Principles of Harmonic Analysis

330 Pages·2014·2.21 MB·English

by  Anton Deitmar

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Anton Deitmar (cid:129) Siegfried Echterhoff
Principles of Harmonic
Analysis
Second Edition
2123
AntonDeitmar SiegfriedEchterhoff
UniversitätTübingenInstitutfürMathematik UniversitätMünsterMathematischesInstitut
Tübingen Münster
Baden-Württemberg Germany
Germany
ISSN0172-5939 ISSN2191-6675(electronic)
ISBN978-3-319-05791-0 ISBN978-3-319-05792-7(eBook)
DOI10.1007/978-3-319-05792-7
SpringerChamHeidelbergNewYorkDordrechtLondon
LibraryofCongressControlNumber:2014934841
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Preface
ThethreadofthisbookisformedbytwofundamentalprinciplesofHarmonicAnal-
ysis: the Plancherel Formula and the Poisson Summation Formula. We first prove
both for locally compact abelian groups. For non-abelian groups we discuss the
Plancherel Theorem in the general situation for type-I groups. The generalization
ofthePoissonSummationFormulatonon-abeliangroupsistheSelbergTraceFor-
mula,whichweproveforarbitrarygroupsadmittinguniformlattices.Asexamples
fortheapplicationoftheTraceFormulawetreattheHeisenberggroupandthegroup
SL (R).IntheformercasethetraceformulayieldsadecompositionoftheL2-space
2
oftheHeisenberggroupmoduloalattice. InthecaseSL (R), thetraceformulais
2
used to derive results like the Weil asymptotic law for hyperbolic surfaces and to
providetheanalyticcontinuationoftheSelbergzetafunction.Wefinallyincludea
chapterontheapplicationsofabstractHarmonicAnalysisonthetheoryofwavelets,
andweincludeachapteronp-adicandadelicgroups,whichareimportantexamples,
astheyareusedinnumbertheory.
Thepresentbookisatextbookforagraduatecourseonabstractharmonicanalysis
and its applications. The book can be used as a follow up of the First Course in
HarmonicAnalysis,[Dei05],orindependently,ifthestudentshaverequiredamodest
knowledgeofFourierAnalysisalready.Inthisbook,amongotherthings,proofsare
given of Pontryagin Duality and the Plancherel Theorem for LCA groups, which
werementionedbutnotprovedin[Dei05].UsingPontryaginduality,wealsoobtain
variousstructuretheoremsforlocallycompactabeliangroups.
Knowledgeofsettheoretictopology,Lebesgueintegration,andfunctionalanalysis
onanintroductorylevelwillberequiredinthebodyofthebook.Fortheconvenience
of the reader we have included all necessary ingredients from these areas in the
appendices.
Differences to the first edition: Many details have been changed, new and better
proofs have been found, some assertions have been sharpened and a few are even
new to this book. Section 1.8 and Chap. 13 have not been part of the first edition.
Whilstfittinginthechanges,wetriedtopreservethenumberingofTheoremsetc.We
apologizeforinconveniencesthatariseatthoseplaceswherethiswasnotpossible.
v
Acknowledgments
Theauthorsthankthefollowingpeopleforcorrectionsandcommentsonthebook:
Ralf Beckmann, Wolfgang Bertram, Robert Burckel, Cody Gunton, Linus Kramer,
YiLi,JonasMorrissey,MichaelMueger,KennethRoss,AlexanderSchmidt,Christian
Schmidt,VahidShirbisheh,FrankValckenborgh,FabianWerner,DanaWilliams.
Chapters3and4arepartlybasedonwrittennotesofacoursegivenbyProf.Eberhard
Kaniuthondualitytheoryforabelianlocallycompactgroups.Theauthorsaregrateful
toProf.Kaniuthforallowingustousethismaterial.
vii
Chapter Dependency
1 2
3 5
4 6
13
7
8 9 12
10 11
Notation
We write N = {1,2,3,...} for the set of natural numbers. The sets of integer,
real, and complex numbers are denoted as Z,R,C. For a set A we write 1 for
A
the characteristic function of A, i.e., 1 (x) is 1 ifx ∈ A and zero otherwise. The
A
Kronecker-deltafunctionisdefinedtobe
(cid:2)
δ =def 1 ifi=j,
i,j 0 otherwise.
Thewordpositivewillalwaysmean≥0.For>0,weusethewordsstrictlypositive.
ix
Contents
1 HaarIntegration.............................................. 1
1.1 TopologicalGroups....................................... 1
1.2 LocallyCompactGroups .................................. 5
1.3 HaarMeasure............................................ 6
1.4 TheModularFunction .................................... 14
1.5 TheQuotientIntegralFormula ............................. 17
1.6 Convolution ............................................. 22
1.7 TheFourierTransform .................................... 25
1.8 Limits .................................................. 26
1.9 Exercises ............................................... 33
2 BanachAlgebras .............................................. 37
2.1 BanachAlgebras ......................................... 37
2.2 TheSpectrumσA(a)...................................... 40
2.3 AdjoiningaUnit ......................................... 43
2.4 TheGelfandMap ........................................ 45
2.5 MaximalIdeals .......................................... 48
2.6 TheGelfand-NaimarkTheorem............................. 49
2.7 TheContinuousFunctionalCalculus ........................ 54
2.8 ExercisesandNotes ...................................... 57
3 DualityforAbelianGroups..................................... 61
3.1 TheDualGroup.......................................... 61
3.2 TheFourierTransform .................................... 64
3.3 TheC∗-AlgebraofanLCA-Group .......................... 66
3.4 ThePlancherelTheorem................................... 69
3.5 PontryaginDuality ....................................... 73
3.6 ThePoissonSummationFormula ........................... 77
3.7 ExercisesandNotes ...................................... 80
xi
xii Contents
4 TheStructureofLCA-Groups.................................. 85
4.1 Connectedness........................................... 85
4.2 TheStructureTheorems ................................... 93
4.3 Exercises ............................................... 105
5 OperatorsonHilbertSpaces ................................... 107
5.1 FunctionalCalculus ...................................... 107
5.2 CompactOperators ....................................... 111
5.3 Hilbert-SchmidtandTraceClass............................ 114
5.4 Exercises ............................................... 119
6 Representations............................................... 123
6.1 Schur’sLemma .......................................... 123
6.2 RepresentationsofL1(G).................................. 127
6.3 Exercises ............................................... 130
7 CompactGroups.............................................. 133
7.1 FiniteDimensionalRepresentations ......................... 133
7.2 ThePeter-WeylTheorem .................................. 135
7.3 Isotypes ................................................ 142
7.4 InducedRepresentations................................... 144
7.5 RepresentationsofSU(2) .................................. 146
7.6 Exercises ............................................... 150
8 DirectIntegrals ............................................... 153
8.1 VonNeumannAlgebras ................................... 153
8.2 WeakandStrongTopologies ............................... 154
8.3 Representations .......................................... 155
8.4 HilbertIntegrals.......................................... 159
8.5 ThePlancherelTheorem................................... 160
8.6 Exercises ............................................... 161
9 TheSelbergTraceFormula..................................... 165
9.1 CocompactGroupsandLattices ............................ 165
9.2 DiscretenessoftheSpectrum............................... 167
9.3 TheTraceFormula ....................................... 172
9.4 LocallyConstantFunctions ................................ 177
9.5 LieGroups.............................................. 177
9.6 Exercises ............................................... 182
10 TheHeisenbergGroup......................................... 185
10.1 Definition............................................... 185
10.2 TheUnitaryDual......................................... 186
10.3 ThePlancherelTheoremforH ............................. 190
10.4 TheStandardLattice...................................... 190
10.5 ExercisesandNotes ...................................... 193