . We refer to ([5], §14) for a proof; here we only point out that if we know that B is uniformly dense in C(M) (resp. C0 (M)), and in particular if B is self-adjoint on M, theorem 3.12 follows from 3.9. In fact, there exists an element x ∈ B such that |ˆ x(M ) − 1| < 14 for M ∈ U 1 and |ˆ x(M )| < 4 for M ∈ M \ U . Defining F (z) as 1 for |z − 1| < 41 and 0 for |z| < 14 , we obtain u as F (x). EXERCISES FOR SECTION 3 1. Show that the distance between any two points of T, considered as the maximal ideal space of C(T), in the metric induced by the dual (in this case M (T)) is equal to 2; hence the norm topology is discrete. 2. We have seen that the maximal ideal space of HC(D) is D = D ∪ T = {z : |z| ≤ 1}. Show that the norm topology on D (i.e., the topology induced by the metric of the dual space on the set of multiplicative linear functionals) coincides with the topology of the complex plane on D and with the discrete topology on T. Hint: Schwarz’ lemma. 3. Show that the relation kw1 − w2 k < 2 is an equivalence relation in the space of maximal ideals (multiplicative linear functionals) of a sup-normed Banach algebra B . The corresponding equivalence classes are called the "Gleason parts" of the maximal ideal space. 4. Let B be an arbitrary Banach algebra and let B1 be a Banach algebra ˜ the orthogonal with trivial multiplication (example 9 of section 1). Denote by B

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direct sum of B and B1 , that is, the set of all pairs (x, y) with x ∈ B, y ∈ B1 with the following operations: (x, y) + (x1 , y1 ) = (x + x1 , y + y1 ) λ(x, y) = (λx, λy)

for complex λ

(x, y)(x1 , y1 ) = (xx1 , 0)

and the norm k(x, y)k = kxkB + kykB1 . Show that B is a Banach algebra without unit and with the same maximal ideal space as B . 5. Let X be a compact (locally compact) Hausdorff space. Show that the maximal ideal space of C(X) (resp. C0 (X)), with the weak-star topology, coincides with X as a topological space. 6. We recall that a set {x1 , . . . , xn } ⊂ B is a set of generators in B if it is contained in no proper closed subalgebra of B , or, equivalently, if the algebra of polynomials in x1 , . . . , xn is dense in B . Show that if {x1 , . . . , xn } is a set of generators in B , then the mapping M 7→ (ˆ x1 (M ), . . . , x ˆm (M )) identifies M, as a topological space, with a bounded subset of C n . 7. Let I be a closed ideal in B . Denote by h(I)–the hull of I –the set of all regular maximal ideals containing I. Show that the maximal ideal space of B/I can be canonically identified with h(I). 8. Let I1 , I2 be (nontrivial) closed ideals in an algebra B such that 1 ∈ B . Assume that B = I1 ⊕ I2 . Show that M is disconnected. 9. Show that not every multiplicative linear functional on M (T) has the form µ 7→ µ ˆ(n) for some integer n. 10. Show that for any LCA group G, L1 (G) and M (G) are semisimple. 11. Let B be a Banach algebra with a unit, realized as a self-adjoint function algebra on a space X . (a) Prove that B is self-adjoint if, and only if, fˆ is real valued on M for every f ∈ B which is real valued on X . (b) Prove that B is self-adjoint if, and only if, 1 + |f |2 is invertible in B for all f ∈ B. 12. Let {wn }n∈Z be a sequence of positive numbers satisfying 1 ≤ wn+m ≤ wn wm for all n, m ∈ Z. Denote§ by A{wn } the subspace of A(T) consisting of the functions f for which kf k{wn } =

∞ X

|fˆ(n)|wn < ∞.

−∞

(a) Show that with the norm so defined, A{wn } is a Banach algebra. (b) Assume that for some k > 0, wn = O |n|k . Show that the maximal ideal space of A{wn } can be identified with T. § Compare

with exercise V.2.7.

228


(c) Under the assumption of (b), show that A{wn } is self-adjoint if, and only if, −1 = O (1) as |n| → ∞. wn w−n 13. Let B be a Banach algebra with norm k k0 and maximal ideal space M. Let B1 ⊂ B be a dense subalgebra which is itself a Banach algebra under a norm k k1 . Assume that its maximal ideal space is again M (this means that every multiplicative linear functional on B1 is continuous there with respect to k k0 ). Assume that k k0 and k k1 are consistent on B1 and denote by k kα , 0 < α < 1, the interpolating norms (see IV.I.2 and exercise 1.4) and by Bα the completion of B1 with respect to the norm k kα . Show that the maximal ideal space of Bα is again M. Remark: If B is semisimple the norm k k0 is majorized by k k1 , (see section 4) and hence by k kα for all 0 < α < 1. 14. Let B1 ⊂ B 0 ⊂ B be Banach algebras with norms k k1 , k k0 and k k respectively. Assume that B1 is dense in B and in B 0 in their respective norms and that B and B1 have the same maximal ideal space M. Show that the maximal ideal space of B 0 is again M. Remark: The assumption that B1 is dense in B 0 is essential; see exercise 11 of Section 9. 4 HOMOMORPHISMS OF BANACH ALGEBRAS

4.1 We have seen (Lemma 2.8) that homomorphisms of any Banach algebra into the field C are always continuous. The Gelfand representation enables us to extend this result: Theorem. Let B be a semisimple Banach algebra, let B1 be any Banach algebra and let ϕ be a homomorphism of B1 into B . Then ϕ is continuous.

P ROOF : We use the closed graph theorem and prove the continuity of ϕ by showing that its graph is closed. Let xj ∈ B1 and assume that xj → x0 in B1 and ϕxj → y0 in B . Let M be any maximal ideal in c B ; the map x 7→ ϕx(M ) is a multiplicative linear functional on B1 , and dj (M ) converges to by Lemma 2.8 it is continuous. It follows that ϕx d0 (M ); on the other hand, since ϕxj → y0 in B , ϕx dj (M ) → yb0 (M ), ϕx d0 (M ) = yb0 (M ). Hence ϕx0 − y0 ∈ M for all maximal ideals so that ϕx M in B and, by the assumption that B is semisimple, ϕx0 = y0 . Thus the graph of ϕ is closed and ϕ is continuous. J Corollary. There exists at most one norm, up to equivalence, with which a semisimple algebra can be a Banach algebra.

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4.2 Let B and B1 be Banach algebras with maximal ideal spaces M and M1 , respectively. Let ϕ be a homomorphism of B1 into B . As we have seen in the course of the preceding proof, every M ∈ M defines c a multiplicative linear functional w(x) = ϕx(M ) on B1 . We denote the corresponding maximal ideal by ϕM . It may happen, of course, that w is identically zero and the "corresponding maximal ideal" is then the entire B1 ; thus ϕ is a map from M into M1 ∪ {B1 }. In terms of linear functionals, ϕ is clearly the restriction to M of the adjoint of ϕ. Theorem. ϕ : M 7→ M1 ∪ {B1 } is continuous when both spaces are endowed with the weak-star topology. If ϕ(B1 ) is dense in B in the spectral norm, then ϕ is a homeomorphism of M onto a closed subset of M1 .

P ROOF : A sub-basis for the weak-star topology on M1 ∪ {B1 } is the collection of sets of the form U1 = {M : x b1 (M1 ) ∈ O}, O an open set in d1 (M ) ∈ O} C and x1 ∈ B 1 . The ϕ pre-image in M of U1 is U = {M : ϕx \ ˆ which is clearly open. If ϕ(B1 ) is uniformly dense in B , the functions d1 (M ), x1 ∈ B1 , determine the weak-star topology on M and it is ϕx obvious that ϕ is one-to-one into M1 and that it is a homeomorphism. \ ˆ What remains to show† is that if ϕ(B 1 ) is uniformly dense in B then ϕ(M) is closed in M1 . We start with two remarks: (a) For M1 ∈ M1 the map ϕx1 7→ x b1 (M1 ) is well defined on ϕ(B1 ) if, and only if, for all x1 ∈ B1 , ϕx1 = 0 implies x b1 (M1 ) = 0. (b) When the above-mentioned map is defined, it is clearly multi\ ˆ plicative and (assuming ϕ(B 1 ) uniformly dense in B ) it can be extended to a multiplicative linear functional on B if, and only if, for all x1 ∈ B1 : |b x1 (M1 )| ≤ kϕx1 ksp .

Assume now that M1 ∈ M1 is in the weak-star closure of ϕ(M). For any x1 ∈ B1 and ε > 0 there exists an M ∈ M such that d1 (M )| < ε. |b x1 (M1 ) − x b1 (ϕM )| = |b x1 (M1 ) − ϕx

d1 (M )| ≤ kϕx1 ksp , it follows that Since ε > 0 is arbitrary and since |ϕx |b x1 (M1 )| ≤ kϕx1 ksp , and by our remark (b) there exists an M0 ∈ M d1 (M0 ) = x such that ϕx b1 (M1 ) for all x1 ∈ B1 ; that is, ϕM0 = M1 and M1 ∈ ϕ(M). J † Notice that if 1 ∈ B then M is compact and ϕ(M) is therefore compact, so that in this case the rest of the proof is superfluous.

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4.3 From 4.2 it follows in particular that if ϕ is an automorphism of B , then ϕ is a homeomorphism of M, and if B is semisimple (so that we can identify it with its Gelfand representation), then ϕ is given by c ϕx(M )=x ˆ(ϕM ). In other words: Every automorphism, of a semisimple Banach algebra is given by a (homeomorphic) "change of variable" on the maximal ideal space. Of course not every homeomorphism of M defines an automorphism of Bˆ (or B ), and the question which homeomorphisms do, is equivalent to the characterization of all the automorphisms of B and can be quite difficult. ?4.4 The following lemma is sometimes helpful in determining the automorphisms of Banach algebras of functions on the line.

Lemma. Let ϕ be a continuous function defined on an interval [a, b] and having the following property: If r0 , r1 , . . . , rN are real numbers such that all the 2N points

(4.1)

ηα = r0 +

N X

εj rj ,

εj = 0, 1

1

lie in [a, b], then the numbers {ϕ(ηj )} are linearly dependent over the rationals. Then the set of points in a neighborhood of which ϕ is a polynomial of degree smaller than N is everywhere dense in [a, b].

P ROOF : Let I be any interval contained in [a, b]; we show that there exists an interval I 0 ⊂ I such that ϕ coincides on I 0 with some polynomial of degree smaller than N . Without loss of generality we may assume that I ⊃ [0, N + 1] so that if 0 ≤ rj ≤ 1, j = 0, 1, . . . , N , all the points ηα defined by (4.1) are contained in I . By the assumption of the lemma, to each choice of (r0 , . . . , rN ) such that 0 ≤ rj < 1, corresponds at least one vector (A1 , . . . , A2N ) with integral entries not all of which vanish, such that N

(4.2)

2 X

Aα ϕ(ηα ) = 0.

α=1

Denote by E(A1 , . . . , A2N ) the set of points (r0 , . . . , rN ) in the N + 1dimensional cube 0 ≤ rj ≤ 1, for which (4.2) is valid. Since ϕ is continuous it follows that E(A1 , . . . .A2N ) is closed for every (A1 , . . . , A2N )

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S and since E(A1 , . . . , A2N ) is the entire cube 0 ≤ rj ≤ 1, it follows from Baire’s category theorem that some E(A1 , . . . , A2N ) contains a box of the form {rj0 ≤ rj ≤ rj1 }, j = 0, . . . , N . Let ε > 0 be smaller than 12 (r01 − r00 ) and letR ψε be infinitely differentiable function carried by (−ε, ε) such that ψα (ξ)dξ = 1. We put ϕε = ϕ ∗ ψε and notice that N

2 X

N

Aα ϕε (ηα ) = ψε

α=1

2 X

α=1

Aα ϕ(ηα )

and consequently N

(4.3)

2 X

Aα ϕε (ηα ) = 0

α=1

for (4.4)

r00 + ε ≤ r0 ≤ r01 − ε,

rj0 ≤ rj ≤ rj1 ,

j = 1, . . . , N.

Now, ϕε is infinitely differentiable and we can differentiate (4.3) with respect to various rj ’s, j > 1. Assume that Aα0 6= 0 and that the coefficient of rj0 in ηα is equal to one. Differentiating (4.3) with respect to rj0 we obtain X (4.5) Aα ϕ0ε (ηα ) = 0 where the summation extends now only over those values of α such that the coefficient of rj0 in ηα is equal to one. Also, (4.5) is nontrivial since it contains the term Aα0 ϕ0ε (ηα0 ). Repeating this argument with ) other rj ’s we finally obtain a nontrivial relation Aα ϕ(M (ηα ) = 0, that ε (M ) is ϕε (ηα ) = 0, with M < N, and it follows that on the range of ηα corresponding to (4.4), say I 0 , ϕε is a polynomial of degree smaller than M − 1 < N . As ε → 0, ϕε → ϕ and ϕ is a polynomial of degree smaller than N on I 0 . J Corollary. If ϕ, as above, is N -times continuously differentiable on [a, b], then it is a polynomial of degree smaller than N on [a, b]. ?4.5 Theorem (Beurling-Helson). Let ϕ be an automorphism of ˆ and let ϕ be the corresponding change of variable on R ˆ (see 4.3). A(R) ˆ Then ϕ(ξ) = aξ + b with a, b ∈ R and a 6= 0.

232


P ROOF : The proof is done in two steps. First we show that ϕ is linear ˆ , and then that the fact that ϕ is linear on some on some interval on R ˆ. interval implies that it is linear on R ˆ . Let First step: By 4.1, ϕ is a continuous linear operator on A(R) N 4 N be an even integer such that 2 > kϕk ; we claim that ϕ satisfies the condition of Lemma 4.4 for this value of N . If we show this, it follows ˆ . ϕ maps I 0 from 4.4 that ϕ is a polynomial on some interval I 0 ⊂ R onto some interval I0 and, since ϕ is an automorphism we can repeat the same argument for ϕ−1 and obtain that the inverse function of ϕ is a polynomial on some interval I00 ⊂ I0 . Since a polynomial whose inverse function (on some interval) is again a polynomial must be linear it follows that ϕ is linear on I 0 . The adjoint ϕ∗ of ϕ maps the unit measure concentrated at ξ to the unit measure concentrated at ϕ(ξ). Since kϕ∗ k = kϕk we obtain that ˆ for every choice of aj ∈ C and ξj ∈ R

X

X

(4.6) aj δϕ(ξ ) ∞ ≤ kϕk aj δ ξ j ∞ j

FL

FL

We remember also that by Kronecker’s theorem (VI.9.2) if {ϕ(ξj )} are P P linearly independent over the rationals, then k aj δϕ(ξj ) kF L∞ = |aj |. ˆ , if the 2N We show now that for every choice of r0 , r1 , . . . , rN ∈ R points ηα given by (4.1) are all distinct, there exists a measure ν carried −N/4 by {ηα } such that kνkM (R) . ˆ = 1 and kνkF L∞ ≤ 2 Put (4.7)

µj =

1 (δ0 + δr2j−1 + δr2j − δ(r2j−1 +r2j ) ) 4

j = 1, . . . , N/2.

The total mass of µj is clearly 1 and (4.8) 1 µ cj (x) = (1 + eir2j−1 x + eir2j x − ei(r2j−1 +r2j )x ) = 4 1 i = eir2j−1 x/2 cos r2j−1 x/2 + e−i(r2j−1 +r2j /2)x sin r2r−1 x/2 2 2

so that (4.9)

|ˆ µ(x)| ≤

1 1 (|cos r2j−1 x/2| + |sin r2j−1 x/2|) ≤ 2− 2 . 2

We now take ν = δr0 ∗ µ1 ∗ · · · ∗ µN/2 . ν is clearly carried by {ηj } and if

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the η ’s are all distinct the total mass of ν is 1. On the other hand N/2

kνkF L∞ ≤

Y

kµj kF L∞ ≤ 2−N/4 .

1

If {ηj } are linearly independent over the rationals it follows first that the η ’s are all distinct, and by (4.6) applied to ν , and Kronecker’s theorem 1 ≤ kϕk2−N/4 which contradicts the assumption 2N > kϕk4 . cλ k ˆ ≤ 3; hence Second step: For all n and λ, keinξ V A(R) cλ (ϕ(ξ))k ˆ ≤ 3kϕk. keinϕ(ξ) V A(R)

cλ (ϕ(ξ)) becomes 1 on larger and larger intervals on R ˆ, As λ → ∞, V ˆ . By VI.2.4 (or by taking which eventually cover any finite interval on R weak limits), einϕ(ξ) is a Fourier-Stieltjes transform of a measure of total mass at most 3kϕk. If we denote by µ1 the measure on R such that µ c1 (ξ) = eiϕ(ξ) , it follows that einϕ(ξ) is the Fourier-Stieltjes transform of µ∗n (n times) 1 = µ1 ∗ · · · ∗ µ1

and we have (4.10)

kµ∗n 1 kM (R) ≤ 3kϕk.

By the first step we know that ϕ(ξ) = aξ + b on some interval I on ˆ . We now consider the measure µ obtained from µ1 by multiplying it R by e−ib and translating it by a. We have µ ˆ(ξ) = e−i(aξ+b) µ c1 (ξ), that is µ ˆ(ξ) = 1 on I (and |ˆ µ(ξ)| = 1 everywhere). It follows from (4.10) that (4.11)

kµ∗n kM (R) ≤ 3kϕk.

Consider the measures νn = 2−n (δ + µ)∗n , (δ = δ0 being the unit mass at ξ = 0). We have n X n ∗j νn = 2−n µ j 0 n 1+ˆ µ(ξ) and consequently kνn k ≤ 3kϕk; also, νc which is equal n (ξ) = 2 to 1 if µ ˆ(ξ) = 1 and tends to zero if µ ˆ(ξ) 6= 1. Taking a weak limit of νn as n → ∞ we obtain a measure ν such that νˆ is equal almost everywhere to the indicator function of the set {ξ : µ ˆ(ξ) = 1} which clearly implies ˆ , hence µ ˆ and νˆ(ξ) = 1 identically on R ˆ(ξ) = 1 almost everywhere on R iϕ(ξ) since µ ˆ is continuous, µ ˆ(ξ) = 1 everywhere. It follows that e ) = i(aξ+b) ˆ e everywhere on R, and ϕ(ξ) = aξ + b. J

234


Remarks: (a) The first step of the proof applies to a large class of algebras. For instance, if B is an algebra whose maximal ideal space is ˆ , B ⊃ A(R) ˆ and for each constant K there exists an integer N such that R every set {ηα } defined by (4.1) (such that the η ’s are all distinct) carries a measure ν such that kνkB ∗ < K −1 kνkM (R) , step 1 goes verbatim for B. (b) If we assume that ϕ is continuously differentiable, step 2 is superfluous. In fact, step 1 shows that the set of points near which ϕ is linear is everywhere dense and if ϕ0 exists and is continuous, the slope must be always the same and ϕ must be linear. This proves that for the algebras discussed in remark (a), the continuously differentiable changes of variable induced by an automorphism must be linear. ˆ (c) We have used the fact that ϕ was an automorphism of A(R) rather than an endomorphism once, when deducing in the first step that ϕ was linear in some interval from the fact that it is a polynomial (on an interval) whose inverse function is also a polynomial. This part of the argument can be replaced (see exercise 12 at the end of this section), and we thereby obtain that every nontrivial endomorphism of ˆ is given by a linear change of variable (and consequently is an A(R) automorphism). EXERCISES FOR SECTION 4 1. Let B be a semisimple Banach algebra with norm k k and B1 ⊂ B a subalgebra of B which is a Banach algebra with a norm k k1 . Show that there exists a constant C such that kxk ≤ Ckxk1 for all x ∈ B1 . 2. Let B be a Banach algebra of infinitely differentiable functions on [0, 1], having [0, 1] as its space of maximal ideals. Show that there exists a sequence (n) {Mn }∞ (x)| ≤ Mn kf k for every f ∈ B . n=0 such that sup|f 3. Show that the space of all infinitely differentiable functions on [0, 1] cannot be normed so as to become a Banach algebra. 4. Let B be a semisimple Banach algebra with maximal ideal space M. Prove that a homeomorphism ψ ˆ ⇒ fˆ ◦ ψ ∈ B ˆ, of M is induced by an endomorphism of B if, and only if fˆ ∈ B where (fˆ ◦ ψ)(M ) = fˆ(ψ(M )). 5. What condition on ψ above is equivalent to its being induced by an automorphism? 6. Construct examples of semisimple Banach algebras B and B1 and a d1 is not dense in Bˆ ) and such that homomorphism ϕ : B1 7→ B (such that ϕB the corresponding mapping ϕ (a) is not one-to-one; (b) is one-to-one but not a homeomorphism; (c) maps M onto a dense proper subset of M1 . 7. Show that a homeomorphism ϕ of T onto itself is induced by an automorphism of A(T) if, and only if, einϕ ∈ A(T) for all n and keinϕ kA(T) = O (1).

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8. (Van der Corput’s lemma): (a) Let ϕ be real-valued on an interval [a, b], and assume that itRhas there a monotone derivative satisfying ϕ0 (ξ) > ρ > 0 on b [a, b]. Show that a eiϕ(ξ) dξ ≤ 2/ρ. (b) Instead of assuming ϕ0 (ξ) > ρ > 0 on [a, b], assume that ϕ is twice differentiable and that ϕ00 > κ > 0 on [a, b]. Show that

b

Z a

1

eiϕ(ξ) dξ ≤ 6κ− 2

Hint: For (a) write eiϕ(ξ) dξ = −i dΦ(ξ)/ϕ0 (ξ), where Φ(ξ) = eiϕ(ξ) . and apply the so-called "second mean-value theorem." For (b), if ϕ0 (c) = 0, write

R

R

b

Z

=

−1 2

c−κ

Z

a

+

a

Z

−1 2

c+κ

−1 c−κ 2

+

Z

b −1 2

c+κ

1

the middle integral is clearly bounded by 2κ− 2 ; evaluate the other two integrals by (a). 9. Let ϕ be twice differentiable, real-valued function on [0, 1] and assume that ϕ00 > κ > 0 there. Show that

Z 0

1

1

ξeiϕ(ξ) dξ ≤ 12κ− 2

Hint: Integrate by parts and use exercise 8. 10. Let ϕ be twice differentiable, real-valued function on [−1, 1] and as sume that ϕ00 > η > 0 there. Put Φn (ξ) = 1 − |ξ| einϕ(ξ) for |ξ| ≤ 1 and Φn (ξ) = 0 for |ξ| ≥ 1 Show that for all x ∈ R,

1

2π

iξx

Z

1

−1

1

1

Φn (ξ)eiξx dξ ≤ 4η − 2 n− 2

Hint: Φn (ξ)e = 1 − |ξ| ei(nϕ(ξ)−ξx) . The second derivative of the exponent is ≥ ηn; use exercise 9. √ 11. Show that for some c > 0, kΦn kA(R) ˆ ≥ c n; Φn being the function introduced in exercise 10. Hint: Use exercise 10, Plancherel’s theorem, and the fact that kΦn kL2 (R) ˆ , is independent of n. ˆ is given by a linear 12. Prove that every nontrivial endomorphism of A(R) change of variable. Hint: See remark (c) of 4.5. If ϕ is the change of variable induced by an endomorphism ϕ, ϕ is a polynomial on some interval and if it is not linear, ϕ00 (ξ) > η (or ϕ00 (ξ) < −η ) for some η > 0 on some interval I . A linear change of variable allows the assumption [−1, 1] ⊂ I . As in the second step of the proof cλ (ϕ(ξ))kA(R) c of 4.5, keinϕ(ξ) V ˆ ≤ 3kϕk, hence kΦn (ξ)Vλ (ϕ(ξ))kA(R) ˆ ≤ 3kϕk.

cλ (ϕ(ξ)) = Φn (ξ) and by exercise 11, kΦn kA(R) For λ sufficiently large Φn (ξ)V ˆ tends to infinity with n, which gives the desired contradiction.

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5 REGULAR ALGEBRAS

5.1 D EFINITION : A function algebra B on a compact Hausdorff space X is regular if, given a point p ∈ X and a compact set K ⊂ X such that p 6∈ K , there exists a function f ∈ B such that f (p) = 1 and f vanishes on K . The algebra B is normal if, given two disjoint compact sets K1 , K2 in X , there exists f ∈ B such that f = 0 on K1 and f = 1 on K2 . Examples: (a) Let X be a compact Hausdorff space. Then C(X) is normal. This is essentially the contents of Urysohn’s lemma (see [15], p. 6). (b) HC(D), the algebra of functions holomorphic inside the unit disc and continuous on the boundary, is not regular. Theorem (partition of unity). Let X be a compact Hausdorff space and B a normal function algebra on X , containing the identity. Let {Uj }nj=1 , be an open covering of X . Then there exist functions ϕj , j = 1, . . . , n, in B satisfying   support of ϕ ⊂ U j j (5.1)  P ϕ = 1. j

P ROOF : We use induction on n. Assume n = 2. Let V1 , V2 be open sets satisfying Vj ⊂ Uj and V1 ∪ V2 = X . There exists a function f ∈ B such that f = 0 on the complement of V1 and f = 1 on the complement of V2 . Put ϕ1 = f , ϕ2 = 1 − f . Assume now that the statement of the theorem is valid for coverings by fewer than n open sets and let U1 , . . . , Un be an open covering of X . Put U 0 = Un−1 ∪ Un and apply the induction hypothesis to the covering U1 , . . . , Un−2 , U 0 thereby obtaining functions ϕ1 , . . . , ϕn−2 , ϕ0 in B, satisfying (5.1). Denote the support of ϕ0 by S and let Vn−1 , Vn be open sets such that Vj ⊂ Uj (j = n − 1, n) and Vn−1 ∪ Vn ⊃ S . Let f ∈ B such that f = 0 on S \ Vn−1 and f = 1 on S \ Vn . Put ϕn−1 = ϕ0 f and ϕn = ϕ0 (1 − f ). The functions ϕ1 , . . . , ϕn satisfy (5.1). J Remark: The family {ϕj } satisfying (5.1) is called a partition of unity in B , subordinate to {Uj }. Partitions of unity are the main tool in transition from "local" properties to "global" ones. A typical and very important illustration is Theorem 5.2 below.

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5.2 Let F be a family of functions on a topological space X . A function f is said to belong to F locally at a point p ∈ X , if there exists a neighborhood U of p and a function g ∈ F such that f = g in U . If f belongs to F locally at every p ∈ X , we say that f is locally in F . Theorem. Let X be a compact Hausdorff space and F a normal algebra of functions on X . If a function f belongs to F locally, then f ∈ F.

For every p ∈ X let U be an open neighborhood of p and g ∈ F such that g = f in U . Since X is compact, we can pick a finite cover of X , {Uj }nj=1 , among the above mentioned neighborhoods. Denote the corresponding elements of F by gj ; that is, gj = f in Uj . Let {ϕj } be a partition of unity in F subordinate to {Uj }. Then X X (5.2) f= ϕj f = ϕj gj ∈ F. J PROOF :

Remark: It is clear from (5.2) that if J ⊂ F is an ideal, and if f belongs to J locally (i.e., gj ∈ J ), then f ∈ J . 5.3 We consider a semisimple Banach algebra B with a unit, and denote its maximal ideal space by M

D EFINITION : The hull, h(I), of an ideal I in B , is the set of all M ∈ M such that I ⊂ M . Equivalently: h(I) is the set of all common zeros of x ˆ(M ) for x ∈ I . Since the set of common zeros of any family of continuous functions is closed, h(I) is always closed in M. D EFINITION : The kernel, k(E), of a set E ⊂ M, is the ideal ∩M ∈E M . Equivalently: k(E) is the set of all x ∈ B such that xˆ(M ) = 0 on E . k(E) is always a closed ideal in B . 5.4 If E ⊂ M, then h(k(E)) is a closed set in M that clearly contains E . One can show (see [15], p. 60) that the hull-kernel operation is a proper closure operation defining a topology on M. Since h(k(E)) is closed in M, the hull-kernel topology is not finer than the weak-star topology. The two coincide if for every closed set E ⊂ M we have E = h(k(E)) which means that if M0 6∈ E there exists an element x ∈ k(E) such that x ˆ(M0 ) 6= 0. Remembering that x ∈ k(E) means x ˆ(M ) = 0 on E , we see that the hull-kernel topology coincides with the weak-star topology on M if, and only if Bˆ is a regular function algebra. In this case we say that B is regular.

238


D EFINITION : A semisimple Banach algebra B is regular (resp. normal) if Bˆ is regular (resp. normal) on M. 5.5 Theorem. Let B be a regular Banach algebra and E a closed subset of M. Then the maximal ideal space of B/k(E) can be identified with E .

P ROOF : The maximal ideals in B/k(E) are the canonical images of maximal ideals in B which contain k(E), that is, which belong to h(k(E)) = E . This identifies M(B/k(E)) and E as sets and we claim that they can be identified as topological spaces. We notice that the Gelfand representation of B/k(E) is simply the restriction of Bˆ to E. A typical open set in a sub-base for the topology of M(B/k(E)) has the form U = {M : M ∈ E, xc E (M ) ∈ O}, O an open set in the complex plane, x ∈ B , and xE = x mod k(E). A typical open set in a sub-base for the topology of M has the form U 0 = {M : x c1 (M ) ∈ O0 } with O0 open in C and x1 ∈ B . If O = O0 and x = x1 then U = E ∩ U 0 and the topology on M(B/k(E)) is precisely the topology induced by M. J

5.6 Theorem. Let B be a regular Banach algebra, I an ideal in B , E a closed set in M such that E ∩ h(I) = ∅. Then there exists an element x ∈ I such that x ˆ(M ) = 1 on E .

P ROOF : The ideal generated by I and k(E) is contained in no maximal ideal since M ⊃ (I, k(E)) implies M ⊃ I and M ⊃ k(E), that is, M ∈ E ∩ h(I). It follows that the image of I in B/k(E) is the entire algebra and consequently there exists an element x ∈ I such that x ≡ 1 ˆ(M ) = 1 on E . J mod k(E), which is the same as saying x Corollary. A regular Banach algebra is normal.

P ROOF : If E1 and E2 are disjoint closed sets in M, apply the theorem to I = k(E1 ), and E = E2 . J 5.7 We turn now to some general facts about the relationship between ideals in regular Banach algebras and their hulls. Theorem. Let I be an ideal in a regular Banach algebra B and x ∈ B . Then xˆ belongs to I locally at every interior point of h(x) and at every point M 6∈ h(I).

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P ROOF : We write h(x) for h((x)), that is, the set of zeros of xˆ in M. If M is an interior point of h(x), x ˆ = 0 in a neighborhood of M and 0 ∈ I . If M 6∈ h(I), M has a compact neighborhood E disjoint from h(I). By Theorem 5.6 there exists an element y ∈ I such that yˆ(M ) = 1 on E . Now xˆ = xˆyˆ on E and xy ∈ l. J Corollary. Let I be an ideal in a regular semisimple Banach algebra B and x ∈ B . If the support of x ˆ is disjoint from h(I), then x ∈ I .

P ROOF : By Theorem 5.7, xˆ belongs to I locally at every point, and by Corollary 5.6 and the remark following Theorem 5.2 it follows that x ˆ ∈ Iˆ hence x ∈ l. J 5.8 Let E be a closed subset of M. The set I0 (E) of all x ∈ B such that xˆ(M ) vanishes on a neighborhood of E is clearly an ideal and if B is regular, h(I0 (E)) = E . It follows from Corollary 5.7 that I0 (E) is contained in every ideal I such that h(I) = E . In other words: I0 (E) is the smallest ideal satisfying h(I) = E , and I0 (E) is the smallest closed ideal satisfying h(I) = E . On the other hand, k(E) is clearly the largest ideal satisfying h(I) = E .

D EFINITION : A primary ideal in a commutative Banach algebra is an ideal contained in only one maximal ideal. In other words, an ideal is primary if its hull consists of a single point. If B is a semisimple regular Banach algebra, every maximal ideal M ⊂ B contains a smallest primary ideal, namely I0 ({M }). We simplify the notation and write I0 (M ) instead of I0 ({M }). The closure, I0 (M ), is clearly the smallest closed primary ideal contained in M . In some cases I0 (M ) = M and we say then that M contains no nontrivial closed primary ideals. Such is the case if B = C(T) (trivial) and also if B = A(T) (Theorem VI.4.11’). On the other hand, if B = C n (T) with n ≥ 1, the maximal ideal {f : f (t0 ) = 0} contains the nontrivial closed primary ideal {f : f (t0 ) = f 0 (t0 ) = 0}. 5.9 D EFINITION : A semisimple Banach algebra B satisfies condition (D) at M ∈ M if, for any x ∈ M there exists a sequence {xn } ⊂ I0 (M ) such that xxn → x in B . We say that B satisfies the condition (D) if B satisfies (D) at every M ∈ M. If B satisfies condition (D) at M ∈ M, M contains no nontrivial closed primary ideal since I0 (M ) is dense in M . It is not known if

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the condition that M contains no nontrivial closed primary ideals is sufficient to imply (D) or not; however, if we know that there exists a constant K such that for every neighborhood U of M there exists y ∈ B such that kyk ≤ K , yˆ has its support in U and yˆ = 1 in some (smaller) neighborhood of M , then we can deduce (D) from I0 (M ) = M . For x ∈ M let zn ∈ I0 (M ) such that zn → x. Let Un be a neighborhood of M such that zn = 0 on Un , and let yn ∈ B such that kyn k ≤ K , yn = 0 outside Un and yn = 1 near M . Put xn = 1 − yn . We have xn ∈ I0 (M ), x − xxn = xyn = (x − zn )yn (since zn yn = 0), and k(x − zn )yn k ≤ Kkx − zn k → 0. J 5.10 Lemma. Let B be a regular semisimple Banach algebra satisfying condition (D) at M0 ∈ M. Let I be a closed ideal in B and x ∈ M0 . Assume that there exists a neighborhood U of M0 such that x ∈ l locally at every M ∈ U \ {M0 }. Then x ∈ l locally at M0 .

P ROOF : Let y ∈ B be such that the support of yˆ is included in U and yˆ = 1 in some neighborhood V of M0 . yx belongs to I locally at every M 6= M0 and yxxn belongs locally to I everywhere ({xn } being the sequence given by (D); remember that xˆn = 0 near M0 ); hence yxxn ∈ I and since xxn → x and I is closed, yx ∈ I . But yc x=x ˆ in V and the lemma follows. J

Theorem (Ditkin-Shilov). Let B be a semisimple regular Banach algebra satisfying (D). Let I be a closed ideal in B and x ∈ k(h(I)) such that the intersection of the boundary of h(x) with h(I) contains no nontrivial perfect sets. Then x ∈ I .

P ROOF : Denote by N the set of M ∈ M such that x does not belong to I locally at M . By Theorem 5.7, N ⊂ (bdry(h)(x)) ∩ h(I) and by the lemma, N has no isolated points; hence N is perfect and since (bdry(h)(x))∩h(I) contains no nontrivial perfects sets, N = ∅ and x ∈ I . J

Corollary. Under the same assumptions on B ; if E ⊂ M is compact and its boundary contains no nontrivial perfect subsets, then I0 (E) = k(E).

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5.11 We have been dealing so far with algebras with a unit element. The definitions and most of the results can be extended to algebras without a unit element simply by identifying the algebra B as a maximal ideal in B ⊕ C. Instead of M we consider its one point compactification M and we say that B is regular on M if B ⊕ C is regular on M. This is equivalent to adding to the regularity condition the following regularity at infinity: given M ∈ M, there exists xˆ ∈ Bˆ such that x ˆ(M ) = 1 and x ˆ has compact support. Similarly, we have to require in defining "x belongs locally to I " not only x ∈ I locally at every M ∈ M, but also x ∈ I at infinity, that is, the existence of some y ∈ l such that x ˆ = yˆ outside of some compact set. The condition (D) at infinity is: for every x ∈ B there exists a sequence xn ∈ B such that xˆn are compactly supported and xxn → x.

EXERCISES FOR SECTION 5 1. Let B be a semisimple Banach algebra, let x1 , . . . , xn ∈ B be generators for B , and assume that

Z

∞

−∞

logkeiyxj k dy < ∞, 1 + y2

j = 1, . . . , n.

Show that B is regular. 2. Describe the closed primary ideals of C n (T), n a positive integer. 6 WIENER’S GENERAL TAUBERIAN THEOREM

In this section we prove Wiener’s lemma stated in the course of the proof of theorem VI.6.1, and Wiener’s general Tauberian theorem. These results are obtained as more or less immediate consequences of some of the material in the preceding section; it should be kept in mind that Wiener’s work preceded, and to some extent motivated, the study of general Banach algebras. 6.1 We start with the analog of Wiener’s lemma for A(T). Lemma. Let f, f1 ∈ A(T) and assume that f is bounded away from zero on the support of f1 . Then f1 f −1 ∈ A(T).

P ROOF : A(T) is a regular Banach algebra. Denote by I the principal ideal generated by f ; then h(I) = {t : f (t) = 0} is disjoint from the support of f1 . By corollary 5.7, f1 ∈ I , which means f1 = gf for some g ∈ A(T). Thus f1 f −1 ∈ A(T) locally and we apply Theorem 5.2. J

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ˆ is locally the 6.2 We obtain Wiener’s lemma by showing that A(R) same as A(T).

Lemma. Let ε > 0 and let ψ be a continuously differentiable function supported by (−π + ε, π − ε). There exists a constant K depending on ψ such that for all −1 ≤ α ≤ 1, keiαt ψkA(T) ≤ K .

P ROOF : We clearly have keiαt ψkC 1 (T) bounded, and the A(T) norm is majorized by the C 1 (T) norm. J Theorem. Let f be a continuous function carried by (−π + ε, π − ε). Then Z X |fˆ(ξ)|dξ < ∞ ⇐⇒ |fˆ(n)| < ∞.

P ROOF : Let ψ ∈ C 1 be carried by (−π + ε/2, π − ε/2), and ψ(t) = 1 P on (−π + ε, π − ε). Assume that |fˆ(n)| < ∞; then f ∈ A(T); hence f eiαt ψ ∈ A(T) and kf eiαt ψkA(T) ≤ Kkf kA(T) for −1 ≤ α ≤ 1. Now P f eiαt ψ = f eiαt and its A(T) norm is (1/2π) |fˆ(n − α)|. Integrating the inequality X X |fˆ(n − α)| ≤ K |fˆ(n)| on 0 ≤ α ≤ 1, we obtain Z

|fˆ(ξ)|dξ ≤ K

X

|fˆ(n)|.

R PR1 ˆ Conversely, if we assume that |fˆ(ξ)|dξ = |f (n − α)|dα < ∞ 0 it follows from Fubini’s theorem that for almost all α, 0 ≤ α ≤ 1, P ˆ |f (n − α)| < ∞, which means eiαt f ∈ A(T). As in the first part of the P proof this implies eiαt f e−iαt ψ = f ∈ A(T) and |fˆ(n)|. J

Corollary. Identifying T with (−π, π], a function f defined in a neighborhood of t0 ∈ T belongs to A(T) locally at t0 if, and only if, it belongs ˆ at t0 . to A(R) ˆ be such that 6.3 Lemma (Wiener’s lemma). Let f and f1 ∈ A(R) the support of f1 is compact and f is bounded away from zero on it. ˆ . Then f1 = gf with g ∈ A(R)

P ROOF : Without loss of generality we assume that the support of f1 is ˆ , ϕ = 1 on included in (−2, 2). Replacing f by f ϕ, where ϕ ∈ A(R) (−2, 2) and ϕ = 0 outside of (−3, 3), it follows from Lemma 6.1 that ˆ . g = f1 f −1 ∈ A(T); hence, by Theorem 6.2, g ∈ A(R) J

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6.4 Theorem (Wiener’s general Tauberian theorem). Let f ∈ ˆ and assume f (ξ) 6= 0 for all ξ ∈ R ˆ . Then f is contained in no A(R) ˆ proper closed ideal of A(R). ˆ has compact supP ROOF : By Lemma 6.2 it follows that if f1 ∈ A(R) ˆ , that is, f1 belongs to (f ) (the principal ideal port, then f1 /f ∈ A(R) ˆ and the generated by f ). By theorem VI.1.12, (f ) is dense in A(R) proof is complete. J

Instead of considering principal ideals, one may consider any closed ˆ , there exists f ∈ I such that f (ξ) 6= 0, then ideal I . If for every ξ ∈ R ˆ I = A(R). As a corollary we obtain again that all maximal ideals in ˆ have the form {f : f (ξ0 ) = 0} for some ξ0 ∈ R ˆ . We leave the A(R) details to the reader. 6.5 The Tauberian character of Theorem 6.4 may not be obvious at first glance. A Tauberian theorem is a theorem that indicates conditions under which some form of summability implies convergence or, more generally, another form of summability. The first such theorem P n was proved by Tauber and stated that if limx→1−0 ∞ n=0 an x = A and P an = A. Hardy and Littlewood, who introduced an = o(1/n), then the term "Tauberian theorem," improved Tauber’s result by showing that Tauber’s condition an = o(1/n) can be replaced by the weaker an = O(1/n), an improvement that is a great deal deeper and harder than Tauber’s rather elementary result. Wiener’s original statement of Theorem 6.4 was much more clearly Tauberian: Theorem (Wiener’s general Tauberian theorem). Let K1 ∈ L1 (R) ˆ and ˆ and f ∈ L∞ (R). Assume K(ξ) 6= 0 for all ξ ∈ R Z Z (6.1) lim K1 (x − y)f (y)dy = A K1 (x)dx. x→∞

Then (6.2)

lim

x→∞

Z

K2 (x − y)f (y)dy = A

Z

K2 (x)dx.

for all K2 ∈ L1 (R). Remark: If f (x) tends to a limit when x → ∞ then (6.1) is clearly satisfied with A = limx→∞ f (x). (6.1) states that f (x) tends to the limit A in the mean with respect to the kernel K1 ; the theorem states that the existence of the limit with respect to the mean K1 implies that of

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ˆ 1 never vanishes. the limit with respect to any mean K2 , provided K We refer to [27], chapter 3, for examples of derivations of "concrete" Tauberian theorems from theorem 6.5.

P ROOF OF THEOREM 6.5: Denote by I the subset of L1 (R) of functions K2 satisfying (6.2). I is clearly a linear subspace, invariant under translation and closed in the L1 (R) norm, that is, a closed ideal in L1 (R). Since K1 ∈ I , it follows from Theorem 6.4 that I = L1 (R) and the proof is complete. J 7 SPECTRAL SYNTHESIS IN REGULAR ALGEBRAS

Let B be a semisimple regular Banach algebra with a unit† . Denote by M its maximal ideal space and by B ∗ its dual.

7.1 D EFINITION : A functional ν ∈ B ∗ vanishes on an open set O if hx, νi = 0 for every x ∈ B such that the support of x ˆ is contained in O. Lemma. If ν ∈ B ∗ vanishes on the open sets O1 and O2 then ν vanishes on O1 ∪ O2 .

P ROOF : Let x ∈ B and assume that the support of xˆ is contained in O1 ∪ O2 . Denote by O3 the complement in M of the support of x ˆ and let ˆ subordinate to Oj , j = 1, 2, 3. ϕˆj , j = 1, 2, 3 be a partition of unity in B Then x = xϕ1 + xϕ2 and hx, νi = hxϕ1 , νi + hxϕ2 , νi = 0, since xϕ3 = 0 and xϕj has its support in Oj . J From the lemma it follows immediately that if ν ∈ B ∗ vanishes on every set in some finite collection of open sets it vanishes also on their union; and since M is compact the same holds for arbitrary unions. The union of all the open sets on which ν vanishes is the largest set having this property and we define the support, Σ(ν), of ν as the complement of this set (compare with VI.4). 7.2 For M ∈ M we denote by δM the multiplicative linear functional associated with M , hx, δM i = xˆ(M ); thus δM is naturally identifiable with the measure of mass 1 concentrated at M .

D EFINITION : A functional ν ∈ B ∗ admits spectral synthesis if ν belongs to the weak-star closure of the span in B ∗ of {δM }M ∈ Σ(ν) . † The standing assumption 1 ∈ B is introduced for convenience only. It is not essential and the reader is urged to extend the notions and results to the case 1 6∈ B .

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Since the subspace of B orthogonal to the span of {δM }M ∈ Σ(ν) is precisely the set of all x ∈ B such that xˆ(M ) = 0 for all M ∈ Σ(ν), that is, the ideal k(Σ(ν)), we see, using the Hahn-Banach theorem as we did in VI.6, that ν admits spectral synthesis if, and only if, it is orthogonal to k(Σ(ν)). 7.3 It seems natural to define a set of spectral synthesis as a set E having the property that every ν ∈ B ∗ such that Σ(ν) = E admits spectral synthesis. If M is very large, however, there may be sets E which are the support of no ν ∈ B ∗ and we prefer to introduce the following.

D EFINITION : A closed set E ⊂ M is a set of spectral synthesis if every ν ∈ B ∗ such that Σ(ν) ⊂ E is orthogonal to k(E). This condition implies in particular that if Σ(ν) = E then ν admits spectral synthesis. It is clear that the condition Σ(ν) ⊂ E is equivalent to the condition that ν be orthogonal to I0 (E). The condition that E is of spectral synthesis is therefore equivalent to requiring that every ν ∈ B ∗ which is orthogonal to I0 (E) be also orthogonal to k(E). By the Hahn-Banach theorem this means I0 (E) = k(E). Thus: E is of spectral synthesis if and only if I0 (E) is dense in k(E). We restate Corollary 5.10 as: Theorem. Assume that B satisfies (D) and let E ⊂ M be closed and its boundary contain no perfect subsets. Then E is of spectral synthesis. 7.4 In some cases every closed E ⊂ M is of spectral synthesis and we say that spectral synthesis is possible in B . Spectral synthesis is possible if B = C(X), X a compact Hausdorff space. Another class of examples is given by Theorem 7.3: B satisfying (D) with M containing no perfect subsets. In particular, if G is a discrete abelian group and B = A(G) (to which we formally add a unit if we want to remain within our standing assumptions), then (D) is satisfied and M contains no perfect subsets. It follows that for discrete G spectral synthesis holds in A(G). We devote the rest of this section to prove: Theorem (Malliavin). If G is a nondiscrete LCA group then spectral synthesis fails for A(G).

The construction is somewhat simpler technically in the case G = D than in the general case and we do it there. For a nondiscrete LCA group G, a Cantor set E on G is a compact set for which there exists

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P a sequence {rj } ⊂ G such that the finite sums n1 εj rj , εj = 0, 1, are all distinct and form a dense subset of E . The construction we give below can be adapted to show that every Cantor set of G contains a subset which is not of spectral synthesis for A(G). Notice that every nondiscrete LCA group has Cantor subsets. We mention, finally, that for G = Rn with n ≥ 3, any sphere is an example of a set which is not of spectral synthesis; this was shown by L. Schwartz (some eleven years before the general case was settled).

7.5 We state the principle on which our construction depends in the general setting of this section, that is, for a semisimple regular Banach algebra with a unit. For typographical simplicity we identify B and Bˆ

and use the letters f, g and so on, for elements of B . We remind the reader that the dual B ∗ is canonically a B module. Theorem. Let f ∈ B and µ ∈ B ∗ , µ 6= 0, and denote

(7.1)

C(u) = keiuf µkB ∗ .

Assume that for an integer N > 1 Z ∞ C(u)|u|N du < ∞. (7.2) −∞

Then there exists a real value a0 such that f0 = a0 + f has the property that the closed ideals generated by f0n , n = 1, ..., N + 1 are all distinct.

P ROOF : We begin with two remarks. First: There is no loss of generality assuming that h1, µi = 6 0. In fact for some h ∈ B hh, µi = h1, hµi = 6 0 and since keiuf hµkB ∗ ≤ khkB keiuf µkB ∗ ,

(7.2) remains valid if we replace µ by hµ. Second: Write Φ(u) = h1, eiuf µi; then |Φ(u)| ≤ C(u) ∈ L1 (R), ˆ Φ(u) is continuous and Φ(0) = h1, µi 6= 0. It follows that Φ(ξ) is well defined and is not identically zero so that there exists a real number a0 ˆ for which Φ(−a 0 ) 6= 0. This is the a0 we are looking for (as we shall see) and again we may simplify the typography by assuming a0 = 0; we simply replace f by a0 + f and notice that eiu(a0 +f ) = eiua0 eiuf so that keiu(a0 +f ) µkB ∗ = keiuf µkB ∗ . Thus we assume Z ∞ (7.3) h1, eiuf µidu 6= 0 −∞

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For p ≤ N the B ∗ -valued integral Z ∞ (7.4) (iu)p (eiuf µ)du ∆p (f, µ) = −∞

is well defined since the integrand is continuous and, by (7.2), norm integrable. Let q ≥ 0 be an integer, g1 ∈ B and consider Z ∞ q (7.5) hg1 f q , eiuf µi(iu)p du. Ip,q = hg1 f , ∆p (f, µ)i = −∞

Integrating (7.5) by parts we obtain (7.6)

Ip,q = −pIp−1,q−1 Ip,q = 0

if p > 0, q > 0, if p = 0, q > 0.

It follows that if q > p we have Ip,q = 0 no matter what is g1 ∈ B . In other words, ∆p (f, µ) is orthogonal to the ideal generated by f p+1 . Now, using (7.6) with p = q, g1 = 1, we obtain Z ∞ (7.7) h1, eiuf µidu 6= 0 hf p , ∆p (f, µ)i = (−1)p p! −∞

by (7.3). Thus f p does not belong to the closed ideal generated by f p+1 and the proof is complete. J Corollary. The sets f −1 (0) and Σ(µ) ∩ f −1 (0) are not sets of spectral synthesis.

The hull of the ideal generated by f p is f −1 (0). Since we found distinct closed ideals having f −1 (0) as hull, f −1 (0) is not of spectral synthesis. The fact that ∆p (f, µ) is orthogonal to the ideal generated by f p+1 implies (see Corollary 5.7) that Σ(∆p (f, µ)) ⊂ f −1 (0). For g ∈ B we have Z ∞ Z ∞ (7.8) hg, ∆p (f, µ)i = hgeiuf , µi(iu)p du hg, eiuf µi(iu)p du = PROOF :

−∞

−∞

so that if the support of g is disjoint from Σ(µ) then hg, ∆p (f, µ)i = 0. This means that Σ(∆p (f, µ)) ⊂ Σ(µ); hence (7.9)

Σ(∆p (f, µ)) ⊂ Σ(µ) ∩ f −1 (0).

J

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7.6 In the case B = A(D) we show that µ and f can be chosen so that C(u), defined by (7.1), goes to zero faster than any (negative) power of |u|. We can take as µ simply the Haar measure of D and we shall have f quite explicitly too, but before describing it we make a few observations. We identify the elements of D as sequences {εn }, εn = 0, 1, the group operation being addition mod 2. Functions on D are functions of the infinitely many variables εn , n = 1, 2, . . . . Denote by xm the element in D all of whose coordinates except the mth are zero. Denote by Em the subgroup of D generated by x2m−1 and x2m , that is, {0, x2m−1 , x2m , x2m−1 + x2m }. Denote by µm the measure having the mass 1/4 at each of the points of Em . µm is the Haar measure on Em and one checks easily that the convolutions µ1 ∗ · · · ∗ µn converge in the weak-star topology of measures to the Haar measure µ of D. We write Q this formally as µ = ∞ 1 ∗µm . Lemma. Let E1 = {x1 , . . . , xk } and E2 = {y1 , . . . , yl } be finite sets on a group G. Let E = E1 + E2 = {xp + yq , p = 1, . . . , k, q = 1, . . . , l} and assume that E has kl points. Let h1 and h2 be functions on E such that h1 (xp + yq ) = g1 (xp ) and h2 (xp + yq ) = g2 (yq ). Then, if µm is a measure carried by Em , m = 1, 2, h1 h2 (µ1 ∗ µ2 ) = (g1 µ1 ) ∗ (g2 µ2 ).

(7.10)

P ROOF : Both sides of equation (7.10) are carried by E and have the mass g1 (xp )g2 (yq )µ1 ({xp })µ2 ({yq }) at xp + yq . J The lemma can be generalized either by induction or by direct verification to sums of N sets Em . The flaw in notation of denoting by Em first specific sets and then, in the lemma, variable sets (and similarly for µ) is forgivable in view of the fact that we use the lemma precisely for the sets Em , and the measures µm , introduced above. Thus, if hm , m = 1, 2, . . . are functions on D and if hm depends only on the variables ε2m−1 and ε2m , we have (7.11)

N Y

hm µ1 ∗ · · · ∗ µN = (h1 µ1 ) ∗ · · · ∗ (hN µN ).

1

P We shall have f = am ϕm with ϕm ∈ A(D), ϕm depending only on the variables ε2m−1 and ε2m , and the series convergent in the A(D) norm.

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Using (7.11) and taking weak-star limits, we obtain the convenient formula: (7.12)

eiuf µ =

∞ Y

∗(eiuam ϕm µm ).

1

We recall that the norm of a measure in A(D)∗ is the supremum of its Fourier transform, and that the Fourier transform of a convolution is the product of the transforms of the factors; thus we obtain (7.13)

keiuf µkA(D)∗ ≤

∞ Y

keiuam ϕm µm kA(D)∗

1

The functions ϕm are defined by: (7.14)

ϕ(x) = ε2m−1 ε2m

for x = {εj }.

If we denote by ξm the character on D defined by (7.15)

< x, ξm >= (−1)εm

for x = {εj },

then (7.16)

ϕm = 41 (1 + ξ2m−1 ξ2m − ξ2m−1 − ξ2m )

so that ϕm ∈ A(D), and kϕm kA(D) = 1. The Fourier transform of the measure eiαϕm µm , can be computed ˆ then explicitly: if ξ = {ζj } ∈ D Z < x, ξ > eiαϕm (x) dµm (x) = (7.17) = 14 1 + (−1)ζ2m−1 + (−1)ζ2m + eiα (−1)ζ2m−1 +ζ2m , which assumes only the three values: that if |α − π| ≤ π/3 mod 2π , then (7.18)

3+eiα 1−eiα eiα −1 4 , 4 , 4 .

It follows

keiαϕm µm kA(D)∗ ≤ 34 .

Theorem. Denote the Haar measure on D by µ. There exists a realvalued function f ∈ A(D) such that, as |u| → ∞, C(u) = keiuf µkA(D)∗ vanishes faster than any power of |u|.

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P P ROOF : Let {Nk } be a sequence of integers such that Nk 2−k < ∞. P P k Write am = π/3 · 2k−1 for k−1 Nj < m ≤ 1 Nj . We clearly have 1 P P P −k am = (2π/3) Nk 2 < ∞, so that writing f = am ϕm as in (7.14), we have f ∈ A(D). For 2k ≤ u ≤ 2k+1 we have 2π/3 ≤ uam ≤ 4π/3 for the Nk values of m such that am = π/3·2k−1 . For these values it follows from (7.18) that keiuam ϕm µm kA(D)∗ ≤

3 4

and consequently, using (7.13), (7.19)

C(u) ≤

∞ Y

keiuam ϕm µm kA(D)∗ ≤ ( 34 )Nk

1

since all the factors in (7.19) are bounded by 1 and at least Nk of them 2 by 34 . If we take Nk = 2k k−2 we obtain C(u) ≤ ( 34 )u log u for u → ∞, and since for real-valued f , C(−u) = C(u), the proof is complete. J Corollary. There exists a real-valued f ∈ A(D) such that the closed ideals generated by f n , n = 1, 2, . . . , are all distinct.

EXERCISES FOR SECTION 7 Prove that for every function a(u) such that u−1 a(u) is monotonic and P 1. −k 2 a(2k ) < ∞ , there exists a real-valued function f ∈ A(D) for which −a(|u|)

C(u) = O e . 2. Denote by Bα , 0 < α < 1, the algebras obtained from A(D) and C(D) by the interpolation procedure described in IV.1. Show that spectral synthesis fails in Bα .

8 FUNCTIONS THAT OPERATE IN REGULAR BANACH ALGEBRAS

8.1 We again consider regular semisimple Banach algebras with unit.

D EFINITION : A function F , defined in a set Ω in the complex plane, ˆ for every x ˆ whose range is included in operates in B if F (ˆ x) ∈ B ˆ∈B Ω. The study of functions that operate in B is also called the symbolic calculus in B . Theorem 3.9 can be stated as: a function F defined and analytic in an open set Ω operates in (any) B . Saying that B is self-adjoint is equivalent to saying that F (z) = z¯ operates in B . If B is self-adjoint and regular, we can prove Theorem 3.9 and a great deal more without the use of Cauchy’s integral formula. We first prove:

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Lemma. Let B be a regular, self-adjoint Banach algebra with maximal ideal space M. Let M0 ∈ M and let U be a neighborhood of M0 . Then there exists an element e ∈ B such that eˆ has its support within U , eˆ = 1 on some neighborhood V of M0 and 0 ≤ eˆ ≤ 1 on M.

P ROOF : By the regularity of B there exists an x ∈ B such that xˆ has its support in U and xˆ ≡ 1 in some neighborhood V of M0 . Take eˆ = sin2 πˆ xx ˆ/2. (Notice that sin2 πˆ xx ˆ/2 is well defined by means of power series.) J Theorem. Let x ∈ B and let f be a continuous function on M such that in a neighborhood of each M0 ∈ M, f can be written as F (ˆ x), where F (ζ) = F (ξ + iη) is real-analytic in ξ and η in a neighborhood of xˆ(M0 ). Then f ∈ Bˆ .

Remark: The two points in which this result is more general than Theorem 3.9 are: (a) We allow real-analytic functions. (b) We allow many-valued functions (provided F (ˆ x(M )) can be defined as a continuous function.)

P ROOF : We show that f (M ) ∈ Bˆ locally at every point. Let M0 ∈ M, x ∈ B , and F such that f = F (ˆ x) in a neighborhood U0 of M0 . Replacing x by x − xˆ(M0 ) and F (ζ) by F (ζ − xˆ(m0 )), we may assume that xˆ(M0 ) = 0, and that near zero, say for |ξ| ≤ 1, |η| ≤ 1, we have P F (ξ + iη) = an,m ξ n η m . Let U ⊂ U0 be a neighborhood of M0 such that |ˆ x(M )| < 12 in U , let ˆ have the properties listed in the lemma and write x eˆ ∈ B c1 = 0 and K > 0 such that if x ˆ(M ) is real valued and kxk < ε, then† kF (x)k < K . F operates boundedly at M ∈ M, if there exists a neighborhood UM of M and constants ε > 0 and K > 0 such that if the support of xˆ is contained in UM and kxk < ε, then kF (x)k ≤ K . Lemma. F operates boundedly if, and only if, it operates boundedly at every M ∈ M.

P ROOF : Replacing F by F − F (0) we may assume F (0) = 0. It is clear that if F operates boundedly, it does so locally at each M . Assume that the operation is bounded locally, and pick M1 , M2 , . . . , Mn such that the corresponding neighborhoods UM1 , . . . , UMn cover M. Let V1 , . . . , Vn be open sets such that Vj ⊂ Umj and such that {Vj } cover M. Let ˆ be real valued with support inside UM and ψj ≡ 1 on Vj , ψj ∈ B j and let {ϕj } be a partition of the unity in Bˆ relative to {Vj }. Let εj and Kj be the constants corresponding to UMj and now take ε > 0 so P ˆ that kεψj k < εj for all j , and K = Kj kϕj k. Assume that x ˆ ∈ B is real valued and kxk < ε; then kˆ xψj k ≤ kˆ xkkψj k < εj and x ˆψj is P supported by UMj , hence kF (ˆ xψj )k ≤ Kj . But F (ˆ x) = ϕj F (ˆ xψj ) so P that kF (x)k ≤ kϕj kKj = K . J 8.4 Lemma. Let B be a regular, self-adjoint Banach algebra and F a function defined on the real line and operating in B. Then there exists at most a finite number of points of M at which F does not operate boundedly.

P ROOF : Again we assume, with no loss of generality, that F (0) = 0. Assume that F operates unboundedly at infinitely many points in M and pick a sequence of such points {Mj } having pairwise disjoint neighborhoods Vj . We now pick a neighborhood Wj of Mj such that Wj ⊂ Vj . Saying that F does not operate boundedly at Mj means that, given any neighborhood Wj of Mj and any constants εj > 0 and Kj > 0, there exists a real-valued fj ∈ Bˆ carried by Wj such that kfj k ≤ εj , and ˆ is kF (fj )k ≥ Kj . We take εj = 2−j and Kj = 2j kϕj k, where ϕj ∈ B P carried by Vj and ϕj ≡ 1 on Wj . We now consider f = fj and F (f ). † We

denote by F (x) the element in B whose Gelfand transform is F (ˆ x).

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By the choice of εj the series defining f converges and consequently ˆ and F (f ) ∈ B ˆ . Now f ∈B kF (f )k ≥

1 1 kϕj F (f )k = kF (fj )k ≥ 2j for all j , kϕj k kϕj k

which gives the desired contradiction.

J

8.5 For some Banach algebras, Lemma 8.4 takes us as far as we can go; for others it can be improved. Consider, for instance, an automorphism σ of B inducing the change of variables σ on M. If f ∈ Bˆ , then F (σf ) = F (f (σM )) = σ(F (f )), which means that the operations by F (on the function) and by σ (on the variables) commute. Since σ is a bounded, invertible operator, it follows that F operates boundedly at a point M ∈ M if and only if it operates boundedly at σM . From this remark and Lemma 8.4 it follows that if F does not operate boundedly at M ∈ M, the set of images of M under all the automorphisms of B is finite. In particular, if for every M ∈ M the set {σM }, σ ranging over all the automorphisms of B , is infinite, then every function that operates in B does so boundedly at every M ∈ M, and consequently, operates boundedly. In particular: Theorem. Let G be a compact abelian group and F a continuous function defined on the real line. If F operates in A(G), it does so boundedly.

P ROOF : The maximal ideal space of A(G) is G. For every y ∈ G the mapping f 7→ fy ‡ is an automorphism of A(G) which carries the maximal ideal corresponding to y to that corresponding to 0 ∈ G. If G is infinite the statement of the theorem follows from the discussion above. If G is finite the operation by F is clearly continuous. J Remark: Since the operation of a function on a Banach algebra is not linear, we cannot usually deduce continuity from boundedness, nor boundedness in one ball in B from boundedness in another (see exercise 3 at the end of this section). 8.6 For some algebras Theorem 8.1 is far from being sharp. For instance, if B = C(M) every continuous function operates in B ; if B = C n (T) every n-times continuously differentiable function operates. For ‡f

y (x)

= f (x − y).

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group algebras of infinite LCA groups Theorem 8.1 is sharp. We shall prove now that for the algebra B = A(T), only analytic functions operate. This is a special case of the following: Theorem. Let G be a nondiscrete LCA group and let F be a function defined on an interval I of the real line. Assume that F operates in A(G). Then F is analytic on I .

Remark: If G is not compact, one of our standing assumptions, namely 1 ∈ B , is not satisfied. Since in this case all the functions in A(G) tend to zero at infinity, we have to add to the statement of the theorem the assumption 0 ∈ I since otherwise every function defined on I operates trivially (the condition of operation being void). The theorem can be extended to infinite discrete groups: we have to assume 0 ∈ I (since discrete and compact implies finite) and the conclusion is that F is analytic at zero (see exercise 1 at the end of this section). As mentioned above we prove the theorem for G = T; the proof of the general case runs along the same lines (see [24], chapter 6).

P ROOF OF THE THEOREM (G = T): Let b be an interior point of I and consider the function F1 (x) = F (x + b). F1 is defined on I − b and clearly operates in A(T). If we prove that F (x) is analytic at x = 0 it would follow that F (x) is analytic at b, so that, in order to prove that F is analytic at every interior point of I we may assume 0 ∈ int(I) and prove the analyticity of F at 0. Once we know that functions that operate are necessarily analytic at the interior points of I we obtain the analyticity at the endpoints as follows (we assume, for simplicity, that I = [0, 1] and we prove that F(x) is analytic at x = 0): consider F1 (x) = F (x2 ). F1 is defined on [−1, 1] and clearly operates in A(T) so P that near x = 0, F1 (x) = bj xj . Now, since F1 (x) = F (x2 ) is even, P P b2j−1 = 0 for all j , so that F1 (x) = b2j x2j and F (x) = b2j xj . The proof will therefore be complete if, assuming 0 ∈ int(I), we prove that F is analytic at 0. By Theorem 8.5, F operates boundedly which means that there exist constants ε > 0 and K > 0 such that if f ∈ A(T) is real valued and kf k < ε, then kF (f )k < K . Pick α > 0 so small that (i) [−α, α] ⊂ I , and (ii) αe5 < ε, and consider F1 (x) = F (α sin x). By (i), F1 is well defined and it clearly is 2π -periodic and operates in A(T). Now if f ∈ A(T) is real valued and kf k < 5, then α sin f is real-valued, and by (ii), kα sin f k < ε so that kF1 (f )k < K .

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In particular, if ϕ ∈ A(T), kϕk ≤ 1, τ ∈ R, |τ | ≤ π , then kF1 (ϕ + τ )k < K

(8.1)

Now α sin x ∈ A(T); hence F1 ∈ A(T) and we can write X (8.2) F1 (x) = An einx ; in particular, F1 is continuous. For real-valued f ∈ A(T), F1 (f ) ∈ A(T) and therefore can be written as X X (8.3) F1 (f (t)) = an (f )eint , |an (f )| < ∞ Since F1 is uniformly continuous on R it follows that Z 1 an (f ) = F1 (f (t))e−int dt 2π depends continuously on f and therefore, for each N , the mapping f 7→

(8.4)

N X

an (f )eint

−N

is continuous from the real functions in A(T) into A(T). We conclude from (8.3) that F1 (f ) is a pointwise limit of continuous functions on A(T), that is, is a Baire function on A(T), and in particular: F1 (ϕ + τ ) considered as a function of τ on [−π, π] is a measurable vector-valued function which is bounded by K if kϕk ≤ 1. It follows that

1 Z

(8.5) F1 (ϕ + τ )e−inτ dτ ≤ K :

2π however, (8.6)

1 2π

Z

F1 (ϕ + τ )e−inτ dτ = An einϕ ,

as can be checked by evaluating both sides of (8.6) for every t ∈ T, and we rewrite (8.5) as (8.5’)

kAn einϕ k ≤ K.

Let us write (8.7)

N (u) = supf real, kf k≤u keif kA(T) ;

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then it follows from (8.5’) that −1 |An | ≤ K N (n)

(8.8)

and if we show that N (u) grows exponentially with u, it would follow that (8.2) converges not only on the real axis, but in a strip around it, so that F1 is analytic on R and, finally, F is analytic at 0. All that we need in order to complete the proof is: Lemma. Let N (u) be defined by (8.7). Then N (u) = eu .

(8.9)

P ROOF : It is clear from the power series expansion of eif that for any Banach algebra N (u) ≤ eu . The proof that, for A(T), N (u) ≥ eu is based on the following two remarks: (a) Let f, g ∈ A(T), then (8.10)

kf (t)g(λt)k → kf k kgk

as λ → ∞

(λ being integer). We prove (8.10) by noticing that if f is a trigonometric polynomial and λ is greater than twice the degree of f then kf (t)g(λt)k = kf k kgk. For arbitrary f ∈ A(T) and ε > 0 we write f = f1 + f2 where f1 is a trigonometric polynomial and kf2 k < εkf k. If λ/2 is greater than the degree of f1 we have kf (t)g(λt)k ≥ kf1 (t)g(λt)k − kf2 (t)g(λt)k ≥ (1 − 2ε)kf k kgk.

(b) If α is positive, then eiα cos t = 1 + iα cos t + . . . so that (8.11) keiα cos t kA(T) = 1 + α + O α2 . P Let u > 0; we pick a large N and write f = (u/N ) cos λj t where the λj ’s increase fast enough to ensure (8.12)

N N

Y

1 Y

i(u/N ) cos t

ei(u/N ) cos λj t > (1 − )

e

N j=1 j=1

f is clearly real valued, kf k = u, and, by (8.11) and (8.12), u2 N 1 u > (1 − ε)eu keif k ≥ 1 − 1+ +O N N N2

if N is large enough.

J

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This completes the proof of the theorem (for G = T). The lemma is not accidental: the exponential growth of N (u) is the real reason for the validity of the theorem. The function as defined by (8.7) can be considered for any Banach algebra B and if, for some B , N (u) does not have exponential growth at infinity, then there exist nonanalytic functions which operate in B . As an example we can take P any F (x) = An einx such that An does not vanish exponentially as P |u| → ∞ but such that for all k > 0, An N (k|n|) < ∞; F operates P in B since for any real-valued f ∈ B , F (f ) = An einf and the series converges in norm. 8.7 We finish this section with some remarks concerning the so-called “individual symbolic calculus” in regular semisimple Banach algebras. Inasmuch as "symbolic calculus" is the study of functions that operate in an algebra and of their mode of operation, individual symbolic calculus is the study of the functions that operate on a fixed element in the algebra. Let us be more precise. We consider a regular, semisimple Banach algebra (identify it with its Gelfand transform) and say that a function F operates on an element f ∈ B if the domain of F contains the range of f and F (f ) ∈ B . It is clear that a function F operates in B if it operates on every f ∈ B with range contained in the domain of F . It is also clear that for each fixed f ∈ B , the set of functions that operate on f is a function algebra on the range of f ; we denote this algebra by [f ]. For F ∈ [f ] we write kF k[f ] = kF (f )kB and with this norm [f ] is a normed algebra. If we denote by [[f ]] the subalgebra of B consisting of the elements F (f ), F ∈ [f ], it is clear that the correspondence F ↔ F (f ) is an isometry of [f ] onto [[f ]]. Since [[f ]] consists of all g ∈ B which respect the level lines of f (i.e., such that f (M1 ) = f (M2 ) ⇒ g(M1 = g(M2 )), [[f ]] and [f ] are Banach algebras. We say that [[f ]] is the subalgebra generated formally by f ; it clearly contains the subalgebra generated by f (which corresponds to the closure of the polynomials in [f ]). It should be noted that the "concrete" algebra [f ] depends on f more than [[f ]]. The latter depends only on the level lines of f and is the same, for example, if we replace a real-valued f by f 3 . Even if the ranges of f and f 3 are the same we usually have [f ] 6= [f 3 ]. If f is real valued, [f ] always contains non-analytic functions. In fact, since keif ksp = 1, it follows from Lemma 3.6 that lim keinf k1/n = 1

n→∞

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so that there exists a sequence {An } that does not vanish exponentially P P such that An einf converges in norm; hence F (x) = An einx belongs to [f ]. The fact that {An } does not vanish exponentially implies that F (x) is not analytic on the entire real line but it can still be analytic on portions thereof which may contain the range of f . So we impose the additional condition that An = 0 unless n = m!, m = 1, 2, . . . , which P implies that An einx is analytic nowhere on R. 8.8 Individual symbolic calculus is related to the problem of spectral synthesis in B . Assume for instance that the range of f is [−1, 1] and that [f ] ⊂ C m ([−1, 1]), m ≥ 1. Since in C m , F (x) = x does not belong to the ideal generated by x2 , and since (Theorem 4.1) the imbedding of [f ] in C m is continuous, f does not belong to the ideal generated by f 2 in [[f ]]. This does not mean a-priori that the same is true in B . We do have a linear functional ν on [[f ]] which is orthogonal to (f 2 ) and such that hf, νi = 6 0 and we can extend it by the Hahn-Banach theorem to a functional on B ; there is no reason, however, to expect that the support of the extended functional should always be contained in f −1 (0). If ν can be extended to B with Σ(ν) ⊂ f −1 (0), spectral synthesis fails in B . Going back to C n one identifies immediately a functional orthogonal to the ideal generated by x2 but not to x; for instance, δ 0 , the derivative (in the sense of the theory of distributions) of the point mass at zero, which assigns to every F ∈ C m the value of its derivative at the origin. In [[f ]] the corresponding functional can be denoted by δ 0 (f ) and remembering that the Fourier transform of δ 0 is δb0 (u) = −iu one may tryR to extend δ 0 (f ) to B using the Fourier inversion formula 1 (iu)eiuf du. Strictly speaking this is meaningless, but it δ 0 (f ) = 2π provided the motivation for Theorem 7.5.

EXERCISES FOR SECTION 8 1. Let B be a semisimple Banach algebra without unit and with discrete maximal ideal space. Show that every function F analytic near zero and satisfying F (0) = 0, operates in B . 2. As in chapter I, Lip1 (T) denotes the subalgebra of C(T) consisting of the (t2 ) functions f satisfying supt1 6=t2 f (t1t1)−f < ∞. −t2 (a) Find the functions that operate in Lip1 (T).

(b) Show that every function which operates in Lip1 (T) is bounded in every ball. (c) Show that F (x) = |x| does not operate continuously at f = sin t.

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3. Assume that F is defined on R and operates in A(T). Assume that for every r > 0 there exists K = K(r) such that if f is real-valued and kf k ≤ r, then kF (f )k ≤ K(r). Show that F is the restriction to R of an entire function. 4. Let B be a regular, p semisimple, self-adjoint Banach algebra with a unit. Assume that F (x) = |x| operates boundedly in B . Prove that B = C(M). Hint: Use Theorem 3.8. 5. Use the construction of section 7 to show that for the algebra B = A(D), N (u) has exponential growth at infinity; hence prove Theorem 8.6 for the case G = D. 6. Let α(u) be a positive function, 0 < u < ∞, such that α(u) ≤ 12 and α(u) → 0 as u → ∞. Show that there exists a real-valued f ∈ A(T) such that keiuf k ≥ euα(u) .

7. (a) Show that if ϕ(t, τ ) ∈ A(T2 ), then for every τ ∈ T, ψτ (t) = ϕ(t, τ ), considered as a function of t alone, belongs to A(T) and kψτ kA(T) ≤ kϕkA(T2 ) . Furthermore: ψτ (t) is a continuous A(T)-valued function of τ . (b) Prove: for every function α(u) as in exercise 6. above, there exists a real-valued g ∈ A(T2 ) whose range contains [−π, π] and such that if F (x) = P inx An e ∈ [g], then An = O e−|n|α(|n|) . Hint: Take g(t, τ ) = f (t)+5 sin τ , where f is a function constructed in exercise 6 above. Apply part (a) and the argument of 8.6. (c) Deduce theorem 8.6 for the case G = T2 from part (b).

9 THE ALGEBRA M (T) AND FUNCTIONS THAT OPERATE ON FOURIER-STIELTJES COEFFICIENTS

In this section we study the Banach algebra of measures on a nondiscrete LCA group. We shall actually be more specific and consider M (T); this in order to avoid some (minor) technical difficulties while presenting all the basic phenomena of the general case. 9.1 We have little information so far about the Banach algebra M (T). We know that for every n ∈ Z, the mapping µ 7→ µ ˆ(n) is a multiplicative linear functional on M (T); this identifies Z as part of the maximal ideal space M of M (T). How big a part of M is Z? We have one negative indication: since M is compact the range of every µ ˆ on M is compact and therefore contains the closure of the sequence {ˆ µ(n)}n∈Z , which may well be uncountable (e.g., if µ ˆ(n) = cos n, n ∈ Z). Thus M is uncountable and is therefore much bigger than Z. But we also have a positive indication: a measure µ is determined by its Fourier-Stieltjes coefficients, that is, if µ ˆ = 0 on Z then µ = 0 and therefore µ ˆ = 0 on M.

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This proves that M (T) is semisimple and may suggest the following question: (a) Is Z dense in M? Other natural questions are: (b) Is M (T) regular? (c) ls M (T) self-adjoint? Theorem. There exists a measure µM (T) such that µ ˆ is real valued on Z but is not real valued on M. Corollary. The answer to all three questions above is "no." PROOF : It is clear that the theorem implies that Z is not dense in M. If M ∈ M is not in the closure of Z, there is no µ ∈ M (T) such that µ ˆ=0 on Z while µ ˆ(M ) 6= 0 (since µ ˆ = 0 on Z implies µ = 0); so M (T) is not regular. Finally: if µ is a measure with real-valued Fourier-Stieltjes coefficients and if for some ν ∈ M (T), νˆ = µ ˆ on M, we have νˆ = µ ˆ on Z, hence ν = µ and µ ˆ=µ ˆ on M which means that µ ˆ is real valued on M. ˆ (T). Thus, if µ has the properties described in the theorem, then µ ˆ 6∈ M J

9.2 In the proof of Theorem 9.1 we shall need Lemma. Let G1 and G2 be disjoint† subgroups of T and let Ej ⊂ Gj , j = 1, 2, be compact. Let µj be carried by Ej , j = 1, 2. Then kµ1 ∗ µ2 kM (T) = kµ1 kM (T) kµ2 kM (T) .

(9.1)

P ROOF R : Let ε > 0 and let ϕj be continuous on Ej , satisfying |ϕj | ≤ 1 and ϕj dµj ≥ kµj k − ε. The function ψ(t + τ ) = ϕ1 (t)ϕ2 (τ ) is well defined and continuous on E1 + E2 (this is where we use the fact that Ej are contained in disjoint subgroups) and Z

ψd(µ1 ∗ µ2 ) =

ZZ

ψ(t + τ )dµ1 dµ2 =

Z

which implies km1 ∗ µ2 k ≥ (kµ1 k − ε)(kµ2 k − ε). † That

is: G1 ∩ G2 = {0}.

ϕ1 dµ1

Z

ϕ2 dµ2 J

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9.3 P ROOF OF 9.1: We construct a measure µ ∈ M (T) with real FourierStieltjes coefficients and such that

(9.2)

keiuµ kM (T) = en

for n ≥ 0.

By Lemma 3.6 it follows that the spectral norm of eiµ is equal to e which means that =(ˆ µ) = −1 somewhere on M. Let E be a perfect independent set on T (see VI.9.4). Let ν be a continuous measure carried by E and ν # the symmetric image of ν , defined by ν # (F ) = ν(−F ) for all measurable sets F . ν # is clearly carried by −E and if we write µ = v + v # we have µ ˆ(n) = 2 0 there exists a K = K(r) such that if kµk ≤ r and µ ˆ(n) ∈ I for all n ∈ Z, then F (ˆ µ(n)) are the Fourier-Stieltjes coefficients of a measure of norm < K . We refer to [24], chapter 6 and to exercises 6 through 9 at the end of this section for further details. 9.5 The individual symbolic calculus on M (T) is also more restrictive than an individual symbolic calculus can be in a Banach algebra considered as function algebra on the entire maximal ideal space. There exist measures µ in M (T) with real-valued Fourier-Stieltjes coefficients such that every continuous function which operates on µ must be the restriction to R of some function analytic in a disc (see exercise 10 at the end of this section). This suggests that portions of the maximal ideal space of M (T) may carry analytic structure.

EXERCISES FOR SECTION 9 1. Let µ ∈ M (T) be such that keαµ k = e|α|kµk for all α ∈ C. Show that {µ }, n = 0, 1, 2 . . . are mutually singular. 2. Let E be a linearly independent compact set on T and let µ be a continuous measure carried by E ∪ −E . Show that {µn }, n = 0, 1, 2 . . . are mutually singular. 3. Show that if µ ∈ M (T), µ ˆ(n) is real for all n ∈ Z and µn are mutually singular for n = 1, 2, . . . then µ is continuous. 4. Deduce Theorem 9.1 from Theorem 9.4. 5. Let rj , j = 1, 2, . . . be positive numbers such that rj /rj−1 < 12 and Q∞ rj /rj−1 → 0 as j → ∞. Show that ϕ(n) = cos rj n, n ∈ Z are the 1 n

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Fourier-Stieltjes coefficients of a measure µ, and that µn are mutually singular, n = 1, 2, . . . . Hint: Show that if rj−1 /rj > M for all j , then {µn }M n=1 are mutually singular. 6. Let F be continuous on [−1, 1] and F (0) = 0. Show that the following two conditions are equivalent: (i) If µ ∈ M (T), kµk < r and −1 < µ ˆ(n) < 1 for all n, then {F (ˆ µ(n))} are the Fourier-Stieltjes coefficients of some measure F (µ) ∈ M (T) such that kF (µ)kM (T) < K . P (ii) If −1 < an < 1 and P = an elnt a polynomial satisfying kP kL1 (T) < r, P int then k F (an )e kL1 (T) < K . Also show that in (ii) we may add the assumption that the an are rational numbers without affecting the equivalence of (i) and (ii). 7. (For the purpose of this exercise) we say that a measure µ contains a polynomial P if for appropriate m and M , Pˆ (n) = µ ˆ(m + nM ) for all integers n which are bounded in absolute value by twice the degree d of P . Notice that if µ contains P then P (M t) = Vd (M t) ∗ (e−imt µ)

and consequently kP kL1 (T) ≤ 2kµkM (T) . Show that there exists a measure µ with real Fourier-Stieltjes coefficients, kµk ≤ 2, and µ contains every polynomial P with rational coefficients such that kP kL1 (T) ≤ 1. Hint: Show that for every sequence of integers {Nj } there exists a sequence of Nj integers {λj } such that, writing Λj = {kλj }k=1 and Λ = ∪j Λj , every P function f ∈ CΛ (see chapter V for the notation CΛ ) can be written f = fj , with P fj ∈ CΛj , and kfj k∞ ≤ 2kf k∞ . Deduce, using the Hahn-Banach theorem, PNj that if the numbers aj,k are such that for each j , k k=1 aj,k eikt k < 1, then there exists a measure µ ∈ M (T) such that kµkM (T) < 2 and µ ˆ(kλj ) = aj,k for appropriate λj and 1 ≤ k ≤ Nj . If the numbers aj,k above are real, one can replace µ by 12 (µ + µ# ). 8. Let F be defined and continuous on R and assume that it operates on Fourier-Stieltjes coefficients. Prove that the operation is bounded on every ball of M (T). Hint: Use exercises 6 and 7 above; show that if kνkM (T) < r then kF (ν)kM (T) ≤ 2kF (rµ)kM (T) . 9. Prove Theorem 9.4. 10. Show that if F is defined and continuous on R and if F (ˆ µ(n)) are Fourier-Stieltjes coefficients, µ being the measure introduced in exercise 7, then F is analytic at the origin. If F (kµ ˆ(n)) are Fourier Stieltjes coefficients for all k, then F is entire. 11. Let µ ∈ M (T) be carried by a compact independent set and assume that µ ˆ(n) → 0 as |n| → ∞. (Such measures exist: see [25].) Let B be the closed subalgebra of M (T) generated by L1 (T) and µ.

264


(a) Check that Theorems 9.1 and 9.4 are valid if we replace in their statement M (T) by B . ˆ to Z is a function algebra on Z, interme(b) Notice that the restriction of B diate between A(Z) and c0 both of which have Z as maximal ideal space, and yet its maximal ideal space is larger than Z. 10 THE USE OF TENSOR PRODUCTS

In this final section we prove a theorem concerning the symbolic calculus and the failing of spectral synthesis in some quotient algebras of A(R). The theorem and its proof are due to Varopoulos and serve here as an illustration of a general method which he introduced. We refer to [26] for a systematic account of the use of tensor algebras in harmonic analysis. 10.1 Let E ⊂ R be compact. We denote by A(E) the algebra of functions on E which are restrictions to E of elements of A(R). A(E) is canonically identified with the quotient algebra A(R)/k(E) (where k(E) = {f : f ∈ A(R) and f = 0 on E}) and is therefore a Banach algebra with E as the space of maximal ideals (see 5.5). The main theorem of this section is: Theorem. Let E1 , E2 be nonempty disjoint perfect subsets of R, such that E1 ∪ E2 is a Kronecker set. Put E = E1 + E2 † . Then: (a) Every function F , defined on R, which operates in A(E) is analytic. (b) Spectral synthesis fails in A(E).

Remarks: (i) We place E on R for the sake of technical simplicity and in accordance with the general trend of this book. Only minor modifications are needed in order to place E in an arbitrary nondiscrete LCA group, obtaining thereby a proof of Malliavin’s theorem 7.4 in its full generality. (ii) We shall actually prove more, namely: A(E) is isomorphic to a fixed Banach algebra (subsections 10.2, 10.3, and 10.4) for which (a) and (b) are valid (subsection 10.5). 10.2 Let X and Y be compact Hausdorff spaces, X × Y their cartesian product. We denote by V = V (X, Y ) the projective tensor product of C(X) and C(Y ); that is, the space of all continuous functions ϕ on X × Y that admit a representation of the form †E

1

+ E2 = {x : x = x1 + x2 with xj ∈ Ej }.

VIII.


(10.1)

ϕ(x, y) =

X

265

fj (x)gj (y)

with fj ∈ C(X), gj ∈ C(Y ), and X (10.2) kfj k∞ kgj k∞ < ∞. We introduce the norm (10.3)

kϕkV = inf

X

kfj k∞ kgj k∞

where the infimum is taken with respect to all possible representations of ϕ in the form (10.1). It is immediate to check that the norm k kV is multiplicative and that V is complete; thus V is a Banach algebra. Lemma. The maximal ideal space of V can be identified canonically with X × Y .

P ROOF : Denote by V1 (resp. V2 ) the subalgebra of V consisting of the functions ϕ(x, y) which depend only on x (resp. only on y ). It is clear that V1 and V2 are canonically isomorphic to C(X) and C(Y ) respectively. A multiplicative linear functional w on V induces, by restriction, multiplicative linear functionals w1 on V1 and w2 on V2 . By Corollary, 2.12, w1 has the form f 7→ f (x0 ) for some x0 ∈ X ; w2 has the form g 7→ g(y0 ) for some y0 ∈ Y , and it follows that if X ϕ(x, y) = fj (x)gj (y), then w(ϕ) =

X

fj (x0 )gj (y0 ) = ϕ(x0 , y0 ).

J

Corollary. V is semisimple, self-adjoint, and regular. 10.3 We assume now that X is homeomorphic to a compact abelian group G (more precisely, to the underlying topological space of G) and that Y is homeomorphic to a compact abelian group H . We denote both homeomorphisms X 7→ G and Y 7→ H by σ . σ induces canonically a homeomorphism of X × Y onto G ⊕ H , and hence an isomorphism of C(G ⊕ H) onto C(X × Y ). Lemma. The canonical isomorphism of C(G ⊕ H) onto C(X × Y ) maps A(G ⊕ H) into V .

266


P ROOF : Let χ be a character on G ⊕ H and let χ be its image under the canonical isomorphism, namely (10.4)

χ(x, y) = χ(σx, σy).

Since χ(σx, σy) = χ(σx, 0)χ(0, σy) we have (10.5)

χ(x, y) = χ(x, 0)χ(0, y)

P so that χ ∈ V and kχkV = 1. If ϕ ∈ A(G ⊕ H) then ϕ = aχ χ P the summation extending over G\ ⊕ H and kϕkA = |aχ |. The image P of ϕ under the canonical isomorphism is ϕ = aχ χ and therefore P kϕkV ≤ |aχ | = kϕkA . J

If ϕ ∈ A(G ⊕ H) depends only on the first variable, that is, if ϕ(x, y) = ψ(x), then ψ ∈ A(G). Assuming G to be infinite we have A(G) 6= C(G) and it follows that the image of A(G ⊕ H) in V does not contain V1 and is therefore a proper part of V . The connection between A(G ⊕ H) and V is only that of (canonical) inclusion, which is too loose for obtaining information for one algebra from the other. A closer look reveals, however, that the structure needed for the lemma is not the group structure on G or on H but only the cartesian structure of G ⊕ H , while in order to show that the image of A(G ⊕ H) is not the entire V we use the group structure of G. The idea now is to keep the useful structure and obliterate the hampering one; this is the reason for the appeal to Kronecker sets. 10.4 Theorem. Let E1 , E2 , and E be as in the statement of Theorem 10.1. Let X and Y be homeomorphic to the (classical) Cantor set. Then A(E) is isomorphic to V (X, Y ).

P ROOF : We begin by noticing that E1 and E2 , being portions of a Kronecker set, are clearly totally disconnected and, being perfect and nonempty, are homeomorphic to the Cantor set. Thus, X, Y, E1 and E2 are all homeomorphic and we simplify the typography by identifying X with E1 and Y with E2 . Since E1 ∪ E2 is a Kronecker set, hence independent, the mapping (x, y) 7→ x + y is a homeomorphism of X × Y on E . We now show that the induced mapping of C(E) onto C(X × Y ) maps A(E) onto V . The fact that A(E) is mapped into V (and that the map is of norm 1) is a verbatim repetition of 10.3. We therefore have only to prove that the mapping is surjective that is, maps A(E) onto V . We shall need:

VIII.


267

˜ can be Lemma. Let E˜ be a Kronecker set. Then every f ∈ C(E) P P iλn x written as f (x) = an e with |an | ≤ 3kf kC(E) ˜ . In particular: ˜ ˜ A(E) = C(E) and kf kA(E) ˜ ≤ 3kf k∞ . ˜ is real P ROOF OF THE LEMMA : It is enough to show that if f ∈ C(E) P P iλn x valued, f (x) = an e with |an | ≤ 3/2kf k∞ . Let p f real valued and assume, for simplicity, that kf k∞ = 1; define g(x) = 1 − (f (x))2 , then g is continuous and Φ = f + ig has modulus 1 on E˜ . Let λ1 be 1 1 such that |Φ − eiλ1 x | < 10 on E˜ ; this implies |f − cos λ1 x| < 10 on E˜ . If −1 kf k∞ is not 1 consider kf k∞ f and obtain λ1 such that |f − kf k∞ cos λ1 x| ≤

1 ˜ kf k∞ on E. 10

We now proceed by induction. Define a1 = kf k∞ , λ1 as above, and f1 = f − a1 cos λ1 x; once we have a1 . . . , an , λ1 . . . , λn , and fn , define an+1 = kfn k∞ , λn+1 by the condition |fn − cos λn+1 x| ≤ an+1 /10, and Pn+1 fn+1 = fn − an+1 cos λn+1 x = f − 1 aj cos λj x. We clearly have an+1 ≤ an /10 ≤ kf k∞ 10−n and it follows that P∞ P P P an ≤ kf k∞ 0 10−n < 3/2kf k∞ . f (x) = an eiλn x with |an | = Writing cos λn x = 12 (eiλn x + e−iλn x ) we obtain f as a series of exponentials. J ˜ is actually isometric to C(E) ˜ ; see exercise 2 at the end Remark: A(E) of this section. PROOF OF THE THEOREM , COMPLETED : We identify X×Y with E , and V with the subalgebra of C(E) consisting of the functions ϕ which admit a representation X (10.6) ϕ(x + y) = fj (x)gj (y), x ∈ E1 , y ∈ E2

where fj ∈ C(E1 ), gj ∈ C(E2 ) such that (10.2) is valid. All that we need to show is that if ϕ ∈ V then ϕ ∈ A(E). Let ϕ ∈ V and consider a representation of the form (10.6) such that X (10.7) kfj k∞ kgj k∞ ≤ 2kϕkV . Using the lemma we write each fj as an exponential series (E1 , being a portion of a Kronecker set, is itself one) and similarly for the gj . Denoting the frequencies appearing in the f ’s by λ, and those appearing in the g ’s by ν , and taking account of (10.6) and (10.7), we obtain X (10.8) ϕ(x + y) = aλ,ν eiλx eiνy , x ∈ E1 , y ∈ E2 λ,ν

268


where X

(10.9)

|aλ,ν | ≤ 18kϕkV .

We now use the fact that E1 ∪ E2 is a Kronecker set: let (λ, ν) be a pair which appears in (10.8) and define ( eiλx x ∈ E1 , h(x) = eiνx x ∈ E2 ; h is clearly continuous and of modulus 1 on E1 ∪ E2 and it follows that there exists a real number ξ such that

(10.10)

|eiξx − h(x)| < 1/200

on E1 ∪ E2 .

We have (for x ∈ E1 , y ∈ E2 ), eiξ(x+y) − eiλx eiνy = (eiξx − e−λx )eiξy + eiλx (eiξy − eiνy )

which means that (with the canonical identifications) keiξ(x+y) − eiλx eiνy kV ≤ 1/100.

We can now write ϕ = ϑ1 + ϕ1 where X ϑ1 (x + y) = aλ,ν eiξ(x+y) and ϕ1 (x + y) =

X

aλ,ν (eiλx eiνy − eiξ(x+y) )

and notice that ϑ1 ∈ A(E) and, by (10.9), X kϑ1 kA(E) ≤ |aλ,ν | ≤ 18kϕkV ; also

18 1 kϕkV ≤ kϕkV . 100 5 Repeating, we obtain inductively kϕ1 kV ≤

ϕn = ϑn+1 + ϕn+1

where ϑn+1 ∈ A(E),

kϑn+1 kA(E) ≤ 18kϕn kV ,

and

kϕn+1 kV ≤

P

ϑn ∈ A(E) and P∞ kϕkA(E) ≤ ( 0 5−n )kϕkV ≤ 25kϕkV .

It follows that ϕ =

1 kϕn kV . 5

J

VIII.


269

10.5 We now show that statements (a) and (b) of Theorem 10.1 are valid for V = V (X, Y ) (X and Y being both homeomorphic images of the Cantor set). This is obtained as a consequence of the fact that (a) and (b) are valid for the algebra A(D)) (see exercise 8.5 for (a) and Theorem 7.4 for (b)). Since X and Y are homeomorphic to D we may consider V as a function algebra on D × D. Using the group structure and the Haar measure on 1) we now define two linear operators M and P as follows:

(10.11)

for f ∈ C(D),

(10.12)

for ϕ ∈ C(D × D),

write Mf (x, y) = f (x + y);

write Pϕ(x) =

Z

ϕ(x − y, y)dy.

D

M maps C(D) into C(D × D), P maps C(D × D) into C(D), and, since for f ∈ C(D); Z Z PMf (x) = Mf (x − y, y)dy = f (x)dy = f (x), D

D

it follows that PM is the identity map of C(D). Lemma. M maps A(D) into V and its norm as such is 1. P maps V into A(D) and its norm as such is 1. P P P ROOF : If f = aχ χ with |aχ | = kf kA(D) < ∞ then Mf =

X

aχ χ(x)χ(y) ∈ V

and

kMf kV ≤

X

|aχ | = kf kA(D)

If ϕ(x, y) = f (x)g(y) then Pϕ =

Z

f (x − y)g(y)dy = f ∗ g

hence kPϕkA(D) =

(10.13)

X

1/2 1/2 X X |ˆ g (χ)|2 |fˆ(χ)|2 |fˆ(χ)ˆ g (χ)| ≤

= kf kL2 kgkL2 ≤kf k∞ kgk∞

By (10.13) and the definition (10.3) of the norm in V it follows that for arbitrary ϕ ∈ V , kPϕkA(D) ≤ kϕkV . J

270


Corollary. Let ϕ ∈ C(D × D) and assume that for some ψ ∈ C(D) we have ϕ(x, y) = ψ(x + y). Then ϕ ∈ V if, and only if, ψ ∈ A(D), and then kϕkV = kψkA(D) .

In other words, M is an isometry of A(D) onto the closed subalgebra V3 of V of all the functions ϕ(x, y) which depend only on x + y . Remark: The subalgebra V3 is determined by the "level lines": x + y = const, which clearly depend on the group structure of D. Instead of D we can take any group G whose underlying topological space is homeomorphic to the Cantor set; for every such G, V has a closed subalgebra isometric to A(G). We are now ready to prove: Theorem. (a) Every function F , defined on R, which operates in V is analytic. (b) Spectral synthesis is not always possible in V .

P ROOF : (a) If F operates in V , so it does in V3 (since the operation by F conserves the level lines), hence in A(D) and by Theorem 8.6 (rather, exercise 8.5), F is analytic. (b) Let H ⊂ D be a closed set which is not a set of spectral synthesis for A(D) (see Theorem 7.4). Define: H ∗ = {(x, y) : x + y ∈ H} ⊂ D × D.

We contend that H ∗ is not a set of spectral synthesis in V . By 7.3 we have a function f ∈ A(D) which vanishes on H and which cannot be approximated by functions in I0 (H) (i.e., functions that vanish in a neighborhood of H ), that is, for some δ > 0, kf − gkA(D) > δ for every g ∈ I0 (H). We show that H ∗ is not of spectral synthesis for V by showing that for Mf , which clearly vanishes on H ∗ , and every ϕ ∈ V which vanishes in a neighborhood of H ∗ , we have kMf − ϕkV > δ . For this we notice that if ϕ vanishes on a neighborhood of H ∗ , then Pϕ vanishes on a neighborhood of H, so that kMf − ϕkV ≥ kPMf − PϕkA(D) = kf − PϕkA(D) > δ.

Theorems 10.4 and 10.5 clearly imply Theorem 10.1.

J

VIII.


271

EXERCISES FOR SECTION 10 1. Let f be a continuous complex-valued function on a topological space X . Assume 0 < |f (x)| ≤ 1 on X . Show that f admits a unique representation in the form f = 12 (g1 + g2 ) such that |gj (x)| = 1 on X and gj are continuous, j = 1, 2. Deduce that if X is homeomorphic to the Cantor set, every continuous complex-valued function f on X admits representations of the form f = 12 kf k∞ (g1 − g2 ) + f1 where gj are continuous, |gj (x)| = 1 on X , and kf1 k∞ is arbitrarily small. 2. Let B be a Banach space with the norm k k0 and let B1 ⊂ B be a subspace which is a Banach space under a norm k k1 such that the imbedding of B1 in B is continuous. Show that if there exist constants K > 0 and 0 ≤ η < 1 such that for every f ∈ B there exist g ∈ B1 and f1 ∈ B satisfying f = g − f1 , kgk1 ≤ Kkf k0 , and kf1 k0 ≤ ηkf k0 , then B1 = B and k k1 ≤ K(1 − η)−1 k k0 . Use this and exercise 1 to prove remark 10.4. Hint: See either proof in 10.4.

Appendix A

Vector-Valued Functions

1 RIEMANN INTEGRATION

Consider a Banach space B and let F be a B -valued function, defined and continuous on a compact interval [a, b] ⊂ R. We define the (Riemann) integral of F on [a, b] in a manner completely analogous to that used in the case of numerical functions, namely: Rb P D EFINITION : a F (x)dx = lim N j=0 (xj+1 − xj )F (xj ) where a = x0 < x1 < · · · < xN +1 = b, +1 and the limit is taken as the subdivision {xj }N j=0 becomes finer and finer, that is: as N → ∞ and max0≤j≤N (xj+1 − xj ) → 0. The existence of the limit is proved, as in the case of numerical functions, by showing that if {xj } and {yj } are subdivisions of [a, b] which are fine enough to ensure that kF (α) − F (β)k ≤ ε whenever α and β belong to the same interval [xj , xj+1 ] (or [yj , yj+1 ]), then N M X

X

(xj+1 − xj )F (xj ) − (yk+1 − yk )F (yk ) ≤ 2(b − a)ε. j=0

j=0

This is done most easily by comparing either sum to the sum corresponding to a common refinement of {xj } and {yk }. The following properties of the integral so defined are obvious: (1) If F and G are both continuous B -valued functions on [a, b], and c1 , c2 ∈ C, then: Z b Z b Z b G(x)dx . F (x)dx + c2 (c1 F (x) + c2 G(x))dx = c1 a

a

a

(2) If a < c < b then Z b Z c Z b F (x)dx F (x)dx + F (x)dx = a

c

a

272

A PPENDIX A. V ECTOR -VALUED F UNCTIONS

(3)

Z

b

a

F (x)dx ≤

273

b

Z

kF (x)kdx .

a

(4) If µ is a continuous linear functional on B , then: Z h

b

F (x)dx, µi =

Z

b

hF (x), µidx .

a

a

2 IMPROPER INTEGRALS

Let F be a B -valued function, defined and continuous in a nonclosed interval (open or half-open; finite or infinite) say (a, b). Rb The (improper) integral a F (x)dx is, by definition, the limit of R b0 F (x)dx where a < a0 < b0 < b and the limit is taken as a0 → a a0 and b0 → b. As in the case of numerical functions the improper integral need not always exist. A sufficient condition for its existence is Z

b

kF (x)kdx < ∞.

a

3 MORE GENERAL INTEGRALS

Once in this book (in VIII.8) we integrate a vector-valued function which we do not know a-priori to be continuous. It is, however, the pointwise limit of a sequence of continuous functions and is therefore Bochner-integrable. We refer the reader to [10], chapter 3, §1, for details on the Bochner integral; we point out also that for the purpose of VllI.8, as well as in other situations where the integral is used mainly to evaluate the norm of a given vector, one can obviate the vector-valued integration by applying linear functionals to the integrand before the integration. 4 HOLOMORPHIC VECTOR-VALUED FUNCTIONS

A B -valued function F (z), defined in a domain Ω ⊂ C is holomorphic in Ω if for every continuous linear functional µ on B , the numerical function h(z) = hF (z), µi is holomorphic in Ω. This condition is equivalent to the apparently stronger one stating P n that for each z0 ∈ Ω, F has the representation F (z) = ∞ n=0 an (z − z0 ) in some neighborhood of z0 ; the coefficients an being vectors in B and the series converging in norm. One proves that, as in the case of complex-valued functions, the power series expansion converges in the largest disc, centered at z0 , which is contained in Ω. These results are

274


consequences of the uniform boundedness theorem (see [10], chapter 3, §2). Many theorems about numerical holomorphic functions have their generalizations to vector-valued functions. The generalizations of theorems dealing with "size and growth" such as the maximum principle, the theorem of Phragmèn-Lindelöf, and Liouville’s theorem are almost trivial to generalize. For instance: the form of Liouville’s theorem which we use in Vlll.2.4 is Theorem. Let F be a bounded entire B -valued function. Then F is a constant.

If F (z1 ) 6= F (z2 ) there should exist functionals µ ∈ B ∗ such that hF (z1 ), µi 6= hF (z2 ), µi However, for all µ ∈ B ∗ , hF (z), µi is a bounded (numerical) entire function and hence, by Liouville’s theorem, is a constant. J PROOF :

Another theorem which we use in IV.I.3 is an immediate consequence of the power series expansion. We refer to Theorem. Let F be a B -valued function holomorphic in a domain Ω. Let Ψ be a B ∗ -valued function in Ω, holomorphic in z¯. Then h(z) = hF (z), Ψ(z)i is a holomorphic (numerical) function in Ω. P Let z0 ∈ Ω; in some disc around it F (z) = an (z − z0 )n , P P P n n Ψ(z) = bn (z − z0 ) , hence h(z) = ( k=0 hak , bn−k i)(z − z0 )n , and the series converges in the same disc. PROOF :

Remark: Φ of IV.1.3 corresponds to Ψ here.

Appendix B

Elementary Probabilistic methods

In this appendix we give a few examples of the power of Probabilistic methods in Harmonic analysis. The approach is to replace the study of particular functions or series, by the study of typical functions or series. There are several ways to define “typical”, 1. The Baire category “typical” in a complete metric space— what happens for all but an exceptional set of the first category. 2. The measure or probabilistic definition. Here one defines a probability measure on a class of objects, say series, and “typical” is what happens for all but an exceptional set of measure zero. A beautiful example of the use of the category method is Kaufman’s theorem VI.9.4. Here we limit ourselves to the second approach in one concrete case, that of series with coefficients that have random signs. 1 RANDOM SERIES

1.1 Independence. We refer the reader to VI.?2.11 for some of the basic terms. Let Ω, B, P be a probability space and Fj , j = 1, . . . , k sub-sigmaalgebras of B.

D EFINITION : then (1.1)

Fj are independent if, whenever Oj ∈ Fj , j = 1, . . . , k,

P

k \ 1

k Y P (Oj ). Oj = 1

The variables X1 , . . . , Xk are independent if FXj are independent, where FXj denotes the field of the variable Xj , that is the sub-σ -algebra of B spanned by the events {Xj ∈ O} = {ω : Xj (ω) ∈ O}, O open. Theorem. The following conditions are equivalent:

275

276


1. The random variables X1 , . . . , Xk are independent. 2. The image of µ under the map X 7→ Rk given by (X1 , . . . , Xk ) is a product measure. 3. If fj are continuous functions on the line such that fj (Xj ) have Q Q expectation (are integrable), then E ( fj (Xj )) = E (fj (Xj )) . Q 4. dFPk X = ∗dFXj (convolution product). 1

j

1.2 Rademacher functions. The Rademacher functions, {rn }, is a sequence of independent random variables, taking the values 1 and −1 with probability 12 for each. A standard concrete realization is to define rn on the interval [0, 1], (endowed with the Lebesgue measure) as follows: let εn (x) be the coefficients in the (non-terminating) binary expansion of x ∈ [0, 1), that P is x = εj (x)2−j , with εj (x) either zero or one, and set rn (x) = (−1)εn (x) . Another common representation of the Rademacher functions is as the characters ξm defined by VIII.(7.15) on the group D (with its Haar measure). P Proposition. Let an be real numbers such that |an |2 = a2 . Then, for all λ > 0, X λ2 an rn > aλ ≤ e− 2 , (1.2) P

and (1.3)

X λ2 P an rn > aλ ≤ 2e− 2 . |a2n | = a2 , X λ2 an rn > aλ ≤ 4e− 2 . P

For complex an with (1.4)

P

P ROOF : For real valued an , P Y Y 1 2 2 Y 1 2 2 E eλ an rn = E eλan rn = cosh λan ≤ e 2 an λ = e 2 a λ . −1 P 2 Write Y = an rn . As E ea λY ≥ eλ P (Y ≥ aλ), we obtain (1.2). Applying the same inequality to −Y , we have (1.3).

A PPENDIX B. P ROBABILISTIC M ETHODS

277

If the an are complex we write aj = cj + idj , the decomposition 2 2 to real and imaginary parts, and notice that |aj |2 = Pcj + d j so that if P 2 P cj = c2 and d2j = d2 , then a2 = c2 + d2 . If aj rj > aλ then either X X cj rj > cλ or dj rj > dλ and we have (1.4)

J

1.3 We denote by Trimλ the operator of trimming at height λ, namely, given a complex valued function g , we define

(1.5)

Trimλ g = min(|g|, λ)· sgn (g),

where sgn (g) = g/|g| (and sgn (0) = 0). P P Lemma. Assume |an |2 = a2 , and set X = an rn . Then, for λ > 0, (1.6)

λ2

kX − Trimλ Xk2L2 ≤ 4(λ2 + 2a2 )e− 2a2 .

P ROOF : If GX (x) = P (|X| > x), then Z Z ∞ kX − Trimλ Xk2L2 = − x2 dGX = λ2 GX (λ) + 2 λ

∞

xGX (x)dx.

λ

x2

λ2

Since GX (x) ≤ 4e− 2a2 , this is bounded by 4(λ2 + 2a2 )e− 2a2 .

J

P P 1.4 Fubini. Let a2n = a2 < ∞, and X(t) = X(t, ω) = an eint rn (ω). Given λ > 0, we have estimate (1.6) for every t ∈ T, and integrating dt we have Z dt λ2 (1.7) E |X(t) − Trimλ X(t)|2 ≤ 4(λ2 + 2a2 )e− 2a2 2π

Reversing the order of integration (Fubini’s theorem) we obtain that there exist choices of ω for which (1.8)

λ2

kX(t) − Trimλ X(t)k2L2 (T) ≤ 4(λ2 + 2a2 )e− 2a2 .

This proves P Theorem. Given complex numbers an such that |a2n | = a2 and given P λ > 0, there exists a choice of εn = ±1 such that, with g(t) = εn an eint ,

(1.9)

λ2

kg(t) − Trimλ g(t)k2L2 (T) ≤ 4(λ2 + 2a2 )e− 2a2

278


When (1.9) is valid, if N is sufficiently large, we can replace Trimλ g by the Fejér sum ϕ = σN (Trimλ g) and still have kϕk∞ ≤ λ and (1.10)

λ2

kg(t) − ϕk2L2 (T) ≤ 4(λ2 + 2a2 )e− 2a2

In the next section we use (1.10) systematically with λk = 10k kgk kL2 , and ϕk = σNk (Trimλk gk ), where Nk is big enough to guarantee (1.11)

kgk (t) − ϕk (t)k2L2 (T) ≤ 5·102k exp(−.5·102k )kgk2L2

which we often replace by the (very wasteful) (1.12)

kgk (t) − ϕk (t)k2L2 (T) ≤ exp(−10k )kgk2L2 .

These inequalities are valid for a proper choice of signs. 2 FOURIER COEFFICIENTS OF CONTINUOUS FUNCTIONS

What we show here is that, in terms of size, Fourier transforms of continuous functions majorize any `2 sequence. 2.1 Theorem (deLeeuw-Kahane-Katznelson). For any sequence {an } ∈ `2 there exist functions f ∈ C(T) such that |fˆ(n)| ≥ |an | for all n ∈ Z. P P ROOF : We may clearly assume that an ≥ 0, and that a2n = 1. The required continuous function f is obtained as a uniformly convergent P sum k ϕk with ϕk defined recursively. P Write g1 = 2 εn an eint , λ1 = 20, and choose εn = ±1 such that (1.11) is valid for g1 , λ1 , and ϕ1 , so that kg1 − ϕ1 k2L2 < 4·500e−50 < 10−19 .

The choice of λ1 = 20 is not optimal; it is done to make obvious the super–exponential decay of kgk kL2 below. Write A1 = {n ∈ Z : |ϕˆ1 (n)| < 3an /2}.

If n ∈ A1 then |ˆ g1 (n) − ϕˆ1 |(n) ≥ an /2, which implies X (2.1) a2n < 4 kg1 − ϕ1 k2L2 < 4·10−19 . n∈A1

P Write g2 = 3 n∈A1 εn,2 an eint where εn,2 = ±1 are chosen such that (1.11) is valid for g2 , λ2 = 100 kg2 kL2 , N2 , and ϕ2 = σN2 (Trimλ2 g2 ).


279

Notice that kg2 kL2 ≤ 6·10−9 and |ϕˆ1 (n) + gˆ2 (n)| ≥ (1 + 2−1 )an for all n.

The inductive step is virtually identical: assuming gj and ϕj known, j ≤ k , and (2.2)

|ϕˆ1 (n) + · · · + ϕˆk−1 (n) + gˆk (n)| ≥ (1 + 21−k )an

for all n, we set Ak = {n ∈ Z : |ϕˆ1 (n) + · · · + ϕˆk (n)| < (1 + 2−k )an } P and define gk+1 = 3 n∈Ak εn,k an eint with {εn,k } such that (1.12) is valid for λk = 10k . Independetly of the choice of signs, (2.2) is valid for k + 1. For n ∈ Ak we have |ˆ gk (n) − ϕˆk (n)| > 2−k an , so

(2.3)

(2.4)

kgk+1 k2L2 ≤ 22k kgk − ϕk k2L2 ≤ 22k exp(−10k )kgk k2L2 .

The norms kgk kL2 decrease super–exponentially; kϕk k∞ are only P exponentially bigger so that the series f = ϕk converges uniformly ˆ and (2.2) implies that |f (n)| ≥ an for all n. J 3 PALEY–ZYGMUND,

when

P

|an |2 = ∞ .

The following special case of a theorem of Paley and Zygmund shows that, as opposed to the “smoothing effect” that adding random signs has on trigonometric series with coefficients in `2 , turning the series a.s. into the Fourier series of a subgaussian function∗ on T, the P P series an rn eint with |an |2 = ∞ is almost surely not a Fourier– Stieltjes series. This and Theorem 2.1 are, in a sense, two sides of the same coin; showing, in particular, that the Hausdorff–Young theorem can not be extended beyond p = 2. √ P P 3.1 Lemma. Assume |a2n | = a2 . Then k an rn (ω)kL1 ≥ a(e 2)−1 .

P ROOF : We may assume, with no loss of generality, that a = 1. If QN an are real-valued, the functions P 2ψN = 1 (1 + ian rn ) are uniformly Q bounded by (1 + |an |2 ) ≤ e |an | = e.

X

Z X X

−1 (3.1) an rn (ω) ≥ e−1 ( a r (ω))ψ dω = e |an |2 , n n N 1 L

∗A

2

random variable X is subgaussian if ec|X| is integrable for some constant c > 0.

280


since the only integrands with non-zero integrals are i|an |2 . For an which are not necessarily real-valued, break the sum into its real and imaginary parts, and apply (3.1) to each. J Remark: The products defining ψN are the Riesz products of the second kind (see V.1.3) for the group D. 3.2 Lemma. Let X be a non-negative random variable, E X 2 < ∞. Then for 0 < λ < 1, P ({ω : X(ω) ≥ λE (X)}) ≥ (1 − λ)2

E (X)2 . E (X 2 )

P ROOF : Denote A = {ω : X(ω) ≥ λE (X)}, a = P (A). The contribution of the set A to E (X) is at least (1 − λ)E (X), which means that average of X on A is at least a−1 (1 − λ)E (X), and the contribution of A to E X 2 is therefore at least a−1 (1 − λ)2 E (X)2 . It follows that 2 2 a−1 (1 − λ)2 E (X) ≤ E X 2 , and a ≥ (1 − λ)2 E (X) /E X 2 . J Corollary. If

(3.2)

|a2n | = a2 , then X P ({ω : an rn (ω) ≥ a/10}) ≥ 2−3 e−2 . P

P ROOF : Take λ = E (X).

1 2,

X = |

P

an rn | and use Lemma 3.1 to estimate J

P Theorem (Paley–Zygmund). If |an |2 = ∞ then the series X (3.3) an rn eint

is almost surely not a Fourier–Stieltjes series.

P ROOF : We use the standard notation for the Fejér kernel, (3.4)

Kn (t) =

n X

j=−n

1−

|j| ijt e , n+1

the de la Vallée Poussin kernel, (3.5)

Vn (t) = 2K2n+1 (t) − Kn (t),


281

and introduce the polynomials (3.6)

WM,N (t) = VN (t) − V2M (t) =

X

cM,N (n)eint .

M 2j }) > c}, we have P (Ωj ) > c. Since Ωj are independent, we have P (lim sup Ωj ) = 1 (Borel–Cantelli), and for no ω ∈ lim sup Ωj can the series (3.3) be Fourier–Stieltjes series. J

Bibliography

[1] N. K. Bary, A treatise on trigonometric series, Macmillan, 1964. [2] H. Bohr, Almost periodic functions, Chelsea, 1947. [3] T. Carleman, L’integrale de fourier et questions qui s’y rattachent, Almqvist and Wiksell, Uppsala, 1944. [4] L. Carleson, On convergence and growth of partial sums of fourier series, Acta Math. 116 (1966), 135–157. [5] Raikov D. A. Gelland, I. M. and G. E. Shilov, Commutative normed rings, Chelsea, 1964. [6] A. Gleason, Function algebras, Seminars on analytic functions II (1967), 213–226. [7] R. Goldberg, Fourier transforms, Cambridge Tracts, No. 52, 1962. [8] G. H. Hardy and W. W. Rogozinski, Fourier series, Cambridge Tracts, No. 38, 1950. [9] E. Hewitt and K. A. Ross, Abstract harmonic analysis, SpringerVerlag, 1963. [10] E. Hille and R. S. Phillips, Functional analysis and semigroups, Amer. Math. Soc. Coll. Publ., Vol. 31, Providence, 1957. [11] K. Hoffman, Banach spares of analytic functions, Prentice-Hall, 1962. [12] R. A. Hunt, On the convergence of fourier series, Proceedings of the Conference on Orthogonal Expansions and Their Continuous Analogues (1968), 234–255. [13] J.-P. Kahane and R. Salem, Ensembles parfaits et series trigonometriques, Hermann, 1963. 283

284


[14] R. Kaufman, A functional method for linear sets, Israel J. Math. 5 (1967), 185–187. [15] L. Loomis, Abstract harmonic analysis, Van Nostrand, 1953. [16] Lee Lorch, The principal term in the asymptotic expansion of the Lebesgue constants, Amer. Math. Monthly 61 (1954), 245 –249. [17] S. Mandelbrojt, Series adherentes, regularisation,. . . , GauthierVillars, 1952. [18] J. G. Mikusinski, Une simple démonstration du theoreme de Titchmarsh, Bull. Acad. Polonaise des Sci. 7 (1959), 715–717. [19] M. A. Naimark, Normed rings, P. Noordhoff N. V., 1964. [20] R.E.A.C. Paley and N. Wiener, Fourier transforms in the complex domain, Amer. Math. Soc. Coll. Publ., vol. 19, 1934. [21] C. E. Rickart, General theory of banach algebras, Van Nostrand, 1960. [22] H. L. Royden, Real analysis, Macmillan, 1963. [23] W. Rudin, Fourier-stieltjes transforms of measures on independent sets, Bull. Amer. Math. Soc. 66 (1960), 199–202. [24]

, Fourier analysis on groups, Interscience, 1962.

[25]

, Real and complex analysis, McGraw-Hill, 1966.

[26] N. T. Varopoulos, Tensor algebra and harmonic analysis, Acta Math. 119 (1967), 51–111. [27] N. Wiener, The fourier integral and certain of its applications, Cambridge, (1967), 1933. [28] A. Zygmund, Trigonometric series, 2nd ed., Cambridge University Press, 1959.

Index

Almost-periodic functions, 170–184, 206 pseudo-measure, 181 Analytic measure, 100

Dini’s test, 63 Distribution function, 73 Ditkin-Shilov, 240 Divergence sets of, 64 Dual group, 204

Banach algebra, 209 homomorphism, 228 normal, 238 regular, 238 self-adjoint, 224 semisimple, 222 Bernstein, 33 inequality, 17, 48 Bessel’s inequality, 29 Beurling-Helson, 231 Blaschke product, 97 Bochner, 109, 150 Bohr, 182 compactification, 207 group, 207

Ergodic theorem, 41 Fejér kernel, 12 means, 13 Fejér’s lemma, 17 Fejér’s theorem, 18 Fourier coefficients of integrable functions, 2 of linear functionals, 35 Fourier transforms on LCA groups, 205 Fourier-Gelfand transform, 221 Fourier-Stieltjes coefficients, 38 series, 38 Function algebra, 218 normal, 236 regular, 236

Carleson, 68 Cesàro means, 13 Character, 203 Characteristic function, 153 Complete orthonormal system, 29 Conjugate function, 71 Poisson kernel, 88 Convolution, 4, 42, 135, 144 of almost periodic functions, 179 of pseudo-measures, 166

Gauss’ kernel, 140 Gaussian variable, 153 Gelfand representation, 220 Gelfand–Mazur, 215 Haar measure, 202 Hardy, 84, 103 spaces, 93 Tauberian theorem, 61

de la Vallée Poussin’s kernel, 15 deLeeuw, 278 Difference equation, 53

285

286


Hausdorff-Young, 111, 132, 157, 206 Herglotz’ theorem, 39 Homogeneous Banach space, 14 Hull-kernel topology, 237 Hunt, 68 Ideal maximal, 213 regular, 215 Jackson kernel, 22 theorem, 49 Jensen’s inequality, 94 Kahane, 278 Kaufman, 199, 275 Kolmogorov, 68 Kronecker set, 199 Kronecker’s theorem, 69, 196, 205 Lacunary (Hadamard), 117 LCA group, 201 Lebesgue, 20 Littlewood, 84 Localization principle of, 63 Locally compact abelian group, 201 Lusin, 68 Malliavin, 245 Marcinkiewicz, 152 Maximal conjugate function, 88 function, 84 Maximal-ideal space, 220 Multiplicative linear functional, 213, 216 Multipliers on Fourier coefficients, 52 universal, 41 Normal variable, 153

Paley–Wiener, 188 Paley–Zygmund, 280 Parseval, 30, 145, 157 Parseval’s formula, 36 Partition of unity, 236 Plancherel, 156 Poisson kernel, 16 summation formula, 142 Poisson-Jensen, 94 Pontryagin duality theorem, 204 Positive definite ˆ , 150 on R sequences, 39 Pseudo-measures, 166 Quasi-analytic class, 127 Rademacher functions, 276 Radical, 222 Resolvent set, 214 Riemann-Lebesgue lemma, 136 Riesz products, 120 Riesz, F., 100 Riesz, M., 81, 100 Riesz-Thorin, 108 Rudin–Shapiro polynomials, 34 Spectral norm, 222 Spectral synthesis, 244 Spectral theorem, 40 Spectrum, 222 norm, 174 weak-star, 184 Summability kernel, 9, 203 positive, 10 Support of a distribution, 46 of a tempered distribution, 164 Symbolic calculus, 250 individual, 257 Tempered distribution, 162 Tensor products, 264

I NDEX Titchmarsh’s convolution theorem, 193 Trigonometric polynomial degree of, 2 frequencies of, 2 on R, 174 on T, 2 Uniqueness theorem, 36, 138 Van der Corput’s lemma, 234 Varopoulos, 264 Vitaly’s lemma, 84 Weierstrass Approximation Theorem, 15 function, 119 Wiener continuous measures, 45 Tauberian theorem, 241 theorem, 217 Wiener-Levy, 225 Wiener-Pitt, 261 Zygmund, 33

287

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