An Introduction to Harmonic Analysis (second corrected edition)

AN INTRODUCTION TO HARMONIC ANALYSIS Yitzhak Katznelson INSTITUTE OF MATHEMATICS THE HEBREW UNIVERSITY OF JERUSALEM Se...

Author: Yitzhak Katznelson

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AN INTRODUCTION TO

HARMONIC ANALYSIS Yitzhak Katznelson INSTITUTE OF MATHEMATICS THE HEBREW UNIVERSITY OF JERUSALEM

Second Corrected Edition

DOVER PUBLICATIONS, INC.

NEW YORK

Copyright © 1968, 1976 by Yitlhak KatzncIson. All rights rescrved under Pan Amcrican and International Copyright Comentions. Publhhed in Canada by General Publishing Compdny, Ltd., 30 Lesmill Road, Don l\1i11s, Toronto, Ontario. Published in the United Kingdom by Constable and Company, Ltd., 10 Orange Street, London WC 2.

This Do\cr edition, first published in 1976, is an unabridged and. corrected republication of the work originally published by John Wiley & Sons, Inc., New York, in 196R

Manufactured in the United States of America Dover Publications, Inc. 180 Varick Street New York, N.Y_ 10014

Preface

Harmonic analysis is the study of objects (functions, measures, etc.), defined on topological groups. The group structure enters into the stud)"" 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. 1n trying to make it clearer, one can proceed in various wayst; 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 concrcte 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 [I], [7]. [8J, and [28J 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 t Hence the indefinite article in the title of the book. vii

viii

An Introduction to Harmonic Analysis

Banach algebras. I t 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 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 boundedness" 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 o,f 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 > I, 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 II I 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 intu chJptcrs, scction~, cll1d subsections. 1 he chapter numbers are denoted by roman numerals and the sections and

Preface

ix

subsections, as well as the exercises, by arabic numerals. I n cross references within the same chapter, the chapter number is omitted; thus Theorem 111.1.6, which is the theorem in subsection 6 of Section I of Chapter III, is referred to as Theorem 1.6 within Chapter III, and Theorem 111.1.6 elsewhere. The exercises are gathered at the end of the sections, and exercise V.I.I is the first exercise at the end of Section I, 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 (~). 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 1 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 10 the production of this book. YI fLIIAK KAfZNELSON

Jerusalem April 1961l

Contents

chapter I

Fourier Series on T I. FOURIER COEFF1CIENTS

2

2. SUMMABILITY IN NORM AND HOMOGENEOUS BANACH SPACES ON T 3. POINTW1SE CONVERGENCE OF

u,,(f)

8 17

4. THE ORDER OF MAGNITUDE OF FOURIER COEFFICIENTS

22

5. FOURIER SERIES OF SQUARE SUM MABLE FUNCTIONS

27

6. ABSOLUTELY CONVERGENT FOURIER SERIES 31

chapter II

7. FOURIER COEFFICIENTS OF LINEAR FUNCTIONALS

34

The Convergence of Fourier Series

46

I. CONVERGENCE IN NORM

46

2. CONVERGENCE AND DIVERGENCE AT A POINT

51

*3.

SETS OF DIVERGENCE

55

chapter II I The Conjugate Function and Functions

Analytic in the Unit Disc

62

l. THE CONJUGATE FUNCTION

62

*2. THE MAXIMAL FUNCTION OF HARDY AND LITTLEWOOD

73

*3. THE HARDY SPACES

81

xi

xii


chapter IV Interpolation of Linear Operators and the Theorem of Hausdorff-Young I. INTER POLA TION OF LINEAR OPERATORS

NORMS

AND

OF 93

2. THE THEOREM OF HAUSDORFF-YOUNG

chapter V

93

98

Lacunary Series and Quasi-analytic Classes

104

I. LACUNARY SERIES

104

*2. QUASI-ANALYTIC CLASSES

112

chapter VI Fourier Transforms on the Line I. FOURIER TRANSFORMS FOR

L '(R)

2. FOURIER-STIELTJES TRANSFORMS 3. FOURIER TRANSFORMS IN

I < p

~

119 120 131

LP(R),

2

139

4. TEMPERED DISTRIBUTIONS AND PSEUDOMEASURES

146

5. ALMOST-PERIODIC FUNCTIONS ON THE LINE

155

6. THE WEAK-STAR SPECTRUM OF BOUNDED FUNCTIONS

170

7. THE PALEY-WIENER THEOREMS *8. THE FOURIER-CARLEMAN TRANSFORM 9. KRONECKER'S THEOREM

chapter VJ] Fourier Analysis on Locally Compact Abelian Groups

173 179 181

186

1. LOCALLY COMPACT ABELIAN GROUPS

186

2. THE HAAR MEASURE

187

3. CHARACTERS AND THE DUAL GROUP

188

4. FOURIER TRANSFORMS

190

5. ALMOST-PERIODIC FUNCTIONS AND THE BOHR COMPACTIFICATION

191

Contents

chapter VlIf

appendix

Commutative Banach Algebras

xiii 194

I. DEFINITION, EXAMPLES, AND ELEME~TARY PROPERTIES

194

2. MAXIMAL IDEALS AND MULTIPLICATIVE LINEAR FUNCTIONALS

198

3. THE MAXIMAL-IDEAL SPACE AND THE GELFAND REPRESENT ATION

205

4. HOMOMORPHISMS OF BANACH ALGEBRAS

213

5. REGULAR ALGEBRAS

221

6. WIENER'S GENERAL TAUBERIAN THEOREM

226

7. SPECTRAL SYNTHESIS IN REGULAR ALGEBRAS

229

8. FUNCTIONS THAT OPERATE IN REGULAR BANACH ALGEBRAS

235

9. THE ALGEBRA M(T) AND FUNCTIONS THAT OPERATE ON FOURIER-STIELTJES COEFFICIENTS

244

10. THE USE OF TENSOR PRODUCTS

249

Vector-Valued Functions

257

Bibliography

261

Index

26]

Symbols

Page

1 2 3 5 12 I3

14 15 16 25 31 34 36 37 47 48 50 55 65 73 82 84 101

104 112 116

Symbol

Page

R,T,Z Ll(T) j(n), sen, S[n !*g Kn, uif), un(j, I) Sn(f), Dn C(T), C(T), U(T) Vn(t), p(r, I) L OO(T), Lip",(T), lip",(T) w(j, h), fl(f, II) A(T) S[P] Sn
mix) Mil) hi!, r) liP, N

IIII~IP 8A> (Ll(T»" C*{Mn}, C#{Mn} .,H(r)

xiv

120 121 123 124 126 127 131 146 147 149 155 156 157 159 160 162 164 187 194 195 205 207 222 223 249

Symbol

Ll(R)

It

A(It) , CoeR) K;., K(,")

V;., p;. 2, 2c M(R) .:FU S(R), S*(R) 1:{v) A{r.,J), AP(R) Wo(f) W(f) a(h) MU)
D

C C(X). Co(X)

9Jl jj

h(l) k(E) A(E)

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/2nZ where, as indicated by the notation, 2nZ is the group of the integral multiples of 2n. There is an obvious identification between functions on T and 2n-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/is integrable on T if the corresponding 2n-periodic function, which we denote again by f, is integrable on [O,2n) and we set ITf(t)dt

= fo2"

f(x)dx.

In other words, we consider the interval [O,2n) as a model for T and the Lebesgue measure dt on T is the restriction of the Lebesgue measure of R to [O,2n). The total mass of dt on T is equal to 2n 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/2n. 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 1/2n in front of every integral. An all-important property of dt on T is its translation invariance. that is, for all to E T and f defined on T,

f

f(t - to)dt

=

f

f(t)drt

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

2


1. FOURIER COEFFICIENTS 1.1 We denote by Ll(T) the space of all complex-valued, Lebesgue integrable functions on T. For f E Ll(T) we put

I f IILI = 2in IT /J(t) / dt , It is well known that JJ(T) , with the norm so defined, is a Banach space. DEFINITION:

A trigonometric polynomial on T is an expression of

the form (1.1)

The numbers n appearing in (1.1) are called the frequencies oj P; the largest integer n such that a" / + a_" / :;f 0 is called the degree of P. The values assumed by the index n are integers so that each of the summands in (Ll) 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 lET by

I

I

2 N

pet)

(1.2)

=

anint.

n= -N

Let P be defined by (1.2). Knowing the function P we can compute the coefficients an by the formula (1.3) which follows immediately from the fact that for integers j,

-f

I..

2n

e'Jtdt

1 if j = 0

={ 0 if

j:;f

o.

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.

1. Fourier Series on T

1.2

3

A trigonometric series on T is an expression of

DEFINITION:

the form

2: ane ,.=-00 Q()

S ~

(1.4)

int

•

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 conjugatet S of the series (1.4) is, by definition, the series

L - i sgn (n)a e 00

S~

int

n

"= -00 where sgn (n)

=0

if

11

=0

and sgn(n)

= nIl n I otherwise.

1.3 Let fELi(T). Motivated by (1.3) we define the nth Fourier coefficient of f by j(n) = 2in

(1.5)

J

f(t)e-intdt.

DEFINITION: T he Fourier series S[jJ of a function j trigonometric series

2:

E

Li(T) is the

00

S[J] ~

inr

j(n)e

•

n= -00

The series conjugate to S[J] will be denoted by S[j] and referred to as the conjugate Fourier series of f. We shall say that a trigonometric series is a Fourier series ifit is the Fourier series of some j E Li(T), We turn to some elementary properties of Fourier coefficients.

1.4

Theorem:

Let j,gELi(T), thell

.

~

(a) (f + g)(n) = j(n) + g(n). (b) For any complex /lumber rx, /'-.,

(rxf)(ll)

= rxj(n).

(c) If J is the complex cOlljugatet of f then /(n) = (d) Denoteft(t)=f(t-T), TET; then h(n) = !(n)e- inr

•

t See,chapter III for motivation of the terminology.

t

Defined by:

jet)

= f(t) for all lET.

!( -n).

4


The proofs of (a) through (e) follow immediately from (1.5) and the details are left to the reader. AssumefjELJ(T),j = 0,1, ... , and !lfj-foIlL'-+O. Then jin) -+ Jo(n) uniformly.

1.5 Corollary: 1.6

Theorem:

LeI fE O(T), assume J(O) = 0, and define

F(/)

= L'f(') dr.

Theil F is continuolls, 2n periodic, and

•

(1.6)

1 •

F(n) = -:- f(l!) , III

n:;z!'O.

The continuity (and, in fact, the absolute continuity) of F is evident. The periodicity follows from PROOF:

F(t

+ 2n;) -

F(/)

J

I+2~

=,

(,)d,

= 2n/(0) = 0,

and (l.6) is obtained through integration by parts: F(n)

i

2~ . - 1 F(t)e-m'dt = -2non

1 = 2-

f

0

2"

1

.

F'(/)--;- e-m1dt • -In

= -;-1 J(n).

...

In

1.7 We now define the convolution operation in Lt(T). The reader will notice the use of the group structure of T and of the invariance of dt in the subsequent proofs. Theorem: Assume f,gE OCT). For almost all t, the function f(t -T)g(r) is integrable (as a (unction of T on T), and, if we write (1.7)

h(t) =

2~

f

f(t - T)g(T)d"

thell hE OCT) and

(1.8)

II h III.• ~ II f

/lL' /I g /ILl.

Moreover (1.9)

Ii(n) = J(n),~(n)

for all n.

I. Fourier

5

Seric~ on T

The functions f(t - T) and g(T) , considered as functions of the two variables (f,.), are clearly measurable, hence so is PROOF:

F(t,.) =f(t-T)g(.).

For almost all " F(t,.) is just a constant multiple off" hence integrable, and

Hence, by the theorem of Fubini, f(t - .)g(r) is integrable (over (0,2n» as a function of T for almost all t, and

~J Ih(t)ldt=~J I 2n~J F(t,.)d'ldt~~ifIF(t")ldtd.= 2n 4n

2n

which establishes (1.8). In order to prove (1.9) we write Ii(n)

= ~ J h(t)e- int dt = ~ i f f(t - T)e-in(t-r)g(T)e-inrdtdT = 2n

4n

As above the change in the order of integration is justified by Fubini's theorem. ... 1.8

DEfiNITION:

The convolution f

* g of the (Ll(T»

functions f and

g is the function h defined by (1.7).

Using the star notation for the convolution, we can write (1.9): (1.10) i..

f-;--g(n) = j(n)g(n).

Theorem: The convolution operation in VeT) is commutative, associatire, alld distributive (with respect to the addition). PRoor:

The change of variable -8 ;n J

that is.

f(l~)g(t -

() d?9

= fI

= 2n J

•

gives

get - fJ)f(fJ)d{).


6

If /lJ /2'/3 ELI(T), then [Ui· /2)· /3](t)

=

=

4!Z

4:2 I I /I(t - u - r;)/z(u)fir;)du dr;

II/l(t-W)fz(w--r)fir;)dwd-r=(Jl*(f2./J)](t).

Finally, the distributive law fl.U2 +/3) =fl.f2 +fl·f3

is evident from (1.7). 1.9 Lemma: AssumefEL1(T) and let ff(t) =einl for some integer n. Theil

PROOF:

(fP·/)(t) =

Corollary:

~

2n

J

ein(t- 0 ijt can have more than 2n zeros on T. Hint: Identify on T with z-nL~najZn+j on Izl = 1. 7. Denote by C* the multiplicative group of complex numbers different from zero. Denote by T* the subgroup of all z E C* such that I z I = 1. Prove that if G is a subgroup of C* which is compact (as a set of complex numbers), then G £ T*.

L:.aje

An Introduction to Harmonic Analysis 8. Let G be a compact proper subgroup of T Prove that G is finite and determine its structure. Hint: Show that G is discrete. 9. Let G be an infmite subgroup of T. Prove that G is dense in T. Hin/: The closure of G in T is a compact subgroup. 10. Let a be an irrational multiple of 2n. Prove that {na(mod2n)}n=0.± 1.±2 .... is dense in T. II. Prove that a continuous homomorphism of T into C* is necessarily given by an exponential function. Hint: Use exercise 7 to show that the mapping is into T*; determine the mapping on "small" rational multiples of 2n and use exercise 9. 12. If E is a subset of T and '0 ET, we define E -t- '0 = {I + '0; lEE}; we say that E is invarianl under Iram,lution by , if E = E +r. Show that, given a set E, the set of rET such that E is invariant under translation by T is a subgroup of T. Hence prove that if E is a measurable set on T and E is invariant under translation by infinitely many rET, then either E or its complement has measure zero. Hinl: A set of E of positive measure has points of density, that is, points T such that (2cf liE n (T - C, T + c) 1--+ 1 as c --+ O. (I Eo 1denotes the Lebesgue measure of Eo.) 13. If E and F are subsets of T, we write E+F={t+r;/EE, rEF}

and call E + F the algebraic sum of E and F. Similarly we define the sum of any finite nwnber of sets. A set E is called a bUJis for T if there exists an integer N such that E + E + ... + E (N times) is T. Prove that every set E of positive measure on T is a basis. H in!: Prove that if E contains an interval it is a basis. Using points of density prove that if E has po~itivc measure then E + E contains intervals. 14. Show that measurable proper subgroups of T have measure zero. 15. Show that measurable homomorphisms of T into C* map it into T*. 16. Let f be a measurable homomorphism of T into T*. Show that for all values of II, except possibly one value.1(II) = O.

2. SUMMABlLlTY IN NORM AND

HOMOGENEOUS

BANACH

SPACES ON T

We have defined the Fourier series of a function fEL\T) as a certain (formal) trigonometric series. The reader may wonder what is the point in the introduction of such formal series. After all, there

2.1

I. Fourier Series on T

9

is no more information in the (formal) expression L~ _ ooJ(n) e1nt than there is in the simpler one U(n)}~=-oo or the even simpler J with the understanding that the function] is defined on the integers. As we shaH see, both expressions, ](n)eint and j, have their advantages; the main advantages of the series notation being that it indicates the way in Vvhich f can be reconstructed from J. Much of this chapter and all of chapter 11 will be devoted to clarifying the sense in which LJ(1I) eint represents f. In this section we establish some of the main facts; we shall see that j determines f uniquely and we show how we can find f if we know j. Two very important properties of the Banach space LI(T) are the following:

L

(H-I') 1f feU(T) and rET, then fit)

=

j (t - r) E Ll(T)

(H-2') The V (T)-valued function for fEV(T) and ToET (2.1)

lim

and

T -+ ft

11ft - fro

IILI

11ft

IILI

=

I f IILI

is continuous

=

011

T, that is,

O.

t-+to

We shall refer to (H-I') as the translation invariance of LI(T); it is an immediate consequence of the translation invariance of the measure dt. In order to establish (H-2') we notice first that (2.1) is clearly valid iff is a continuous function. Remembering that the continuous functions are den~e in LI (T), we now consider an arbitrary / e Ll(T) and e > O. Let g be a continuous function on T such that

h - f IILI < 1:/2; thus 11ft =

fto

IILI ~ I ft -

I (f -

Hence lim

IILI + I gt - gto IILI + I gto - fro IILI = g)t 11,.' + I gt - gro IILI + II (g - f)ro IILI < e + I gt - gto IILI. Iii fto ilL' < I: and, e being an arbitrary positive number, gt

t -

r-to

(H-2')

~s

established.

2.2 DEFII'ITJOI': A summability kernel is a sequence {k n } of continuous 21r-periodic functions satisfying: (S-l)

~Jk(t)dt=l 217 n

lO


2~

(S-2) (S-3) For all 0 < 0
0 so that (2.3) is bounded bye, and keeping this 0, it results from (S-3) that (2.4) tends to zero as n -+ co so that (2.2) is bounded by 2e for large n. ...

2.3 For fe VeT) we put rp(T) = f.(t) = f(t - T), then by (H-I') and (H-2') rp is a continuous L1(T)-valued function on T and f{l(0) = f. Applying lemma 2.2 we obtain Theorem:

Let feL1(T) and {k.} be a summability kernel; then

(2.5)

f

=

lim 21

n~OO

7t

f

k.(T)f.dT

ill the LI(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. The ambiguity, however, is harmless.

Lemma: Let k be a continuous function on T and feL1(T). Then (2.6) PROOF:

Assume first that f is continuous on T. We have (see Ap-

pendix)

1f l . "

2rr

k(T)f.dT

=

2rr hm L,.- (Tj+l - Tj)k(T)fTj' J

the limit being taken in the LI(T) norm as the subdivision {Tj} of [0,2n) becomes finer and finer. On the other hand,

uniformly and the lemma is proved for continuous f. For arbitrary fe OCT), let e >0 be arbitrary and let g be a continuous function on T such that II f - g IILI < e. Then, since (2.6) is valid for g,

2~

f

1

k(T)frdr: - k*f= 2n:

f

k(T)(f-g)rdr:

+ h(g-f)

12


and consequently

Using lemma 2.4 we can rewrite (2.5):

I

(2.5')

=

lim k.

*1

in the LI(T) norm.

2.5 One of the most useful summability kernels and probably the best known is Fejer's kernel (which we denote by {K.}) defined by

K"(t)

(2.7)

=

'"

=-.

.L

J

(1 - ~) e 11+1

iiI.

The fact that K. satisfies (S-l) is obvious from (2.7); that Kit) and that (S-3) is satisfied is clear from

Lemma:

. n

~

0

T1 }2

SIn .-2-t

K.(t) PROOF:

{ SIn

tt

Recall that • 2

(2.8)

n +1

Sin

-

t = -1(1 2

2

COS

t) = - -1e -it 4

+ -21 - -41e it .

A direct computation of the coefficients in the product shows that

, We shall adhere to the generally used notation and write a.(/) and a.(J, t) = (K. * f)(t). It follows from corollary 1.9 that (2.9)

ail,t)

=

=

K. *1

~ (1 - nl~II)J(j)eiit.

2.6 The fact that an(f)~1 in the Ll(T) norm for every feLJ(T), which is a special case of(2.5'), and the fact that an (/) is a trigonometric polynomial imply that trigonometric polynomials are dense in OCT).


13

Other immediate consequences are the following two important theorems.

2.7 Theorem: (The Uniqueness Theorem): Let ](n) =0101' all n. Then I = O. By (2.9) a.U)

PROOF:

that 1=

=

IE Ll(T) and assume

0 for all n. Since a.U) -+ I, it follows

o.

~

An equivalent form of the uniqueness theorem is: Let and assume j (n) = g(n) for aJl 11, then I = g.

2.8 Theorem: (The Riemann-Lebesgue Lemma): then liml.l-+oo](n) = O.

l,gE Ll(T)

LeI IELI(T),

PROOF: Let e > 0 and let P be a trigonometric polynomial on T stich that /I I < e. If I n 1 > degree of P, then

pilL'

I](n) 1

= I (J-:>P)(n) I

~ I 1- P /ILl < e.

~

Remark: Jf K is a compact set in Ll(T) and e > 0, there exist a finite nunlber of trigonometric polynomials PI> ... , PN such that for every IE K there exists a j, I ~ j ~ N, such that Pj < e. If Inl is greater than maXI~j~N (degree of P) then 1](n)1 <e for all f E K . Thus, the Riemann-Lebesgue lemma holds uniformly on compact subsets of Lt(T).

II! - IILI

2.9 For IELl(T) we denote by S.U) the nth partial sum of S[J], that is, (2.10)

(S.(f»)(t)

~= S.U, I)

=

i

jU)eijt •

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

a.U)

1 =

--

n+

-1 (SoU) + StU) + ... + S.U» ,

in other words, the a.U) are the arithmetic means of S.U). It follows that if S[J] convergest in Ll(T) , then the limit is necessarily I· From coroJlary 1.9 it follows that SnU) = D" *1 where D" is the Dirichlet kernel defined by (2.12)

~

D (t) •

=

~ eijt

~

t That is, if SnU) converge as

=

~n (~__±j2.~. SIn

II

-+ CO.

it

14


It is important to notice that {Dn} is not a summability kernel in our sense. It does satisfy condition (S-l); 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.tO DEFINITION: A homogeneous Banach space 01/ T is a linear subspace B of L'(T) having a norm I lin ~ I IILI under which it is a Banach space, and having the following properties:

lin

(H-t)

IffEB and tET, thenfTEB and lifT

(H-2)

J.(t) = J(t For all fEB,
(3.7)

which proves part (a). Part (b) follows from the uniform continuity off on 1; we can pick [} so that (3.5) is valid for all to E I, and no depends only on {) (and e). Part (c) depends only on the fact that K.(t) ~ 0; if m ;;;; f then (1,,(j,I) -

m

=

;n:

f

K,,(r)(j(t - r) - m)dr

~ 0,

the integrand being nonnegative. If f;;;; M then M-(1,,(f,t)=;n:f K.(r)(M-f(t-r»dr

~

0,

for the same reason.

Corollary: If to is a poin! of continuity of f and if the Fourier series of f converges at to then its sum is f(to) (cf. 2.9).

3.2

Fejer's condition v

j(t o)

=

+

f(to h) + f(to - II) lim - - - 2 - h-O

implies that (3.8)

lim h .... O

~ fh I f(~ -+ r) _2~lV~ ~ -~) - /(to) I d't h

0

=

°.

Requiring the existence of a number fit) such that (3.8) is valid is far less restrictive than Fejer's condition and more natural for sum mabie functions. It does not change if we modify f on a set of measure zero and, although for some function f Fejcr's condition may hold for no value to, (3.8) holds with ](to) = f(to) for almost all to (cf. [28], Vol. 1, p. 65).


20

Theorem (Lebesgue): If (3.8) holds, then (T"U,/ o)-+1(10 ), In particular (T"U, t) -+ f(t) almost everywhere. As in the proof of F6jer's theorem,

PROOF:

(3.9) (TltU,/ o) -l(to) =MLiJo + .

Remembermg that K,,('t) for 0 < 't

< 1t,

=

fo") Klf) [f(to-'t)+!(to+'t)

1 (Sin(n + 1)'t/2)' 2 --1 ---'-/2- and that

n

+

sm't

(

n + 1, en +7t 1)i2)

J(t o)] d't. •

't

SID -

2

't

> -

1t

we obtain 2

• I

obtain _1t_

(3.11)

2

we

rli If(to + :L+f(to - :) _ J(t o) I d: =

n +1 )

__ 1t_

- n

+1

't

2

1/"

[1I>('t)] li 't 2 JIll

+ ~ n +1

r

)

0 1I>('tl d't 't 3 •

II"

For ll> 0 and n > nell) we have by (3.8) ({>('t)
0 (4.1) Then there exists a /lonnegative fut/ctioJl

IE Ll(T) such

that

J(II) =

an'


23

PROOF: We remark first that the convexity condition (4.1) implies that (an - an+ I) is monotonically decreasing with n, hence

lim n(a n

0,

-

an + 1)

=

a o - aN -

=

,,-+00

and consequently N

L

n(a n - I

+ a n + 1 -2a n )

N(a N - aN+ I)

n=1

converges to ao as N -). (1). Put 00

f(l) =

(4.2)

L u(a

n- 1

+an + 1

-

2a.) Kit-IV),

n=1

Kn denoting, as usual, the Fejer kernel of order n. Since

I Kn IILI =

1, the series (4.2) converges in LI(T) and, all its terms being nonnegative, its limit f is nonnegative. Now 00

l(j)

=

L

n(a"_1 +a n +l - 2a n )

Kn - 1(j)

n= J

and the proof is complete. 4.2 Comparing theorem 4.1 to our next theorem shows the basic dilference between sine-series (a -n = -an) and cosine-series (a -n = an)·

Theorem:

Let rEVeT) lind assume that

1(1,,1)

=

-J(-IIlI)~o.

Tlrell

'":" -f(n) 2, thenfEe(T) and consequently I < co; this is all that we can assert. There exist continuous functions f such 2 that I -. = OCJ for all 8> O.

IJ(nW

1/(,,)1


26

EXERCISES FOR SECTION 4 l. Given a sequence {ca.} of positive numbers such that ca. - 0 as In 1_ 00. show that there exists a sequence {a.} satisfying the conditions of theorem

4.1 and

for all

11.

L

2. Show that if IJ(n) I I nl 1< 00, then f is [-times continuously differentiable. Hence, if ](11)= O(jllr k ) where k > 2. and if 1-

'{k-2 [k]-l

k integer

otherwise

then fis [-times continuously differentiable. Remarks: Properly speaking the elements of LI(T) are equivalence classes of functions any two of which differ only on a set of measure zero. Saying that a function IELI(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 1 is analytic on T if in a neighborhood of every toE T, /(t) can be represented by a power series (of the form Ln~oa.(t-/o)n). Show that 1 is analytic if, and only if, f is infinitely differentiable on T and there exists a number R such that

n>O. 4. Show that 1 is analytic on T if, and only if, there exist constants K> 0 and a> 0 such that IJ(j)1 ~ Ke -ulj~ Hence show that 1 is analytic on T if, and only if, J(j) e lj • converges for IIm(z)l< a for some a> O. 5. Let/be analytic on T and let g(e/I) = f(t). What is the relation between the Laurent expansion of g about 0 (which converges in an annulus containing the circle Izi = 1) and the Fourier series of I? 6. Let 1 be infinitely differentiable on T and assume that for II;C I sup,If(II)(t)1 < Kn"l1 with IX> 0, Show that

L

liml 7.Assume and

li(m ~ Kexp( -

$ Kex p ( _

~ /jIl''').

/jIll,,). SholV that

1 is

infinitelyditferentiable


1/",(/)1
M, •

IISN-s~112

02

N

=

~2 a"1',, I M+l

L f{

=

10

, 1 2-+ 0

as M-+oo.

M+l

5.2 Lemma: Let.}(' be a Hilbert space. Let {1',,}:=1 be a finite orthonormal system in .}('. For j E.}(' write a" = (f, tp.). Then

PROOF:

=

N

N

N

N

1

1

1

1

IIfl12 - L a,,(J,f/J.) - 2: o"(tp,,,f) +L la,,12 = Ilfl2- L la,,12....

Corollary (Bessel's inequality): Let.}(' be a Hilbert space and {1'..} an orthonormal system in .}('. For je.}(' write aa. = (/,1'11.)' Then (5.3)

The family {1'.. } 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 {!pa.} we have (5.2). In particular acr = 0 except for countably many values of ex and the series I 10 .. 12 converges. If .}(' = L2(T) all orthonormal systems in .}(' are finite or countable (cf. exercise 2 at the end of this section) and we write them as sequences

{1',,} • 5.3 DEFINITION: A complete orthonormal system in .}(' is an orthonormal system having the additional property that the only vector in .Jf' orthogonal to it is the zero vector.


29

Lemma: Let {9',,} be an orthonormal system in .JF. Then the lollowing statements are equivalent: (a) {9'.} is complete. (b) For every IE.JF we hqve

11/112

(5.4)

1

(c)

=

2 1 O. Then f E A(T).

011

T and as-

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

*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 l. For n = 1,2, ..• , let f,. E A(T) and 1If,. IIA(T) ~ I. Assume that f" converge to I uniformly on T. Show that IEA(T) and II/IIA(T) ~ l. 2. Show that the conditions in exercise I are not sufficient to imply lim,,_oo III- f,. IIA(1) = 0; however, if we add the assumption that

II I

IIA(T) = lim 3. For 0 < a

II 1", IIA(T)' then III - In II.~(T) ~ O. ~ Il

define

Ail)

=

{

1-a- 11 1 1

for

It! ~a

o

for

a ~It! ~ Il

Show that A/JEA(T) and 1I11/JIL-I(T)= l. Hint: i,,(n) ~O for all n. 4. Let IE C(T) be even on (- 1l,1l), decreasing on [O,Il] and convex there (Le.,/(1 + 211) + f(1) ~ 2f{1 + h) for 0 ~ 1 < t + 2h ~ Il). Show that IE A(T) and, if I ~ 0, II/IIA(T) =/(0). Hint: I can be approximated uniformly by positive combinations of A". Compare with theorem 4.1. 5. Let f/1 be a "modulus of continuity," that is, an increasing concave function on [0,1] with q>(0) = 0, Show that if the sequence of integers {,til} increases fast enough and if I(t)

=

~ ~ eUnf

L... n 2

then ro(f,II)¥= O( q>(h» as h ~ O.

ro(f,h) is the modulus of continuity of I (defined in 4.6). 6. (Rudin, Shapiro.) We define the trigonometric polynomials Pm and Qm inductively as follows: Po = Qo = I and Pm + 1(1) = P".(t)

+ ei2mfQm(l)

12 Q",+I(1) = P".(t) - e "" Q",(t).

34


(a) Show that IPm + 1(t)1

2

+IQm+l(t)1

2

=2(IPm(t)1 2 +IQm(t) 12)

hence and

lipmIIc(T) ;;;; 2(m+ I )/2. (b) For I nl < 2"', Pm+1(n) = P",(n), hence there exists a sequence {En}~o such that En is either 1 or -1 and such that Pm(t) = I~'"-l Ene1nl • (c) Write 1m = P",-P",_I = e i2 "'-" Q",-1 and 1= 2- m/",. Show that k 1 k leLipl/2(T) and/¢A(T). Hint: For 2-

N

39

We have Ln,ma"-mz,,zm = Lj Cj,l,Qje iJt where C i .N is the number of ways to write j in the form n - m where I n I ~ Nand 1m I ~ N, that is, C i.N = max (0, 2N + 1 j It follows that

-I I).

CT2N(S,t)

=

2N

1

+1

4

ijt CJ.l',aje

~0

J

and the theorem follows from 7.5. 7.7 An important property of Fourier-Stieltjes coefficients is that of being "universal multipliers." More precisely: Theorem: Let B be a homogeneous Banach space on T and let J.l E M(T). There exists a unique linear operator on B having the following properties:

Jl

I JlII

(i) (ii)

~ II Jt IIM(T)

A

Jlf(n)

•

=

fi(n)f(n) for all fE B.

Jl

If an operator satisfies (ii), then for f = L~N!(n)eint int we have = L~N {l(n)!(n)e , that is, Jl is completely determined on the polynomials in B. If Jl is bounded it is completely determined. In order to show the existence of Jl it is sufficient to show that if we define PROOF:

Jlf

Jlf = ')'

(7.7) for all polynomials

f

{l(n)!(n)i

nt

then

(7.8) since Jl would then have a unique extension of norm ~"It IIM(T) to all of B. If J.l = (1/211:) get) dt with g E C(T), it is clear that Jlf, as defined in (7.7), is simply g *f which we can write as a B-valued integral (see 2.4) (7.9)

Jlf

=

g*f

=

;1[ I g(r)ft dr

and deduce the estimate

I

I JlfliB ;;:; IIfllB ~1[ Ig(r) Idr

=

I JtllMmllfliB'

40


For arbitrary p.eM(T), (j,,(p.) has the form (1/2n)gll(t)dt, where g,,(t) = }::,,(1 j I/(n + 1» AU) eijl and

-I

2~

JI

g,,(t) I dt

=

I (1,,(p.) IIM(T) ~ I p.IIM(T)'

By our previous remark Ilg,,*JIIB ~ 1Ip.IlM(T)IIJ liB, and since I1J = lim" ... 00 gIl * J (remember that J is a trigonometric polynomial) ~o~~~~

~

Corollary: LetJeB and p.eM(T). Then {{i(n)](n)} is the sequence oj Fourier coefficients oj some Junction in B. In view of (7.9) we shall write p. *J instead of I1J, and refer to it as the convolution of p. and J. With this notation, our earlier condition (ii) becomes a (formal) extension of (1.10).

7.8 For p.eM(T) we define p.#eM(T) by (7.10)

/-/(E) = p.( -E)

for every Borel set E (recall that - E by (7.11)

=

{t; - teED, or equivalently,

JJ(t)dp.# JJ( -t)dp. =

for all Je C(T). It follows from (7.11) that --

/".

(7.12)

P.#(II)

=

pen).

7.9 By Parseval's formula, the adjoint of 11 is the operator which assigns to a v e B* the element of B* whose Fourier series is Iii(n)v(n)ei"t = I?(n)ll(n)e illt • We extend the notation of7.7, write this element of B* as p.# * v, and refer to it as the convolution of p.# and v. We summarize:

Theorem: Let B be a homogeneous Banach space on T and B* its dual. Let p.eM(T), veB*; then IA(II)v(n)e int is the Fourier series oj an element p. *V e B*. Moreover II p. *V liB' ~ II J.L IIM(T) II V IIB 0 and integer m, Jim - - - - - = 0 r--+1

(J,21r - J).

uniformly in

Chapter 11

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 Fejer's or Poisson's. As in the case of summability, problems of convergence "in norms" are usually easier than those of pointwise convergence. Many problems, concerning pQjntwise convergence for various spaces, are still unsolved and the convergence almost everywhere of the Fourier series of square .summable functions was pIoved only recently (L. Carleson 1965). Convergence seems to be closely related to the existence and properties of the so-called 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 ITI. ]. CONVERGENCE IN NORM

1.1

Let B be a homogeneous Banach

(1.1)

Sif)

=

Sn(f, t)

=

~pace

on T. As usual we write

2:" l(j) iiI. -"

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

lim

"-00

1/

SnCf) - f liB

=

O.

Our purpose in this section is to characterize the spaces B which

have this property. 46

II. The Convergence of Fourier Series

47

We have introduced the operators Sn:f -+ Sin in chapter I. Sn is well defined in every homogeneous Banach space B; we denote its norm, as an operator on B, by I Sn I B • Theorem: A homogeneous Banach space B admits cOllvergence in norm if, and only if, I Sn I B are bounded (as n -+ <Xl), that is, if there exist a constant K such that

(1.3)

for all feB and n

~

0,

PROOF: If Slf) converge to f for all fe B, then Sif) arc bounded for every fEB. By the uniform boundedness theorem, it follows that I S~ liB = 0(1). On the other hand, if we assume (1.3), let fEB, B > 0, and let P be a trigonometric polynomial satisfying Ilf - P liB ~ s/2K. For If greater than the degree of P, we have Sn(P) = P and hence

I Sif) - f liB

=

I Sn(f) -

SiP)

+ P - f liB

~ IISn(f-P)IIB+llp-fIIB~K2~ t-2~-~e. 1.2 The fact that Sif) (1.4)

D (1) n

=

Dn * f, where Dn is the Dirichlet kernel

=~,

~

yieJdsa simple bound for 1 Sn it follows that

~

eijt = sin ~n~ !~ SID

t/2

liB. Tn fact, since I Dn*flIB ~ I Dn IlL' Ilfll8'

(1.5)

D"IILI

The numbers Ln = 1 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 = Lf(T) the inequality (1.5) becomes an equality. This can be seen as follows: as in chapter I we denote by Kn the F6jer kernel and remember that I KN I Ll = 1. We clearly have lI[,ItT ) ~ I SnCKN) 111.1 = I CfN(Dn) and since (1N(Dn)-+Dn as N -+00, we obtain

liS"

ilL'

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


48

1.3 In the case B = e(T), convergence in norm is simply uniform convergence. We show that Fourier series of continuous functions need not converge uniformly by showing that ~ S" IIC(T) are unbounded; more precisely we show that S" = L". For this, we consider continuous functions "'.. satisfying

I Ile(T)

""'.1100

=

sup r

,,,,.(t)1 ~ 1

and such that "'it) = sgn(D.{t» except in small intervals around the points of discontinuity of sgn(D.(t». If the sum of the lengths of these intervals is smaller than e/2n. we have

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 Ianeintto be the series -i Isgn(n)a.e i • t. If/eL1(T) and if the series conjugate to Il(n) einr is the Fourier series of some function g e Ll(T) , we call g the conjugate lunction ofI and denote i! by J This definition is adequate for the purposes of this section; however, it does not define I for all Ie Ll(T) and we shall extend it later. DEFINITION:

for every IE B,

A space of functions B c Ll(T) admits conjugation if is defined and belongs to B.

I

If B is a homogeneous Banach space which admits conjugation, then the mapping I -+ lis 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 I -+ I is closed, that is, that if linif. = I and lim!.. = g in B, then .g = I. This follows from the fact that for every integer j g(j)

=

lim f.(j)

=

lim -

j

sgn (j)l. (j) =

-

i sgn (j) lim lnU)

.-+00

= -

i sgn (j)l(j)

If B admits conjugation then the mapping 00

(1.6)

=

.-+00

1-+ P

=

tl(O) + ·HI + if) '"

L j(j) eiJt o

=

lU) .

II. 'the Convergence of Fourier Series

49

is a well-defined. bounded linear operator on B. Conversely. if the mapping f -TP is well-defined in a space B. then B admits conjuga. tion since 1 = - i(2P - f - 1(0».

Theorem: Let B be a homogeneous Banach space on T and assume that for feB and for all n. t!"'feB and (1.7)

Then B admits conjugation if, and only if. B admits convergence in norm. PROOF: By theorem 1.1 and the foregoing remarks, it is clearly sufficient to prove that the mappingf -T is well defined in B if, and only if. the operators S" are uniformly bounded on B. Assume first that there exists a constant K such that ~ S.IIB ~ K. Define

r

(1.8) by (1.7) we have ~ S!~B ~ K. Let feB and I: > 0; let P e B be a trigonometric polynomial satisfying ~f-pIIB ~ t/2K. We have (1.9)

I S~W -

S~(P) liB

=

I S~U -

P}

II. ~ i·

If nand m are both greater than the degree of p. s!(P} it follows from (1.9) that

I S!(f) - s!.(f) ~. ~

= S!,(P) and

1:.

The sequence S:(f) is thus a Cauchy sequence in B; it converges and its limit has the Fourier series L;J(j) eijt • This means = lim S!W E B.

t

Assume conversely that

f ..... f~ is well defined, hence bounded, in

8. Then

S!W

= f~

-

i(2"+1)t(e-1I2.. +1)'l)~

S!

which means that II liB is bounded by twice the norm over B of the mapping f -T f~ . Since, by (1.1) and (1.8), I Sn liB = I S! liB, the theorem ~ follows. 1.5 We shall see in chapter III that, for 1 < P < conjugation, hence:

00,

e'(T) admits


50

Theorem: For 1< p < 00, tile Fourier series of every fE I!(T) converges to f in the I!(T) norm. EXERCISES FOR SECTION I

I. Show that the Lebesgue constants L" L"

Hint:

=

2

4/1t log n

=

liD"

ilL'

satisfy

+ 0(1).

t

L =~ "

1t

(" / s~n(~2}2'

Jo

stnlj2

/dl = ~ ~ 1t

J7t

remember that 0+ I)"

"+ 1/2

J

I sin(n+!)I1 dt

In ~-+112

2. Show that if the sequence Fourier series of the function

{Ni } tends to infinity fast enough, then the

.L 200

((t) =

iKN/t)

1

does not converge in 1}(T). 3. Let {an} be an even sequence of positive numbers,convex on (0, 00) and vanishing at infinity (cf. 1.4.1). Prove that the partial sums of the series La"eint are bounded in Ll(T) if, and only if, anlogn = 0(1) and the series converges in LI(T) if, and only if, lim an logn = O. 4. Show that B = Cm(T) does not admit convergence in norm. Hint: S. commute with dcrivation. 5. Let 'I' be a continuous, concave (i.e., rp(h) -1- rp(h + 20) ~ 2rp(h + 0», and increasing function on [0, I], satisfying !p(O) = O. Denote by A. the subspace of C(T) consisting of all the functions I for which w(f,h) = O( rp(h» as h-+o and by Aop the subspace of Aop consisting of all the functions I for which w(J,h)=o«p(h» as h-+O. (w(f.h) is the modulus of continuity of I; see 1.4.6.) Consider the following statements: (a) rp(h) = O( - (logh)_l) as h -+ O. (b) For every IE ). 0 there exists a A. > 1 such that

(25')

lim sup ,,-' co. infinitely often for every teE. We now pick a sequence of integers {.tj} such that (3.6)

and then integers Pj such that (3.7)

CO"l

> 2 sup S*(O')i/), I) t

and write P j = VpJ + t *(f - 0').1(f» where as usual VI' denotes de la Vallee Poussin's kernel (see 1.2.13). It follows immediately from (3.6) that 1/ Pj B < 00. If 1 E E and n is an integer such that S,,(J, I) > COn' then for some j, Pj < n ~ Pj+ 1 and

I

L

SiP}>t)

I

=

SIt(f - O')'J(f),t)

I

Hence, by (3.7), IS,,(Pj,t) >

1COn'

=

I

S,,(/,t) - S ..(O').//),t).

and (3.5) follows.

Theorem: Assume (3.4). Let Ej be sets of divergence for B, j = 1,2, ... ; then E = U j~ I Ej is a set of divergence for B 0

Let {p!} be the sequence of polynomials corresponding to Omitting a finite number of terms does not change (3.5), but permits us to assume LJ... p!IIB < 00 which shows, by the lemma, that E is a set of divergence for B. ~ PROOF:

Ejo

I

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 (j. There exists a trigonometric polynomial rp such that S*( 9',1) >

(3.8)

;n

log

(3~)

on

E

1/9' 1100 ~ 1.

It is convenient to identify T with the unit circumference {z; z = I}. Let I be a (sma))) interval on T, 1 = {elt , t ~ e}; the function "', =(1 +e- ze-itO)-l has a positive real part throughout the unit disc, its real part is larger than 1/3e on I, and its value at the origin (z =0) is (1 +e)-I. We now write E S;U~Ij' the I j being sma)) intervals of equal length 2& such that Ne < (j, and consider the function PROOF:

II

I

tol

58


1/1 has the following properties: Re(I/I(z) > 0

1/1(0)

(3.9)

II/I(z) I ~ Re(I/I(z»

Iz I ~ 1

for =

1 1

1

> 3Nc > 3c5

on E.

The function log 1/1 which takes the value zero at z = 0 is holomorphic in a neighborhood of {z; z ~ 1} and has the properties

II

I1m (log I/I(z» I < Ilogl/l(z) I > log(3c5)-1

on T

1t

(3.10)

on E.

Since the Taylor series of log 1/1 converges uniformly on T. we can take a partial sum (z) = I ~ a.z· of that series such that (3.10) is valid for in place of log 1/1. We can now put g;(t)

=

!e-iMtlm((eit» 1t

=

~e-iMt( '5' a 27ft......., I

e int n

'5'

(ine-int)

.l.....-

I

and notice that

Theorem:

Every set of mea~ure zero is a set of divergence for C(T).

If E is a set of measure zero, it can be covered by a union the In being intervals of length ITnl such that I < 1 and such that every tEE belongs to infinitely many In's. Grouping finite sets of intervals we can cover E infinitely often by U En such that every En is a finite union of intervals and such that En < e - 2". Let rrn be a polynomial satisfying (3.8) for E = En and put p. = n - 2 rr., n 1 2 We clearly have I P n 1100< 00 and S*(p.,t»2 - j21t1l on En· Since every tEE belongs to infinitely many E.'s, our theorem follows from lemma 3.3. ... PROOf:

IT.I

U In.

I I

I

3.5 Theorem: Let B be a homogeneolls Banach space 011 T satisfying the condition (3.4). Assume B 2 C(T); then either T is a set of divergence for B or the sets of divergellce (or B are precisely the set s of measure zero.

II. The Convergence of Fourier Series

59

PROOF: 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 sct 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 C( eT denote by Ea, the translate of E by a; Ea. is clearly a set of divergence for B. Let {an} be the sequence of all rational multiples of 2n: and put £ = U Ea n • By theorem 3.2 .£ is a set of divergence, and we claim that T".£ is a set of measure zero. In order to prove that, we denote by X the characteristic function of E and notice that

x(t - IX.)

This means or

2, 1

~

X(t)

Hj) e -i7"1 eijt

for all t and ct •• =

.2:

xU) eijl

j

(all IX.) If j ~ 0, this implies xU) ,= 0; hence X(t) = constant almost every where and, since X is a characteristic function, this implies that the measure of E is either zero or 2n:. Since E:::> E, E is almost all of T. Now T',jJ is a set of divergence (being of measure zero) and .£ is a set of divergence, hence T is a set of divergence. ~ 3.6 Thus, for spaces B satisfying the conditions of theorem 3.5, and in particular for B = JJ'(T) , 1 ~ P < 00, or B = C(T), either there exists a function feB whose Fourier series divergcs everywhere, or the Fourier series of every fe B converges almost everywhere. In the case B = IJ(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 alll!(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 el'erywhere.

PROOF: For arbitrary K > 0 we shall describe a positive measure 11" of total mass one having the property that for almost all t eT

(3.11)

60


Assume for the moment that such /l" exist; it follows from (3.11) that there exists an integer N" and a set E" of (normalized Lebesgue) measure greater than 1- 1/", such that for teE"

I

(3.12)

I

sup S.{jt", I) > ". " ... ,t-xN,narelinearly independent over the rationals. By Kronecker's theorem there exist, for each such I. integers n such that

I

i(n+!)('-XJ) -

isgn (sin I -;

Xi)

I ~ISint-Xjl-1 sm!(t-xJ}

2

for all j.

2

It follows that

(3.13)

S,,{j1, t} > 2 ~_

~

Isin t -; x j I-I

j=1

and since the x /s are so close to the roots of unity of order N. the sum in (3.13) is bounded below by t!:/N 1 sin t/21- 1dt > log N > K, provided we take N large enough. ~ EXERCISE FOR SECTION 3

1. Let B be a homogeneous Banach space on T . Show that for every Ie B there exist ge Band heL'm such that 1= g* h. Hinl: Use lemma 3.2 and theorem 104.1.

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 I 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 _=e il } in the complex plane. The unit disc {;;; z < I} is denoted by D and the closed unit disc, {z; z ~ I}, by [). For f E Ll(T) we denote by j(ri'), r < I, the Poisson integral of f,

II

(1.1)

f(re il )

II

00

=

(P(r,)*f)(t)

=

)'

........

rlnY'(n)einl .

-00

In chapter f 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

62

63

Ill. The Conjugate Function

this section is to consider it as a function of the complex variable

z=rei'inD. The functions rlllleln"

-

00

0

cr"

tnl/I(A.) ~ 2x -

(1.4)

00,

All

(or equivalently, j{t;lf(t)1 ~ ~ Cr"). Every fe J!(T) is clearly of weak I!' type. In fact, for all A> 0

IIf~fp

=

;x f:"dml/I(X)

~ ;x

f"XPdm,fl(X)

1 (00 A." ~ 2x A." d ml/I(x) = 2x (2x - mlfl(A.»)

JI

I IIfp.

hence (1.4) is satisfied with C = 2x f It is equally clear that there are functions of weak I! type which are not in J!(T); sin t t;" is a simple example.

I

I-

Lemma: Iff is of weak I! type then feL"'(T) for every p' < p. PROOF:

f IfI

,,'dt

=

=

L"x "dml/l(x)

~

ml/P) - [x 1"(2 It -

mln(l) +

m,

Jl oox'" dmlfl(x)

ll (x»]f

+ 1l (2It -

=

mlfl(x»d(xP ,)

66


1.6

II IE Lt(T) then / is 01 weak JJ :type.

Theorem:

PROOF:

II f II u

=

We assume first

that/~O;

also, we normalizelby assuming

1. We want to evaluate the measure of the set of points where

/1/ > A.. The function Hiz) = 1 + n~arg z -=+ ~~ = 1 + ~Im (lOg z - ~,l) n z + l,l Z

III.

is clearly harmonic and nonnegative in the half plane Rc (z) > 0, and its level lines are circular arcs passing through the points i,l and - i,l. The level line H;,,(z) =! is the half circle z = ,leiS, -n12 ).}I ~~. il

)

Since the mapping I -+ / is linear it is clear that if we omit the normalization II I II Ll = 1 we obtain, letting r -+ 1 , that for I ~ 0 in /J(T)

I{t; l!C I > ,l}/ ~ 811 IlIu ri . eil

)

Every IE Ll(T) can be written as I = II - f2 I- if) - i/4 where Ii ~ 0 and ~ We have / =/1 -/2 +i}~ - if4 and consequently

IIfillu II/lIu.

{t;I/(i') I >).} It follows that for c

=

£

~1 {t;lh(/I)1

> :}.

128 and every IE Ll(T)

(1.6) Corollary: IlleLI(T) thell /EL"(T) lor all rx < t . PROOF:

*1.7

t

Lemma 1.5.

Theorem:t

If flog+lfl

EI.J(T) ,

Log + X = sup (log x, 0) for x ~ O.

thell

/E /.J(T).

67

Ill. The Conjugate Function PROOF:

We shall use the fact that for geL2(T) we have geL2(T)

I IIL2 I IIL2.

and g ~ g This is an immediate corollary of theorem 1.5.5. As we have seen in 1.5, this implies

(1.7)

It)..

We have to prove that dmlll()..) < if.) which is the same thing as IIR)'dmIIP) = 0(1) as R-+ 00. Integrating by parts and remembering (1.6) we see that the theorem is equivalent to

lR

(1.8)

(21t - mlll()..» d)"

=0

as R

(1)

-+ 00.

In order to estimate 2rr - mlIP) we write f = g + h, where g =1 when ~). and h = I when I>).. We have 1 = g + ii and consequently

III

(1.9)

If

{t;ll(eit)I>).}c{t;jg(eit)1 >~}

U {t;lii(eit)1

>;}.

By (1.7) (1.10)

I {t;lg(it)1 >~} I ~ 81tr 1Jg11i2 2

= 81trZL' x2dmlfl

and by (1.6)

I {t;1 ii(e I >~} I ~ 2cr Ilhllu it

= 2cJ.

t

)

-I

I

oo

xdmlfl;

for x ~ )., (log x) 1/2> (log ).)1/2 and we obtain (1.11)

- . I > 2")..}

I {t; Ih(e")

I I

2c

=f).

~ ---~ .. ). jlog).

xv'log X dtntf(

= 0(1)

and the proof is complete.

1.8 TlJeorem (M. Riesz): For l

(1.20)

I, it follows

J

< = - 1- .

S"I imi arIy, -1 cos :x 2rr Adding and letting r --+ 1 we obtain (1.19).

2rr

~

2rr (1 - CO~.J2

e-;' )

fl + if2 where flJ2 are real valued. We have

I/(eil) I > A happens only if either IJI( /') I > rt A or lJie I > r! A

and consequently

it

Now, by (1.19) with rx

)

=.J2, 4rr

< -·--e cos \/2

-;.

j

=

1,2

and (1.20) follows.

* 1.10

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, m 1(x) and mix), of real-valued functions on T are equal if /~ 1

(1.27) EXERCISES FOR SECTION I

1. Show that there exists a constant A such that for all n, 1 and every 11/1100 ;&; I,

IE C(T),

2. Show that for I;&; p < co there exist constants Ap such that for all n,l and IELP(T), ;&; I

II/IILI'

I {t;ISil,t)1 > l} 1< ::.

73

III. The Conjugate Function 3. Prove that if leLPm, 1 < p< co, then lim IIJ(ri') -J(e")IILP

=

O.

r- la,; let n 2 be the subfamily of all the intervals in o which do not intersect Jl ' Denote a2 = sup fell> and let 12 E O 2 such that > ~ a 2' We continue by induction; having picked I" ... ,lk we consider the family OH 1 of the intervals of n which inter-

I

III

1/21

Ih+11

> iak+! sect none of Ij,j ~ k, and pick IHI EnH ! such that where (/k+' = sUPfell .. We claim that the sequence so obtained satisfies (2.2). In fact, denoting by I n the interval of length

,III·

{In}

41/nl of which In is the center part, we claim that U I n 2 U la' clearly implies (2.2). We notice first that ak ..... 0 and consequently

n Ok n

=

0.

For lEO

let k denote the first index such that I ¢ Ok; then I Ik - 1 -:;t. since /I k - , ~ Is; J k - , • and the lemma is proved.

I ll/l,

2.2

Theorem:

which

0

and, I

l2n +8 Joor

it

l

tooYdm(Y)dA

.1I1d integrating by parts again we finally obtain:

p

=

I


76

2.4 Lemma: Let k be a nonnegative even function on (-n,1t), monotone non increasing on (0,1t) and such that f~- k(t) dt = 1. Then for all f E IJ(T)

II

(2.7)

k(t - t)f(r) dt

I; ;

Mit).

The definition (2.1) is equivalent to

PROOF:

M

~t)

= sup 0 0 and IX > 0

IJ(t + h) - J(t) I ~ K Ih I'"

for all Ie E,

then the integrals (2.21) are uniformly bounded and

2~

L'

U(t - -r) - J(t

+ t")}cotidt" -+ I(e i -)

uniformly in teE as B -+ O. It follows, rexamining the proof of 2.7, that /(r t!') -+ I(eit) uniformly for teE as r -+ 1. In particular, if E is an interval, it follows that I(i') is continuous on E. 2.10 Conjugation is not a local operation; that is. it is not true that ifJ(t) = g(t) in some interval I. then /(t) = get) on I. or equivalently. that if J(t) = 0 on I, then I(t) = 0 on I. However,

Theorem: IJ J(t) =0 on an interval I, then I(t) is analytic on I . PROOF: By the previous remarks I is continuous on I. Thus the function F = J + il 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(it ) = il(e1 on 1, the theorem follows. rJ) ~ hp(fl,r)

=

have If(z)

1=

hp(j,r).

Since hp(f, r) is a continuous function of r, the same is true even if does have zeros on z \ = r, and the lemma is proved. (2'"'' counting each one according to its multiplicity. Then L (1 - I < 00.

q)

Remark: The convergence of the series L (1 -I (n I) is equivalent I (n I hence to the boundedness to the convergence of the product (below) of the series I log I (n I (not counting the zeros at the origin, if any).

n

PROOF: We may assume f(O);& o. By Jensen's inequality (3.3), if M is fixed and r sufficiently close to 1 }.{

10gl!(0) I-II f 11.v ~ Letting r

L logl ~n I - Mlogr. I

-+ 1 we

obtain

and since M is arbitrary the lemma follows.

3.6 If we combine lemma 3.3 with 3.5 we see that iffE N, the Blaschke product corresponding to the sequence of 7..eros of f is a well-defined holomorphic function in D, having the same zeros (with the same mUltiplicities) as f and satisfying I 8(z) I < 1 in D. If we write F(z) = j(z)(B(z)f 1 then F is holomorphic and satisfies I F(z) I > If(z) I in D. We shall refer to f = BF as the callonical factorization of f· Theorem:

Let f

E

H P, P > 0, and let f = BF be its canonica I fac-

torization. Then FE H P and II F IIHP ~ II f IllfP' PROOF:

The Blaschke product B has the form N

B(z)

=

lim N _ ",zmTIb(z, (n) . 1

If we write Fiz) ~f(z)(zmn~ b(z,(,))-t,then F",convergesto Funiformly on every disc of the form I z I ~ r < 1. Since the absolute value of the finite product appearing in the definition of F N tends to one uniformly as I z 1, it follows from lemma 3.2 that F N E /fP and

1-+

"FN IIHP

=

IlfIIHP' Let r hp(F, r)

=

< 1, then

lim h/FN, r) ~ lim N-oo

N-OO

I F IIJp = I f II,fp· N


86

r)

f II:p

r

FilliP

filliP,

Now h/F, ~ I for all < 1 is equivalent to I ~I and since the reverse inequality is obvious, the theorem is proved.

~

Theorem 3.6 is a key theorem in the theory of H P spaces. It allows us to operate mainly with zero-free functions which, by the fact of being zero-free, can be raised to arbitrary powers and thereby move from one HP to a more convenient one. This idea was already used in the proof of lemma 3.2. Our first corollary to theorem 3.6 deals with Blaschke products. 3.7 Corollary: Let B be a Blaschke product. Then I B(e it ) 1 = I almost everywhere. it PROOF: Since I B(z) 1 < 1 in D it follows from lemma 1.2 that B(e )

exists as a radial (actually: nontangentiaI) .Iimit for almost all t E T. The canonical factorization of f = B is trivial, the function F is identically one, and consequently

Since 1 B(eit)1 ~ 1, the above equality can hold only if I B(il) 1 = 1 almost everywhere. ~ 3.8 Theorem: Assume fE lIP, p> O. Then the radial limit lim, .... d(reil ) exists for almost all tE T and, denoting it by f(e· t ), we have

PROOf: The case p = 2 follows from 3.4. For arbitrary p > 0, let f = BF be the canonical factorization of f, and write G(z) = (F(z)Y/2. Then G belongs to H2 and consequently G(rei,) -> G(ei~ for almost all t E T; ~t every such t, F(rit) converges to some F(e i ,) such that 1FCit) Ip/2 = 1G(e il ) I. Since B has a radial limit of absolute value one almost everywhere we see that f(i') = limf(r eit ) exists and /fCe i ,) IP/2 = I G(eit ) 1 almost everywhere. 1 il IIGII ~2 ~II ) 1P dt Now I "~p = I F _n c(i') 12 dt = -2n and the proof is complete.

f

Ilftp=

Ilf(e

3.9 The convergence assured by theorem 3.8 is pointwise convergence almost everywhere. For p;:;; 2 we know that f is the Poisson

87

III. The Conjugate Function "

,

,

integral of f(e') and consequently f(re") converges to f(e't) in the J!(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:

Theorem: fEH P' ,

Let 0 < p < p' and supposefEHP andf(e i') ElJ"(T), Then

PROOF: As before, if we writef = BF, G(z) = (F(z)Y12, then G E H2 and G(ei') E L2P 'IP(T). G is the Poisson integral of G( eit ) and consequently GEHzp'Ip which means FEHP', hencefEHP'. ~

Corollary: Let fELI(T) alld assume JELI(T); then (f + iJ)EHI, PROOF: We know (corollary 1.6) that (f+ij)EW for all cx< 1 and by the assumption (f. + ij)(eit ) E LI(T). ~

3.10 Theorem: Every function f ill HI can be written as a product f = fd2 with fl.JZ E HZ . Let f = BF be the canonical factorization of Fl/z, fz =BFl/z.

PROOF:

fl

=

3.11

f.

Write ~

We can now prove:

Theorem: Let fE H' and let fCe i ,) be its boul/dary value. Then f is tlte Poisson integral of f(e it ). We prove the theorem by showing that f(re i,) converges to fe/') in the L' norm, This implies that if fez) = Lanzn, then an is the Fourier coefficient of f(i') which is clearly equivalent to f being the Poisson integral of f( it). Write f =fdz with f j EH 2, j = 1,2; we know that as r~ 1, PROOF:

lifiri\- fieit) Ilu -+ O.

f(re it ) - f(e it )

Now, =

f,(reit)fz(re i')

-

fl(eit)/z(/');

adding and subtracting 1,(ei')/2(r eit ) and using the Cauchy-Schwarz inequality, we obtain

IIf(reit ) - f(i')

liLt

it ~ IIJzIIL2I1fl(re ) - fl(eit)lIu

+ II II IILllllireit ) - f2(i t) IIL2 and the proof is complete.

88


Remark: See exercise 2 at the end of the section for an extension of the theorem to the case 0 < p < 1. Corollary: Let feL1(T) and leL1(T). Then S[l]

=

S[J].

From 3.9 and the theorem above follows that l(r eit) is the .... Poisson integral of l( i~ (see remark 1.4). PROOF:

3.12 Theorem: Assllme p ~ I . A function f belongs to H P if, and only if, it is the Poisson illtegral of some f(i t ) E l!'(T) satisfying j(lI)

(3.8)

=

0

for all n < O.

Let f E H P ; by theorem 3. I 1, f is the Poisson integral of f(it) and (3.8) is clearly satisfied. On the other hand, let fE IJ(T) and assume (3.8); the Poisson integral off, PROOF;

00

f(re it )

=

2:o j(lI)r" e

is hoIomorphic in D, and since that fez) E HP.

1/

00

int

=

fer e

2: j(n)z" 0

it

) 1/ LP

~

1/

f( eit ) II LP, it follows ....

3.13 For p > 1, we can prove that every fEll P is the Poisson integral of f(e it) 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 =00 of 3.12): if fEH P , IIf(re it )/lLP is bounded as r .....d; we it can pick a sequence rn ~ I such that f.(e ) = fern eit ) converge weakly in I!(T) to some f(e it ). Since weak convergence in IJ implies convergence of Fourier coefficients, it is clear that (3.8) is satisfied and that the function f with which v.e started is the Poisson integral of f(i t ). For p = 1 the proof as given above is insufficient. Ll(T) is a subs pace 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 E H 1 is the Poisson integral of some measure Jl on T. This measure has the property (3.9)

{l(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 = I is to prove that the measures satisfying (3.9), often called allalytic measures, are absolutely continuous with respect to the Lebesgue measure on T.

III. The Conjugate Function

Theorem: (F. and M. Riesz): fying (3.9)

lien)

=

89

Let Jl be a Borel measure on T satisfor all n

which proves the continuity of (1.3). To prove the better estimate (1.4), we consider the function eQ(z-a) F(z), where ell = I S 110 II S II; I. We have

I s/II:

~ "S(e (Z-II)F(z» Q

=

II'

sup{e-Q<xIISF(it)II~, e

Q

(1-

lIl

IlSF(1 + it)

II:}

t

~ (1 + 8)sup{e-

=

QII

Q

II Silo' e (1-II)!! Sill}

(1+e)IISII~-allslI~·

..

Remark: The idea of using the function eD (Z-II) goes back to Hadamard (the "three-circles theorem"); it can be used to show that, for every /E B, (1.5)

1.3 A very important example of interpolation of norms is the following: let (X, dx) be a measure space, let I ~ Po < PI ~ CIJ, and let B be a subspace of J!o (') U'(dx). We claim that the norms II 110 and II III induced on B by 1.!'°(dx) and U'(dx), respectively, are consistent. By lemma 1.1, all we have to show is that, given / E B, / # 0,

96


there exists a linear functional J1, continuous with respect to both norms, such that =jfgd1 where gEVnL"'(dl) has the propertyt that fg> 0 whenever

\II> o.

B LPO (")

Let (X, dl) be a measure space, = I!'(dl) (with t ~ Po < PI ~ (0). Denote by " the norms induced by J!'(dl) , and by II 1111 the interpolating norms. Then II II. coincides with the norlll induced all B by l!'a(dl) where Theorem:

t

_

(1.6)

___ J'OPI_ __

Pil - poa

+ PI(t

(

_ a)

Po

= 1_ a

of

I

PI

Let fEB and IlfIILPa~l. Consider F(z) = i = e ., and

PROOf:

where

f

= 00

\II

a=poai°;'&I~)

(= I-J--a

WehaveF(a)=fand consequently Ilfll.~ = so that

\llp·,PO

II F(iy)

110

=

)

\11

• 0

(:-11)+

lei.,

if PI =(0).

IIFII;nowIF(iy)1 = \III-a.

(f \lIPa dl

t

Po

~ I;

similarly 111'(1 + iy) III ~ 1 (use the same argument if PI < 00 and check directly if PI = (0); hence ~ 1. This proves II II,,;?;; II P • ° In order to prove the reverse inequality, we denote by qo, ql the conjugate exponents of Po and PI and notice that the exponent conjugate

Ilfllll

IIL

to p" is

(1.6') We now set B' = Co (") Lq'(dl) and denote by go' the corresponding space of holomorphic B'-valued functions. LetfE B and assume 111 IIL P • > I; then, since 8' is dense in C·, there exists agE B' such that IILO. ~ 1 and such that j fgdl > 1. As in the first part of this proof, there exists a function G E go' such that G(a) = g and II G II (with respect to qo, q I) is bounded by t. Let FE tB such that F(a) = j. The function h(z) = j F(z)G(z)dl (remember that for each ZEn, F(Z)EB and G(z)EB') is holomorphic and bounded in n (see Appendix). Now Il(a) > 1, hence, by the Phragmen-LindelOf

h

t If we writef= I f I eicp with real-valued

!P. we may takeg = min (l,lfIPo)eiCP.

IV. Interpolation of Linear Operators

I I I I I

97

theorem, h(z) must exceed 1 on the boundary, However, on the boundary lI(z) ~ F" ~ F!! so that 1/ F 1. This proves ~ 1 and it follows that and 1/ are identical. ..

l/fll"

IIG I I I II"

I/u.

II>

1.4 As a corollary to theorems 1.2 and 1.3, we obtain the RieszThorin theorem. Theorem: Let (X, dr) and (~, dl}) be two measure spaces. Let B = !!,o fl lJ"(dl) and 8' = fl I!".(dl}) alld let S be a linear transformation from B to B', cOlltinuous as (B, (B', j = 0, I, where (resp. II~) is the norm induced by I!"(dx) (resp. I!'J(dlJ». Then S is contilluous as

eo

I Ilj

I Ilj);

I

(8,11

II,,); (B', II

I Iii),

II~),

where II 1111 (resp. II II~) is the norm induced by J!·(dx) (resp. I!~(dl}), p" and p~ are defined in (1.6».

A bounded linear transformation S from one normed space 8 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 be extended as a transformation from If·(dr) into IT; (dl}) with norm satisfying (1.4). The same remark is clearly valid for theorem 1.2. *1.5 As a first application of the Riesz-Thorin theorem we give Bochner's proof of M. Riesz' theorem III.1.8. We show that !!'(T) admits conjugation in the case that p is an even integer. It then follows by interpolation that the same is true for all p ~ 2. Using duality we can then obtain the result for all p > 1. Let f be a real- valued trigonometric polynomial and assume, for simplicity, 1(0) = O. As usual we denote the conjugate by f and put f" = t(f + if)· P is a Taylor polynomiaIt and its constant term is zero; the same is clearly true for (10) P, P being any positive integer. Consequently 1 2n (f°(tW dt = O.

f

Assume now that p is ev:!n, p = 2k, and consider the real part of the identity above; we obtain: t We use the term "Taylor polynomial" to designate trigonometric polynomials of the fonn r~aneinl.

98


~

2n

f

(J)2k

dt - (2k) ~ f (J)2k- 2r dt + (2k) ~ f (/)2k-Y4 dt 2 2n 4 2n Z

_ ... =0. By HOlder's inequality

hence

or, denoting we have

which implies that Y is bounded by a constant depending on k (i.e., on p). Thus the mapping i ~ J is bounded in the l!(T) norm for all polynomials i, and, since polynomials are dense in L"(T), the theorem follows. EXERCISES FOR SECfION I I. Prove inequality (1.5).

2. Let {an} be a sequence of numbers. Find min(L conditions L an l 2 = I, L an 14 = a.

I

I

Iani) under the

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 J.4 (theorem 1.4.7); it is known as the Hausdorff-Young theorem:

Theorem: Let 1 ~ p ~ 2 alld let q be the conjugate exponent, that is, q = p/(p-l). If fEI!(T) then L Il(n)lq < 00. More pre-

2.1

cisely

(L

Il(n)lq)l,q~

"i"V'. :F

The mappingf ~ {/(n)} is a transformation of functions ~n the measure space (T,dt) into functions on (Z,dll), Z being the group of integers and £ill the so-called counting measure, that is, the PROOf:


99

measure that places a unit mass at each integer. We know that the norm of the mapping as LI(T)! L""(Z) = t'" is 1 (1.1.4) and we know that it is an isometry of VeT) onto U(Z) = t2 (1.5.5). It follows from the Riesz-Thorin theorem that :F is a transformation of norm ~ 1 from J!(T) into I!(Z) = t q > which is precisely the statement of our theorem. We can add that sinC£ the exponentials are mapped with ~ no loss in norm, the norm of f! 1 on H(T) into t q is exactly 1. 2.2 Theorem: Let 1 ~ P ~ 2 and let q be the cOlljugate exponent. If {an} E t P thell there exists a fUllction fE JJ(T) such that an = 1(11) • Moreover, Ilflll,. ~ la n lp)l/P.

(2:

PROOF: 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} E t 1 then f(t) = 2: anein , is continuous on T and l(n) = an' The case p = 2 is again given by theorem 1.5.5 and the case 1 < P < 2 is obtained by interpolation. ~

*2.3 We have already made the remark (end of ].4) that theorem 2.1 cannot be extended to the case p > 2 since there exist continuous functions f such that 2: 11(11) 12 -. = 00 for all & > O. An example of "" ein logn . such a function isf(t) = 2: --1/2 -. -)2 ern, (see [2H] , vol. I, p. 199); n=2 11 (ogll another example is get) = I m- 2r m/2fm(t) where fm are the RudinShapiro polynomials (see exercise 1.6.6, part c.) 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 that, given p < 2, there exist functions g such that II g II 00 ~ 1 and I I§(II) IP is arbitrarily big. If we assume that, we may assume that our functions g are polynomials (replace g by (Jig) with sufficiently big n) and then, taking a sequence Pr-+2, gj satisfying II gill", ~ 1 and Ln",=_",\§/II)lpJ>2 J , we can writef= Lj=lr2eimj'gj(t) where the integers m j increase fast enough to ensure that eimj' git) and eimktgit) have no frequencies in common if j i:- k. The series defining f converges uniformly and for any p < 2 we have

100


One way to show the existence of the functions g above is to show that, given e > 0, there exist functions g sati:;fying (2.1)

IIgll",~l,

IldL2~L

I

I

sup g(lI) <e. n

In fact, if (2.1) is valid then

2:

Ig(n)lp~

eP-zL Ig(n)12>i ep - 2 ,

and if e can be chosen arbitrarily small, the corresponding g will have I 1g(n)IP 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 E IJ(T) and }.. > 2N is an integer, then the Fourier coefficients of ~t) = f(}..t)P(t) are either zero or have the form j(m)P(k). This follows from the identity 97(n) = I;'m+k=J(m)P(k) and the fact that there is at most one way to write n = Am + k with integers m, k such that 1 k 1 ~ N < A/2. Consider now any continuous function of modulus 1 on T, which is not an exponential (of the form ein .); for example t{!(t)=eicos t. Since $(n) 12 = 1 and the sum contains more than one term, it follows that suP. 1$(n) I = p < 1. Let M be an integer such that pM < E. Let 'I < 1 be such

II

that '1M> t. Let fji = O"N(t{!) , where the order N is high enough to ensure tI < 1 2 we have

2:

1iifl)jP ~ e P - z

L

1gj(fl) 12 ~

r

j(p-2)

Ij

and consequently, for any choice of the integers mi , eimj'g/t) = Iane int does satisfy 1an Ip < ro for all p > 2. We now choose the integers

L


101

I1!J increasing very rapidly in order to well separate the blocks corresponding to j eimJ'g, in the series above. If we denote by N j the degree ofthe polynomial gj' we can take mjso that 3Nj > mj-1 + 3N j _ 1 • If anein'is the Fourier-StieItjes series of a measure Jl then

In, -

L

Jl

* eim,lyNJ = j eimjlgj

(YNJ being de la Vallee Poussin's kernel)

and consequently

which is impossible. We have thus proved

Theorem: (a) There exists a continuous fUllction f such that for all p < 2, l1(n)lp = roo (b) There exists a trigonometric series, I an e,n" which is not a p Fourier-Stieltjes series, such that anl < ro for all p > 2.

L

LI

*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 t P for any p < 2. Such a set clearly must be totally disconnected and therefore carries no continuous functions. Its characteristic function, however, is a bounded function the Fourier coefficients of which belong to no t P , P < 2.

Theorem: There exists a compact set Eon T such that E has positive measure alld such that' the only functi01l f carried by E with I IJ(j) IP < ro for some p < 2, is f == 0 . First, we introduce the notation t

and prove:

Lemma: Let e> 0, 1 ~ p < 2. There exists a closed set E.,p c T havi1lg the following properties: (1) The measure of E•. p is > 21< - e. (2) Iff is carried byE, p then

11I1!st

OO

PROOF:

ellflls,p.

Let y> O. Put

epi t) = t

~

Notice that

y - 21< {

-1-y-

iI lis!' is the same

o< t < y y~ t

as II

IIA(T) •

~

mod 2IT

2IT mod 2IT .

102

An rntroduction to Harmonic Analysis

Then, by theorem 2.1 !..-I

I tpy lI,to ~ I tpyllu ~

(2.3)

1 1 where - + - = 1.

2rry

p

q

We notice that ~/O) = 0 so that, if we choose the integers )..1' A2 , ••• , AN increasing fast enough, every Fourier coefficient of L~ tp/AA is essentially a Fourier coefficient of one of the summands. It then follows that 1 N n q.!.-I 9 ' (2.4) tp,(Ai) II ~ 2N tpy II N

2:

I lI,t

Ft9

1

We take a large value for N and put y =

=

$(t)

1 N

e/ Nand

N

2:

tppjt).

1

Then, by (2.3) and (2.4), it follows that .!..- I !..-I .!.-I .!.-.!. q q N = 4rrE" N "

II $IIFto~ 4rrl

so that if N is large enough

I $11:r(0
e -1 and that if mn increases fast enough, then g, defined by: g(t) =eiNmntfor 2nn/N ~!~2n(n + I)/N, n=l, ... ,N sat isfies (2.1). 3. Let {all} be an even sequence of positive numbers. A closed set E c T is a set of type U(a,,) if the only distribution Jl carried by E and satisfying {1(n) = o(a,,) as I n I -+ (», is Jl = O. Show that if a" -+ 0 there exist sets E of positive measure which are of type U(a,,). Hint: For 0 < a < 1r we write (see exercise 1.6.3):

t..(t)

__ {01-a- 1I

t

l

III a

~

~

a

It I

;5

n.

We have t.. E A(T), II t.. I AU) = I, and ~a(O) = a/2n. Choose nj so that I n I> nJ implies aj < 1O-~; put E j = {t; t. 2 -)(2n/) = o} and 1 j E = E j • Notice that lEI;;;; 2n 2 - > O. If Jl is carried imr by E we have, for all m and j, < e t. 3-) (2nJ'), Jl > = 0 since t. 3-) (2n/) vanishes in a neighborhood of E. By Parseval's formula

L

ni"'=1

3- i - - -

.

0= <e,mtt.3._;(2ni),Jl)

=

-fi.(m)

21[

.......--..... -----j (k){1(m +2nj k)

+ I t. 3 k*O

if nj > I m I and if I{1(n)l ~ an we have for k :f- 0, I p(m + 2njk) I < 10- J, hence (3- i /2n) I fi.(m) I < lO- i . Letting j-+ (» we obtain p(m) = 0, and, m being arbitrary, Jl = O.

Chapter V


The theme of this chapter is that of 1.4, namely, the study of the ways in which properties of functions or of classes of functions are reflected by the Fourier series. We consider important special case~ of the following general problem: let A be a sequence of integers and B a homogeneous Banach space on T; denote by B" the closed subspace of B spanned by {i).I}A e" or, equivalently, the space of all IE B with Fourier series of the form e" a l eill • Describe the properties of functions in B" in terms of their Fourier serics (and A). An obvious example of the above is the case of a finite A in which ~1I the functions in B" are polynomials. If A is the sequence of nonnegative integers and B = J!(T), 1 ~ P < if.), then B" is the space of boundary values of functions in the corresponding HP. Tn the first section \\<e consider lacunary sequences A and show, for instance, that if A is lacunary a la Hadamard then (V(T»" = (V(T»" and every bounded function in (V(T»" 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.

LA

l. LACUNARY SERIES

1.1 A scquence of positive integers Un} is said to be Hadamardlacunary, or simply lacunary, if there exists a constant q > 1 such that }'n+l > qAn for all n. A power series LanzA .. is lacunary if the 104

V. Lacunary Series and Quasi-analytic Classes

105

sequence P.} is, and a trigonometric series is lacunary if all the frequencies appearing in it have the form ± A. where Pn} is lacunary. The reason for mentioning Hadamard's name is his classical theorem 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. OUf 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 fE Ll(T) and assume that 1(j) = 0 for all j satisfying 1 ~ I no - j I ~ 2N. Assume that f(t) = O(t) as t -+ O. Then

(1.1)

IJ(no)1 ~ 2rr4(N-l

It- f(l) I + N-zllfIIL')' I

sup Irl 4A." and /(1) = L~ anCOSA"/, is real-valued, then sup IJ(/) I >

(2: I ani·

Hint: Consider the sets {t; an cos Ant> I an 1/2} . 5. If dn ~ 0 as n ~ <X> we can write d n = on"''' where {on} is bounded 1 and = ~(n) for some IJI EL (T). (See theorem 1.4.1 and exercise 1.4.1.) Deduce that if A is a lacunary sequence and d;. ~ 0 as I AI ~ <x> , there exists a function gEL l(T} such that

"'n

for AEA.

I

6. If 2: dn 12 < <X> there exist sequences {o,,} and {"'n} such that 2 d n = 0""',,, 2:,1 < <x> , and = v;(n) for some", EL1(T). Remembering that the convolution of a summable function with a bounded function is continuous, prove theorem 1.6. 7. Assume Aj+l/)'j ~ q > J. Show that there exists a number M = Mq such that every integer n has at most one representation of the form n = 2: l1/mj + jM where '1j = -1,0,1, and 1 ~ mj ~ M. Use this to show that the product TIi=l (1 + 2:~=1 dm+jMCOS(Am+jMt + 'l'm+jM» has (formally) the Fourier coefficient t dk e i , . at the point Ak' Show that if 0 < d" ~ II Mq for all k, then the product above is the Fourier Stieltjes series of a positive measure which interpolates {t dk ei'k} on{ Ak } • 8. Assume Aj+t!A.j ~ q> 1 and d j l 2 < <x>. Find a product analogous to that of exercise 7), which is the Fourier series of a bounded function, and which interpolates {dj } on pj}. *9. Show that the following condition is sufficieI1t to imply that the sequence A is a Sidon set: to every sequence {d.} such that d. = I there exists a measure Ji E M(T) such that

onl

"'n

LI

I I

AeA.

112


*10. Show that a finite union of lacunary sequences is a Sidon set. *2. QUASI-ANALYTIC CLASSES

2.1

We consider classes of infinitely differentiable functions on T. Let {Mn} be a sequence of positive numbers; we denote by C*{Mn} the class of all infinitely differentiable functions j on T such that for an appropriate R > 0 (2.1)

n = 1,2, ....

We shall denote by C# {M.} the class of infinitely differentiable functions on T satisfying: (2.2)

n

=

1,2, .. ,

for some R (depending on f). The inclusion C*{Mn} £; C#{Mn} is obvious; on the other hand, since the mean value of derivatives on T is zero, we obtain SUPt eT/J(n)(t) 1 ;;;; IIf(n+l>IIL2 and consequently C# {Mn} £; C*{M"+I'} Thus the two classes are fairly close to each other. Examples: If Mn = 1 for all n, then C#{Mn} is precisely the class of all trigonometric polynomials on T. If Mn = n 1, C# {Mn} is precisely the class of all functions analytic on T. (See exercise 1.4.3.) We recall that a sequence {cn } , Cn > 0, is log convex if the sequence {log cn } is a convex function of n. This amounts to saying that, given k < I < m in the range of n, we have (2.3)

log c1

;;;;

m-I --k log Ck

m-

I-k

+ -m--k

log Cm

or equivalently (2.4) 2.2 The identity IIf(n>IIL2 =(2: 11(j)12/n)t allows an expression of condition (2.2) directly in terms of the Fourier coefficients of f. Also it implies

Lemma: Let f be N times differentiable on T. Then the sequence {IIf(n) IIu} is monotone increasing and log cOl/vex for 1 ;;;; n;;;; N.

IIfn)IILl (L

The fact that = IJ(j)12/n)~ is monotone increasing is obvious. In order to prove (2.4) we write p = (m -k)/(m -I), q = (m-k)j(l-k); then lip + lIq = 1 and by Holder's inequality PROOF:

V. Lacunary Series and Quasi-analytic Classes

L 1a j 12j21 = L (I a

j

12/p/

k P /

)(1 a jI2, q/m, q) ~

113

IIf(k)IIi'! IIf m)W q

which is exactly (2.4). rt follows from lemma 2.2 (cf. exercise 2 at the end of this section) that for every sequence {M~} there exists a sequence {Mil} which is monotone increasing and log convex such that C#{Mn} = C#{M~}. Thus, when studying classes C# {M.} 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 < I < m, (2.5)

2.3 For a (monotone increasing and log convex) sequence {Mn} we define the associated function T(r) by (2.6)

T(r) = inf Mnr-n n~O

We considersequencesM. which increase fasterthatRnfor all R>O; hence the infimum in (2.6) is attained and we can write T(r) = minn~oMnr-n. If we write 111 = M~ I, and 11. = M._ tlMn for /l > 1; then 11ft is a monotone-decreasing sequence since by (2.5), 11.+dl1n = M;/Mn_IMn+ 1 < 1; we have M"r -n = (11 jr) - 1 and consequently

n;

(2.6')

T(r) =

11

(l1jr)-l.

Ili'> 1

The function T(r) was implicitly introduced in 1.4; thus it follows from f.4.4 that if fE C# {Mn} then, for the appropriate R > 0

11u)1 ~ ,(jR-I), and exercise 1.4.6 is essentially an estimate for oCr) in the case Mn = /lIZ•• 2.4 An analytic function on T is completely determined by its Taylor expansion around any point to E T, that is, by the sequence {J(n)(to)}:'=o' In particular if j +y

IX

.

1 log 1F(e") 1dt > - 00; 20

-

00

and the lemma is proved.

...

Theorem: A sufficient condition for the qllasi-al/alyticity or en logr(r) . the class c# {'I'fn } i~ that -1--- ' dr = - 00 where r{r) IS defined , + r2

2.6

uy

(2.6).

f

v. PROOF:


Let fE

ell' {M.}

and assume that fn)(O) =

°

lIS

n = 0,1, ....

Define p(z)

1

fh

= 2n Jo

Integrating by parts we obtain, z p(z) =

-.!.

27t

e-z'l"(t) dt.

*0

[-=-!z e-z'l'(t)] 2" + 2nz 1_ 0

°

fo2l< e-Z'!'(t)dt

Jc

*

and since 1(0) = 1(27t} = the first term vanishes for all z 0 (we have used the same integration by parts in 1.4; there we did not assume 1(0) = but considered only the case z = irn, that is, e-ztl is 2x-periodic). Repeating the integration by parts n times (using/(j)(O) = fj)(27t)

°

=

°

for j ~ n), we obtain

1 2

p(z) =

1 2nz.

0 "

e-zy 2 JLO=JL.=4, JLj=-M;'

-I

L AlMn

). Write

n+.

M" = 16)Jli 1

an d t h e

function f defined by lemma 2.7 has a zero of infinite order (actually vanishes outside of[ -1,1]), is not identically zero, andlE C# {Mn } . 1 (JLjr)-I; hence - logr(r) =

L JIlT> 1

10g(J1.f)~

.L pjr~e

log (JLjr) ~ ..#(r).

V. Lacunary Series and Quasi-analytic Classes Thus for k = 2,3, ... (2.7)

I

eHI

ek

k

:-.I

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