An Introduction To Harmonic Analysis

299 Pages·2004·1.047 MB·English

by  Katznelson Y.

AN INTRODUCTION TO
HARMONIC
ANALYSIS
Yitzhak Katznelson
Third Corrected Edition

Preface
Harmonicanalysisisthestudyofobjects(functions,measures,etc.),
definedontopologicalgroups. Thegroupstructureentersintothestudy
byallowingtheconsiderationofthetranslatesoftheobjectunderstudy,
thatis,byplacingtheobjectinatranslation-invariantspace. Thestudy
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 ob-
ject under study (harmonic or spectral analysis); and second: finding
a way in which the object can be construed as a combination of its
elementarycomponents(harmonicorspectralsynthesis).
The vagueness of this description is due not only to the limitation
of the author but also to the vastness of its scope. In trying to make it
clearer,onecanproceedinvariousways†;wehavechosenheretosac-
rifice generality for the sake of concreteness. We start with the circle
group T and deal with classical Fourier series in the first five chap-
ters, 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
philosophybehindthechoiceofthisapproachisthatitmakesiteasier
forstudentstograspthemainideasandgivesthemalargeclassofcon-
creteexampleswhichareessentialfortheproperunderstandingofthe
theoryinthegeneralcontextoftopologicalgroups. Thepresentationof
Fourierseriesandintegralsdiffersfromthatin[1],[7],[8],and[28]in
being, I believe, more explicitly aimed at the general (locally compact
abelian)case.
The last chapter is an introduction to the theory of commutative
Banach algebras. It is biased, studying Banach algebras mainly as a
toolinharmonicanalysis.
This book is an expanded version of a set of lecture notes written
†Hencetheindefinitearticleinthetitleofthebook.
iii
IV ANINTRODUCTIONTOHARMONICANALYSIS
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 fa-
miliarwiththebasicnotionsandfactsofLebesgueintegration,themost
elementary facts concerning Borel measures, some basic facts about
holomorphicfunctionsofonecomplexvariable,andsomeelementsof
functionalanalysis,namely: thenotionsofaBanachspace,continuous
linearfunctionals,andthethreekeytheorems—"theclosedgraph",the
Hahn-Banach, and the "uniform boundedhess" theorems. All the pre-
requisites can be found in [23] and (except, for the complex variable)
in [22]. Assuming these prerequisites, the book, or most of it, can be
covered in a one-year course. A slower moving course or one shorter
than a year may exclude some of the starred sections (or subsections).
Aimingforaone-yearcourseforcedtheomissionnotonlyofthemore
generalsetup(non-abeliangroupsarenotevenmentioned),butalsoof
many concrete topics such as Fourier analysis on Rn,n > l, and finer
problemsofharmonicanalysisinTorR(someofwhichcanbefound
in[13]). Also,someimportantmaterialwascutintoexercises,andwe
urgethereadertodoasmanyofthemashecan.
Thebibliographyconsistsmainlyofbooks,anditisthroughthebib-
liographiesincludedinthesebooksthatthereaderistobecomefamil-
iar with the many research papers written on harmonic analysis. Only
some,morerecent,papersareincludedinourbibliography. Ingeneral
wecreditauthorsonlyseldom—mostoftenforidentificationpurposes.
With the growing mobility of mathematicians, and the happy amount
of oral communication, many results develop within the mathematical
folklore and when they find their way into print it is not always easy
to determine who deserves the credit. When I was writing Chapter Ill
ofthisbook,Iwasverypleasedtoproducethesimpleelegantproofof
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
Carlesonindicatedtomethissameproof.
The book is divided into chapters, sections, and subsections. The
chapter numbers are denoted by roman numerals and the sections and
subsections, as well as the exercises, by arabic numerals. In cross ref-
erences within the same chapter, the chapter number is omitted; thus
Theorem llI.1.6, which is the theorem in subsection 6 of Section 1
of Chapter Ill, is referred to as Theorem 1.6 within Chapter IlI, and
PREFACE V
TheoremIll.1.6elsewhere. Theexercisesaregatheredattheendofthe
sections,andexerciseV.1.1isthefirstexerciseattheendofSection1,
Chapter V. Again, the chapter number is omitted when an exercise is
referredtowithinthesamechapter. Theendsofproofsaremarkedby
atriangle(J).
The book was written while I was visiting the University of Paris
andStanfordUniversityanditowesitsexistencetothemoralandtech-
nicalhelp1wassogenerouslygiveninbothplaces. Duringthewriting
Ihavebenefittedfromtheadviceandcriticismofmanyfriends;1would
liketothankthemallhere. ParticularthanksareduetoL.Carleson,K.
DeLeeuw, J.-P. Kahane, O.C. McGehee, and W. Rudin. I would also
liketothankthepublisherforthefriendlycooperationintheproduction
ofthisbook.
YITZHAK KATZNELSON
Jerusalem
April1968
The 2002 edition
The second edition was essentially identical with the first, except for
the correction of a few misprints. The current edition has some more
misprints and “miswritings” corrected, and some material added: an
additionalsectioninthefirstchapter,afewexercises,andanadditional
appendix. The added material does not reflect the progress in the field
inthepastthirtyorfortyyears. Almostallofitcould,andshouldhave
beenincludedinthefirsteditionofthebook.
Stanford
March2002

Contents
I FourierSeriesonT 1
1 Fouriercoefficients . . . . . . . . . . . . . . . . . . . . 2
2 Summabilityinnormandhomogeneousbanach
spacesonT . . . . . . . . . . . . . . . . . . . . . . . . 8
3 Pointwiseconvergenceofσ (f). . . . . . . . . . . . . . 17
n
4 Theorderofmagnitudeof
Fouriercoefficients . . . . . . . . . . . . . . . . . . . . 22
5 Fourierseriesofsquaresummablefunctions . . . . . . 27
6 Absolutelyconvergentfourierseries. . . . . . . . . . . 31
7 Fouriercoefficientsoflinearfunctionals . . . . . . . . 35
8 Additionalcommentsandapplications . . . . . . . . . 48
II TheConvergenceofFourierSeries 55
1 Convergenceinnorm . . . . . . . . . . . . . . . . . . . 55
2 Convergenceanddivergenceatapoint . . . . . . . . . 60
?3 Setsofdivergence . . . . . . . . . . . . . . . . . . . . . 64
III TheConjugateFunction 71
1 Theconjugatefunction . . . . . . . . . . . . . . . . . . 71
2 ThemaximalfunctionofHardyandLittlewood . . . . 84
3 TheHardyspaces . . . . . . . . . . . . . . . . . . . . . 93
IV InterpolationofLinearOperators 105
1 Interpolationofnormsandoflinearoperators . . . . . 105
2 ThetheoremofHausdorff-Young . . . . . . . . . . . . 111
V LacunarySeriesandQuasi-analyticClasses 117
1 Lacunaryseries . . . . . . . . . . . . . . . . . . . . . . 117
?2 Quasi-analyticclasses. . . . . . . . . . . . . . . . . . . 125
vii
VIII ANINTRODUCTIONTOHARMONICANALYSIS
VI FourierTransformsontheLine 132
1 FouriertransformsforL1(R) . . . . . . . . . . . . . . . 133
2 Fourier-Stieltjestransforms. . . . . . . . . . . . . . . . 144
3 FouriertransformsinLp(R);1<p≤2. . . . . . . . . . 155
4 Tempereddistributionsandpseudo-measures . . . . . 162
5 Almost-Periodicfunctionsontheline . . . . . . . . . 170
6 Theweak-starspectrumofboundedfunctions . . . . . 184
7 ThePaley–Wienertheorems . . . . . . . . . . . . . . . 188
?8 TheFourier-Carlemantransform. . . . . . . . . . . . . 193
9 Kronecker’stheorem . . . . . . . . . . . . . . . . . . . 196
VII FourierAnalysisonLocallyCompact
AbelianGroups 201
1 Locallycompactabeliangroups . . . . . . . . . . . . . 201
2 TheHaarmeasure . . . . . . . . . . . . . . . . . . . . 202
3 Charactersandthedualgroup . . . . . . . . . . . . . . 203
4 Fouriertransforms . . . . . . . . . . . . . . . . . . . . 205
5 Almost-periodicfunctionsandtheBohr
compactification . . . . . . . . . . . . . . . . . . . . . . 206
VIIICommutativeBanachAlgebras 209
1 Definition,examples,andelementaryproperties . . . . 209
2 Maximalidealsandmultiplicative
linearfunctionals . . . . . . . . . . . . . . . . . . . . . 213
3 Themaximal-idealspaceandthe
Gelfandrepresentation . . . . . . . . . . . . . . . . . . 220
4 HomomorphismsofBanachalgebras . . . . . . . . . . 228
5 Regularalgebras. . . . . . . . . . . . . . . . . . . . . . 236
6 Wiener’sgeneralTauberiantheorem. . . . . . . . . . . 241
7 Spectralsynthesisinregularalgebras . . . . . . . . . . 244
8 Functionsthatoperateinregular
Banachalgebras . . . . . . . . . . . . . . . . . . . . . . 250
9 ThealgebraM(T)andfunctionsthatoperateon
Fourier-Stieltjescoefficients . . . . . . . . . . . . . . . 259
10 Theuseoftensorproducts . . . . . . . . . . . . . . . . 264
A Vector-ValuedFunctions 272
1 Riemannintegration . . . . . . . . . . . . . . . . . . . 272
2 Improperintegrals . . . . . . . . . . . . . . . . . . . . . 273
3 Moregeneralintegrals . . . . . . . . . . . . . . . . . . 273
CONTENTS IX
4 Holomorphicvector-valuedfunctions . . . . . . . . . . 273
B ProbabilisticMethods 275
1 Randomseries . . . . . . . . . . . . . . . . . . . . . . . 275
2 Fouriercoefficientsofcontinuousfunctions . . . . . . 278
3 Paley–Zygmund, when |a |2 =∞ . . . . . . . . . 279
n
(cid:0) P (cid:1)
Bibliography 283
Index 285
Symbols
HC(D),210 K ,12
n
A(T),31 P(r,t),16
B ,14 V (t),15
c n
C(T),14 Trim ,277
λ
Cn(T),14 r ,276
n
Cm+η(T),48 S[f],3
D ,13 fe∗g,5
n
E (ϕ),48 f ∗ g,179
n
M
M(T),38 f ,4
τ
Pinv,41 D,202
S[µ],35 R,1
S[f],3 T,1
S (µ),36 Z,1
n
S (µ,t),37 Dˆ,205
n
S (f),13
n
Lip (T),16
α
Ω(f,h),25
H,28
H ,40
f
δ,38
δ ,38
τ
fˆ(n),3
χ
,153
X
lip (T),16
α
L1(T),2
L∞(T),16
Lp(T),15
µ ,40
f
ω(f,h),25
σ (µ),36
n
σ (µ,t),37
n
σ (f),12
n
σ (f,t),12
n
x