An introduction to harmonic analysis

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 ba

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 In- ternational Copyright Comentions

Publhhed in Canada by General Publishing pdny, Ltd., 30 Lesmill Road, Don l\1i11s, Toronto, Ontario

Com-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

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

turning then to the real line in Chapter VI and coming to locally

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

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

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

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

W Rudin I would also like to thank the publisher for the friendly

Jerusalem

April 1961l

YI fLIIAK KAfZNELSON

Contents

2 SUMMABILITY IN NORM AND

3 POINTW1SE CONVERGENCE OF u,,(f) 17

4 THE ORDER OF MAGNITUDE OF FOURIER

5 FOURIER SERIES OF SQUARE SUM MABLE

6 ABSOLUTELY CONVERGENT FOURIER SERIES 31

7 FOURIER COEFFICIENTS OF LINEAR

*2 THE MAXIMAL FUNCTION OF HARDY AND LITTLEWOOD

*3 THE HARDY SPACES

xi

73

81

xii An Introduction to Harmonic Analysis

and the Theorem of Hausdorff-Young 93

I INTER POLA TION OF NORMS AND OF

2 THE THEOREM OF HAUSDORFF-YOUNG 98

I FOURIER TRANSFORMS FOR L '(R) 120

*8 THE FOURIER-CARLEMAN TRANSFORM 179

1 LOCALLY COMPACT ABELIAN GROUPS 186

3 CHARACTERS AND THE DUAL GROUP 188

5 ALMOST-PERIODIC FUNCTIONS AND THE

Contents xiii

I DEFINITION, EXAMPLES, AND

2 MAXIMAL IDEALS AND MULTIPLICATIVE

3 THE MAXIMAL-IDEAL SPACE AND THE

4 HOMOMORPHISMS OF BANACH

10 THE USE OF TENSOR PRODUCTS

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 quo-tient 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 im-

plicit 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

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,

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

2 An Introduction to Harmonic Analysis

The numbers n appearing in (1.1) are called the frequencies oj P;

the largest integer n such that I a" / + I 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

N

n= -N

Let P be defined by (1.2) Knowing the function P we can compute

the coefficients an by the formula

in (1.4) may be infinite and there is no assumption whatsoever about

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

00

S ~ L -i sgn (n)a n e int

"= -00

where sgn (n) = 0 if 11 = 0 and sgn(n) = nIl n I otherwise

to as the conjugate Fourier series of f We shall say that a

We turn to some elementary properties of Fourier coefficients

1.4 Theorem: Let j,gELi(T), thell

h(n) = !(n)e-inr •

t See,chapter III for motivation of the terminology

t Defined by: jet) = f(t) for all lET

4 An Introduction to Harmonic Analysis

The proofs of (a) through (e) follow immediately from (1.5) and the details are left to the reader

PROOF: The continuity (and, in fact, the absolute continuity) of F

is evident The periodicity follows from

I Fourier Seric~ on T 5

PROOF: The functions f(t - T) and g(T) , considered as functions of

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

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

(0,2n» as a function of T for almost all t, and

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

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

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

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):

associatire, alld distributive (with respect to the addition)

PRoor: The change of variable -8 = f - • gives

that is

6 An Introduction to Harmonic Analysis

EXERCISES FOR SECTION 1

where An = len) + j( - n) and Bn = i {J(n) - l( -n) } Equivalently:

I(t) = -j( -t), then An = 0 for all n

3 Show that if S - L aj cos jt, then S , , L aj sin jt

4 Let IE Ll(T) and letP(t) ~ L~.'Van eil/ t • Compute the Fouriercoefficients

6 The trigonometric polynomial cosnt='Hin'+e- in,) is of degree nand

has 2n zeros on T Show that no trigonometric polynomial of degree n > 0

can have more than 2n zeros on T Hint: Identify L:.ajeijt

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*

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 1 denotes the Lebesgue measure of Eo.)

13 If E and F are subsets of T, we write

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

inter-val 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

2.1 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

I Fourier Series on T 9

the understanding that the function] is defined on the integers As

advan-tages; the main advantages of the series notation being that it indicates

and all of chapter 11 will be devoted to clarifying the sense in which

LJ(1I) e int represents f In this section we establish some of the main

following:

fit) = j (t - r) E Ll(T) and 11ft IILI = II f IILI

t-+to

immediate consequence of the translation invariance of the measure

valid iff is a continuous function Remembering that the continuous

and e > O Let g be a continuous function on T such that

h - f IILI < 1:/2; thus

11ft - fto IILI ~ II ft - gt IILI + II gt - gto IILI + II gto - fro IILI =

= II (f - g)t 11,.' + II gt - gro IILI + II (g - f)ro IILI < e + II gt - gto IILI

Hence lim Iii t - fto ilL' < I: and, e being an arbitrary positive number,

lO An Introduction to Harmonic Analysis

(S-3) For all 0 < 0 < 11:,

lim r2~-.I1 k.Ct)Jdt = O

,,~oo Jo

and we replace in (S-3), as well as in the applications, the limit "lim"

by "lim"

The following lemma is stated in terms of vector-valued integrals

We refer to the Appendix for the definition and relevant properties

Lemma: Let B be a Banach space, g; a continuous B-valued function

on T, and {k.} a summability kernel Then:

I Fourier Series on T II

By (S-2) and the continuity of f{I( T) at T = 0, given e > 0 we can find

o > 0 so that (2.3) is bounded bye, and keeping this 0, it results from

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

n~OO 7t

ill the LI(T) norm

2.4 The integrals in (2.5) have the formal appearance of a tion although the operation involved, that is, vector integration, is

[0,2n) becomes finer and finer On the other hand,

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,

12 An Introduction to Harmonic Analysis

and consequently

Using lemma 2.4 we can rewrite (2.5):

2.5 One of the most useful summability kernels and probably the

and that (S-3) is satisfied is clear from

and a.(J, t) = (K * f)(t) It follows from corollary 1.9 that

1 Fourier Series on T 13 Other immediate consequences are the following two important theorems

2.7 Theorem: (The Uniqueness Theorem): Let IE Ll(T) and assume ](n) =0101' all n Then I = O

PROOF: By (2.9) a.U) = 0 for all n Since a.U) -+ I, it follows

and assume j (n) = g(n) for aJl 11, then I = g

2.8 Theorem: (The Riemann-Lebesgue Lemma): LeI IELI(T),

PROOF: Let e > 0 and let P be a trigonometric polynomial on T stich that /I I - pilL' < e If I n 1 > degree of P, then

I](n) 1 = I (J-:>P)(n) I ~ II 1-P /ILl < e ~

Remark: Jf K is a compact set in Ll(T) and e > 0, there exist a

every IE K there exists a j, I ~ j ~ N, such that II! - P j IILI < e

f E K Thus, the Riemann-Lebesgue lemma holds uniformly on

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

(2.11) a.U) = - -1 -1 (SoU) + StU) + + S.U» ,

n+

From coroJlary 1.9 it follows that SnU) = D" *1 where D" is the Dirichlet kernel defined by

t That is, if SnU) converge as II -+ CO

14 An Introduction to Harmonic Analysis

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 II lin ~ II IILI under which it is a Banach space, and having the following properties:

(H-t) IffEB and tET, thenfTEB and lifT lin = Ilflln (where

J.(t) = J(t - t»

Remarks Condition (H-I) is referred to as translation invariance and (H-2) as continuity of the translation We could simplify (f-I-2)

somewhat by requiring continuity at one specific <0 E T, say to = 0 rather than at every toET, since by (H-I)

Also, the method of the proof of (H-2') (see 2.1) shows that if we have

a space B satisfying (H-I) 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 Be Ll(T) be a Banach space sati~fying (H-l) Denote

by Be the set oj all fE B such that < + fT is a continuous B-valued Junction Then Be is a closed subspace oj B

Examples of homogeneous Banach spaces on T

(a) C(T)-the space of all continuous 2rr-periodic functions with the nonn

I Fourier Series on T 15

The validity of CH-l) for all three examples is obvious The validity

of (H-2) for (a) and (b) is equivalent to the statement that continuous

is identical to that of (H-2') (see 2.1)

2.11 Theorem: Let B be a homogeneous Banach space on T; let fE B and let {k"} be a summability kernel Then

II k" * f - f II B 0 as II -+ 00

1

PROOF: Since II lIB ~ II IILI' the B-valued integral 21r Jk.(r)/Jr

is the same as the U(T)-valued integral which, by lemma 2.4, is equal

2.12 Theorem: Let B be a homogeneous Banach space on T Then the trigonometric polynomials in B are everywhere dense

PROOF: For every fE B, CT"(f) -+ f

Corollary (Weierstrass Approximation Theorem): Every uous 2n:-periodic function can be approximated uniformly by trigonometric polynomials

kernels

(S-l), (S-2) and (S-3) are obvious from (2.15) V" is a polynomial

of degree 2n + 1 having the property that V"U) = 1 if I j I ~ n + 1;

pre-scribed intervals (namely V" * f)

(2) The Poisson kernel: for 0 ~ r < 1 put

(2.16)

16 An Intcod uction to Harmonic Analysis

(2.17) P(r,t)*! = 2: ]en)r,n'e int •

Compared to the Fejer kernel, the Poisson kernel has the disadvantage

of not being a polynomial; however, being essenti ally the real part

(1 + re't)

for 0 ~ I ~ 1!

t -+ int where n is a rational integer lIint: Use Exercise 1.16

with the norm

3 Show that for B = C' (T), Be (see lemma 2.10) is C(T)

4 Assume 0 < a < 1; show that for B = Lip {T)

e a: 'h-O t I " I a

6 Let B be a Banach space on T, satisfying (H-J) Prove that Be is the

I Fourier Series on T 17

7 Use exercise 1.1 and the fact that step functions are dense in LI(1')

to prove the Riemann-Lebesgue lemma

9 Show that for fe L leT) the norm of the operator f: g + I * g on L 1(1')

It For n = 1,2, • , let kll be a nonnegative, infinitely differentiable function on T having the properties (i) Jkit)dt = 1, (ii) k.(t) = 0 if I tl > lin

close to the support of f

that SUptIP'(t)1 ~2n sup,1 P(t)l Hint: Show that p' = - P*2n K.-l(t) sin nt

and use the fact that 112n K,,_I(t)sin nt IIL'(T) < 2n

and fEB then g * I E B, and

IIg*/IIB ~ IIg liuli/IiB'

translation-invariant subspace Show that H is spanned by the exponentials

every n e Z such that len) ¢ 0, there exists g E H such that g(n) ¢ o

3 POINTWISE CONVERGENCE OF er.(f)

of er,,(/) from its convergence in norm, nor can we relate the limit

t Bernstein's inequality is: sup Ip'l ~ nsup Ipl ' and can be proved similarly

18 An Introduction to Harmonic Analysis

of (fiJ, to), in case it exists, to J(to), We therefore have to reexamine the integrals defining (fn(f) for pointwise convergence

3.1 Theorem (FeNr): Let fEL1(T)

(a) Assume that 1imh~OU(to + h) + J(to - h» exists (we allow the values -00 and + 00); then

(3.1) (fn(f,t O) ~ -1 lim U(to + h) + f(to - h»

h O

In particular, if to is a point of continuity of f, then (fn(f, to) ~ f(to)

Jor f, (fnU,t) converges tof(t) uniformly on 1

(c) If for allt, m ~f(t), then m ~ (f"U,t); if for alll,f(t) ~ M, then (fn(f,t) ~ M

Remark: The proof will be based on the fact that {K,,(t)} is a positive summability kernel which has the following properties:

(3.2) For 0 < f} < 1t

lim (sup Kn(t» = O

n OO I}< t <2x-1}

(3.3)

The statement of the theorem remains valid if we replace (fnU) by

k n * f, where {k n} is a positive summabHity kernel satisfying (3.2) and (3.3) For example: the Poisson kernel satisfies all the above re-quirements and the statement of the theorem remains valid if we replace (fnU) by the Abel means of the Fourier series of f

PROOF OF FEJER's THEOREM: We assume for simplicity that

J(/ o ) = lim f(to + h) ~f(to - h) is finite, the modifications needed for

I Fourier Series on T

Given e > 0, we choose {} > 0 so small that

and then no so large that n > no implies

{}<T<2n-{}

From (3.4), (3.5), and (3.6) we obtain

(3.7) I (1.(f, to)- J(to)1 < e + e Ilf -/(to) ilL'

which proves part (a)

19

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

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)

no value to, (3.8) holds with ](to) = f(to) for almost all to (cf [28],

Vol 1, p 65)

20 An Introduction to Harmonic Analysis

Theorem (Lebesgue): If (3.8) holds, then (T"U,/ o )-+ 1(10 ), In cular (T"U, t) -+ f(t) almost everywherẹ

parti-PROOF: As in the proof of F6jer's theorem,

(3.9) (TltU,/o) -l(to) =MLiJo + fo") Klf) [f(to-'t)+!(tớt) J(to)] d't

I<,,('t) ~ mm n + 1, en + 1)i2

the first integral

I Fourier Series on T 21

Corollary: If the Fourier series of f E Ll(T) converges on a set E

of positive measure, its sum coincides with f almost everywhere on E

In particular, if a Fourier series cOllverges 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

3.3 The need to impose in theorem 3.2 the strict condition (3.8) rather than the weaker condition

comes from the fact that in order to carry the integration by parts we

the condition (3.8') is sufficient Thus we obtain:

Theorem (Falou): If (3.8') holds, then

00

lim L J(j)r1J1e IJto = J(to)

r-+l -co

EXERCISES FOR SECfION 3

1 Let 0 < a ;;;:; I and let IE L 1(T), Assume that at the point 10 ET, f

satisfies a Lipschitz condition of order IX, that is, 1/(10 + t) -/(10) 1 < KI T la for 1 T 1 ;5 It Prove that for ex < I

Un , to) - 1(1 0} ;5 - - -Kn

J - ex

while for IX = I

Hint: Use (3.10) and (3.4) with {} = lin

2 If IE Lip (T), 0 < a;;;:; I, then

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

22 An Introduction to Harmonic Analysis

0<'1<1

'1=1

3 LctfELco(T) and assume li(n)/ ~ K/ nl-I• Prove that for all nand t,

ISi!,t)l;S Il/lIco + 2K Hint:

4 Show that for all nand t, IL~ r l

sinjtl ~ 1rr + I

H in/: Consider r(t) ~ 1/2 in (0,2rr)

4 THE ORDER OF MAGNITUDE OF FOURIER COEFFICIENTS The only things we know so far about the size of Fourier coeffi-cients {j(n)} of a function IE LI(T) is that they are bounded by

llfllL' (1.4(e» and that limlnl_>?i(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 cer-tain rate of vanishing of J( n) as 1 n 1-+ OC!?

We show that the answer to (a) is negative; len) can go to zero arbitrarily slowly (see 4.1)

(b) In view of the negative answer to (a), is it true that any sequence

{all} which tends to zero as In 1-+ OC! is the sequence of Fourier efficients of some f E LI(T)?

co-The answer to (b) is again negative (see 4.2)

(c) How are properties like bounded ness, continuity, smoothness, etc of a function f reflected by {j(Il)}?

Question (c) is one of the main problems of harmonic analysis

In the second half of this section we show how various smoothness conditions affect the size of the Fourier coefficients The effect of square integrability will be discussed in the following section

(4.1)

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)·

Tlrell

'" :" -f(n)<OC! 1

~n

117:-0

PRoor: Without loss of generality we assume 1(0) = O Put

F(/) = J~f(r) dr; then FE C(T) and, by theorem 1.6,

F(n) = -;-f(ll) , n :;60

w

Since F is continuous, we can apply Fejer's theorem for 10 = 0 and obtain

(4.3) lim 2 ')' (1 - ~-- ) 101 = i(F(O) - teO»~

N-oo n~ N + 1 n

and since l~n] ~ 0, the theorem follows

n

24 An Introduction to Harmonic Analysis

Fourier series Hence there exist trigonometric series with coefficients tending to zero which are not Fourier series

4.3 We turn now to some simple results about the order of tude of Fourier coefficients of functions satisfying various smoothness conditions

-PROOF: By theorem 1.6 we have fen) = (1/in)l'(n) and by the

Rie/ '

integration by parts) we see that if f is k-times differentiable and

I Fourier Series on T

I J( ) n I = < Var(f) 2xl n I PROOF: We integrate by parts using Stieltjes integrals

Theorem: For n =F 0, IJ(n) I ~ ~ n(f,,:J

PROOF: j(n) = ~ J f(t)e-intdt = -1 J f(t)e1n(l+tr/n)dt;

by a change of variable,

hence

Corollary: IffELiplT),thenj(lI) = O(n-«)

4.7 Theorem: Let 1 < p ~ 2 and let q be the conjugate expollent

(i.e., q = p ~ 1)' If fE1!'(T) then I 11(n)lq < OCJ

1 < p < 2 will be proved in chapter IV

Remark: Theorem 4.7 cannot be extended to p > 2 Thus, if

fE1!'(T) with p> 2, thenfEe(T) and consequently I IJ(nW < co;

that I 1/(,,)12

- = OCJ for all 8> O

26 An Introduction to Harmonic Analysis

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

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 con-venient 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, L 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 con-taining 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

liml $ Kexp ( _ ~ /jIl''')

7.Assume li(m ~ Kexp( - /jIll,,) SholV that 1 is infinitelyditferentiable and

I Fourier Series on T 27

1/",(/)1 < Kl ea"n all

for some constants a and K 1• Hint: 1 1(1I)(t) 1 ~ 2 KLUlllexp( -I jill a)

8 Prove that If 0 < IX ~ I then 1(1)= L,.1 jn.x belongs to Lipi );

"\''''' sin nt

9 Show that the serIes L,.n=2 - - - converges lor all IE T

iogn

5 FOURIER SERIES OF SQUARE SUMMABLE FU NCTlONS

In some respects the greatest success in representing functions by means of their Fourier series happens for square summable functions

defined by

(5.1) <f.g) = 2n 1 f -J(t) g(t) dt ,

and in this Hilbert space the exponentials form a complete orthogonal system We start this section with a brief review of the basic properties

If E is a subset of Yf' we say that fE f f is orthogolla I to E iff is

is one that is if whenever f gEE <f g) = 0 iff:F- g and <f,1) = 1

be complex numbers Then

PROOF:

28 An Introduction to Harmonic Analysis

Corollary: Let {lP,,}:=l be an orthonormal system in }(' and let {a"}:=l be a sequence of complex numbers such that L 10,,12 < 00

Then L • = 1 O"tp" converges In }('

PROOF: Since.}(' is complete all that we have to show is that the partial sums Sf{ = L~ a.tp" from a Cauchy sequence in }(' Now for

(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 ing 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

(a) => (c) From Bessel's inequality and corollary 5.1 it follows that

L (J,9',,)9' converges in JF If we denote g = L (/,9'.)9' we see

PROOF: If 1 is a finite linear combination of {9'.}, (5.5) is obvious

In the general case

orthonormal system The orthonormality is evident:

(in' iml) - ~ f ei(,,-m)/ dt =- (j

The completeness is somewhat less evident; it follows from theorem 2.7 since

(J,e int ) = ~1tf I(t)ein'dt =l(n)

The general results about complete orthonormal systems in Hilbert space now yield

Theorem: (a) Let IEL2(T) Then