Introduction to harmonic analysis and generalized Gelfand pairs

234 Pages·2009·1.05 MB·English

by  Gerrit van Dijk

de Gruyter Studies in Mathematics 36
Editors: Carsten Carstensen · Nicola Fusco
Niels Jacob · Karl-Hermann Neeb
de Gruyter Studies in Mathematics
1 Riemannian Geometry, 2nd rev. ed., Wilhelm P.A.Klingenberg
2 Semimartingales, Michel Me´tivier
3 Holomorphic Functions of Several Variables, Ludger Kaup and Burchard Kaup
4 Spaces of Measures, Corneliu Constantinescu
5 Knots, 2nd rev. and ext. ed., Gerhard Burde and Heiner Zieschang
6 Ergodic Theorems, Ulrich Krengel
7 Mathematical Theory of Statistics, Helmut Strasser
8 Transformation Groups, Tammo tom Dieck
9 Gibbs Measures and Phase Transitions, Hans-Otto Georgii
10 Analyticity in Infinite Dimensional Spaces, Michel Herve´
11 Elementary Geometry in Hyperbolic Space, Werner Fenchel
12 Transcendental Numbers, Andrei B. Shidlovskii
13 Ordinary Differential Equations, Herbert Amann
14 Dirichlet Forms and Analysis on Wiener Space, Nicolas Bouleau and
Francis Hirsch
15 Nevanlinna Theory and Complex Differential Equations, Ilpo Laine
16 Rational Iteration, Norbert Steinmetz
17 Korovkin-type Approximation Theory and its Applications, Francesco
Altomare and Michele Campiti
18 Quantum Invariants of Knots and 3-Manifolds, Vladimir G.Turaev
19 Dirichlet Forms and Symmetric Markov Processes, Masatoshi Fukushima,
Yoichi Oshima and Masayoshi Takeda
20 Harmonic Analysis of Probability Measures on Hypergroups, Walter R.Bloom
and Herbert Heyer
21 Potential Theory on Infinite-Dimensional Abelian Groups, Alexander Bendikov
22 Methods of Noncommutative Analysis, Vladimir E. Nazaikinskii,
Victor E. Shatalov and Boris Yu. Sternin
23 Probability Theory, Heinz Bauer
24 Variational Methods for Potential Operator Equations, Jan Chabrowski
25 The Structure of Compact Groups, 2nd rev. and aug. ed., Karl H.Hofmann
and Sidney A.Morris
26 Measure and Integration Theory, Heinz Bauer
27 Stochastic Finance, 2nd rev. and ext. ed., Hans Föllmer and Alexander Schied
28 Painleve´ Differential Equations in the Complex Plane, Valerii I.Gromak,
Ilpo Laine and Shun Shimomura
29 Discontinuous Groups of Isometries in the Hyperbolic Plane, Werner Fenchel
and Jakob Nielsen
30 The Reidemeister Torsion of 3-Manifolds, Liviu I. Nicolaescu
31 Elliptic Curves, Susanne Schmitt and Horst G. Zimmer
32 Circle-valued Morse Theory, Andrei V. Pajitnov
33 Computer Arithmetic and Validity, Ulrich Kulisch
34 Feynman-Kac-Type Theorems and Gibbs Measures on Path Space, Jo´zsef Lörinczi,
Fumio Hiroshima and Volker Betz
35 Integral Representation Theory, Jaroslas Lukesˇ, Jan Maly´, Ivan Netuka and Jiri
Spurny´
Gerrit van Dijk
Introduction to Harmonic Analysis
and Generalized Gelfand Pairs
≥
Walter de Gruyter
Berlin · New York
Author
Gerrit van Dijk
Mathematical Institute
Leiden University
2300 RA Leiden, Netherlands
E-Mail: [email protected]
Series Editors
Carsten Carstensen Niels Jacob
Department of Mathematics Department of Mathematics
Humboldt University of Berlin Swansea University
Unter den Linden 6 Singleton Park
10099 Berlin, Germany Swansea SA2 8PP, Wales, United Kingdom
E-Mail: [email protected] E-Mail: [email protected]
Nicola Fusco Karl-Hermann Neeb
Dipartimento di Matematica Department of Mathematics
Universita` di Napoli Frederico II Technische Universität Darmstadt
Via Cintia Schloßgartenstraße 7
80126 Napoli, Italy 64289 Darmstadt, Germany
E-Mail: [email protected] E-Mail: [email protected]
Mathematics Subject Classification 2000: 43-01, 43-02, 43A85, 22Bxx, 22Dxx
Keywords: Fourier theory, harmonic analysis, Gelfand pairs, locally compact groups, Haar
measures
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Preface
Thisbookisintendedasanintroductiontoharmonicanalysis,andespeciallytomy
favorite topic, generalized Gelfand pairs. It is aimed at advanced undergraduates
or beginning graduate students. The scope of the book is limited, with the aim of
enabling students to reach a level suitable for starting PhD-research. It is based
on lectures I have given in several places, most recently at Kyushu University in
Fukuoka, Japan (2008). Student input has strongly influenced the writing, and I
hope that this book will help students to share my enthusiasm for the beautiful
topicsdiscussed.
Starting with the elementary theory of Fourier series and Fourier integrals, I
proceed to abstract harmonic analysis on locally compact abelian groups. Here I
follow the classical paper [7] of H. Cartan and R. Godement. It turns out that the
vi Preface
technique they developed works as well for Gelfand pairs .G;K/, where G is a
not necessarily abelian locally compact group, and K is a compact subgroup of
G. This approach is based on my thesis [51]. Finally I develop part of the theory
of generalized Gelfand pairs .G;H/ where H is a closed, possibly noncompact,
subgroupofG. ThebasicideasareduetoE.G.F.Thomas, seeforexample[49],
and several applications are from my own work [52]. I also draw on papers by
J. Faraut and V.F. Molchanov to deal with examples related to the generalized
Lorentzgroup.
There is relatively little expository literature on generalized Gelfand pairs, and
thereisnostandardreference. Forfurtherreading,IrecommendmyKyushuLec-
tureNote[53].
The main prerequisites for the book are elementary real, complex and func-
tional analysis. In the later chapters we shall assume familiarity with some more
advancedfunctionalanalysis,inparticularwiththespectraltheoryof(unbounded)
self-adjoint operators on a Hilbert space. Some knowledge of distribution theory
andLietheoryisalsoassumed. Referencestothesetopicsaregiveninthetext.
ForterminologyandnotationswegenerallyfollowN.Bourbaki. Proofsfollow-
ing theorems, propositions and lemmas are written in small print. The index will
behelpfultotraceimportantnotionsdefinedinthetext.
Thanks are due to my colleagues and students in several countries for their re-
marks and suggestions. Especial thanks are due to Dr. J.D. Stegeman (Utrecht)
whosehelpindevelopingthefinalversionofthemanuscripthasgreatlyimproved
thepresentation.
Leiden,September2009
GerritvanDijk
The picture shows Snellius’ quadrant. Snellius (1580–1626), best known for Snell’s law on the
breaking of light, used this quadrant to measure the earth. It is now on display in the Museum
Boerhaave at Leiden. The building of the Leiden Mathematical Institute is named after Snellius.
Areplicaofthequadrantwasofferedtomeontheoccasionofmyretirementandplacedinthehall
oftheSnelliusbuilding.
Contents
Preface v
1 FourierSeries 1
1.1 Definitionandelementaryproperties . . . . . . . . . . . . . . . . 1
1.2 Convergence . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
1.3 Uniformconvergence . . . . . . . . . . . . . . . . . . . . . . . . 3
1.4 SomefactsaboutconvergenceofFourierseries . . . . . . . . . . 5
1.5 Parseval’stheorem . . . . . . . . . . . . . . . . . . . . . . . . . 5
1.6 Generalization . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
2 FourierIntegrals 7
2.1 Theconvolutionproduct . . . . . . . . . . . . . . . . . . . . . . 7
2.2 ElementarypropertiesoftheFourierintegral . . . . . . . . . . . . 8
2.3 Theinversiontheorem . . . . . . . . . . . . . . . . . . . . . . . 9
2.4 Plancherel’stheorem . . . . . . . . . . . . . . . . . . . . . . . . 12
2.5 ThePoissonsummationformula . . . . . . . . . . . . . . . . . . 13
2.6 TheRiemann–Stieltjesintegralandfunctionsofboundedvariation 13
2.7 Bochner’stheorem . . . . . . . . . . . . . . . . . . . . . . . . . 19
2.8 ExtensiontoRn . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
3 LocallyCompactGroups 25
3.1 Groups. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
3.2 Topologicalspaces . . . . . . . . . . . . . . . . . . . . . . . . . 25
3.3 Topologicalgroups . . . . . . . . . . . . . . . . . . . . . . . . . 27
3.4 Quotientspacesandquotientgroups . . . . . . . . . . . . . . . . 29
3.5 Someusefulfacts . . . . . . . . . . . . . . . . . . . . . . . . . . 30
3.6 Functionsonlocallycompactgroups . . . . . . . . . . . . . . . . 31
4 HaarMeasures 32
4.1 Measures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32
4.2 Invariantmeasures . . . . . . . . . . . . . . . . . . . . . . . . . 34
4.3 Weil’sformula. . . . . . . . . . . . . . . . . . . . . . . . . . . . 39
4.4 Haarmeasuresforspecificgroups . . . . . . . . . . . . . . . . . 41
4.5 Quasi-invariantmeasuresonquotientspaces . . . . . . . . . . . . 41
4.6 TheconvolutionproductonG. PropertiesofL1.G/ . . . . . . . . 45
viii Contents
5 HarmonicAnalysisonLocallyCompactAbelianGroups 49
5.1 Positive-definitefunctionsandunitaryrepresentations . . . . . . . 49
5.2 Somefunctionalanalysis . . . . . . . . . . . . . . . . . . . . . . 55
5.3 Elementarypositive-definitefunctions . . . . . . . . . . . . . . . 60
5.4 Fouriertransform,Riemann–LebesguelemmaandBochner’sthe-
orem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62
5.5 Theinversiontheorem . . . . . . . . . . . . . . . . . . . . . . . 66
5.6 Plancherel’stheorem . . . . . . . . . . . . . . . . . . . . . . . . 69
5.7 Pontryagin’sdualitytheorem . . . . . . . . . . . . . . . . . . . . 70
5.8 Subgroupsandquotientgroups . . . . . . . . . . . . . . . . . . . 73
5.9 Compactanddiscreteabeliangroups . . . . . . . . . . . . . . . . 74
6 ClassicalTheoryofGelfandPairs 75
6.1 Gelfandpairsandsphericalfunctions. . . . . . . . . . . . . . . . 75
6.2 Positive-definitesphericalfunctionsandunitaryrepresentations. . 79
6.3 Representationsofclassone . . . . . . . . . . . . . . . . . . . . 82
6.4 HarmonicanalysisonGelfandpairs . . . . . . . . . . . . . . . . 83
6.5 CompactGelfandpairs . . . . . . . . . . . . . . . . . . . . . . . 86
7 ExamplesofGelfandPairs 89
7.1 Euclideanmotiongroups . . . . . . . . . . . . . . . . . . . . . . 89
7.2 Thesphere . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95
7.3 Sphericalharmonics. . . . . . . . . . . . . . . . . . . . . . . . . 96
7.4 Sphericalfunctionsonspheres . . . . . . . . . . . . . . . . . . . 105
7.5 Realhyperbolicspaces . . . . . . . . . . . . . . . . . . . . . . . 106
8 TheoryofGeneralizedGelfandPairs 131
1
8.1 C vectorsofarepresentation . . . . . . . . . . . . . . . . . . . 131
8.2 InvariantHilbertsubspaces . . . . . . . . . . . . . . . . . . . . . 135
8.3 GeneralizedGelfandpairs . . . . . . . . . . . . . . . . . . . . . 141
8.4 InvariantHilbertsubspacesofL2.G=H/ . . . . . . . . . . . . . . 145
9 ExamplesofGeneralizedGelfandPairs 152
9.1 Non-Euclideanmotiongroups . . . . . . . . . . . . . . . . . . . 152
9.2 Pseudo-Riemannianrealhyperbolicspaces. . . . . . . . . . . . . 158
A TheAveragingMappingontheSpaceRnC1 195
A.1 SpecialcaseofatheoremofHarish-Chandra. . . . . . . . . . . . 195
A.2 ResultsofMéthée . . . . . . . . . . . . . . . . . . . . . . . . . . 197
A.3 ResultsofTengstrand . . . . . . . . . . . . . . . . . . . . . . . . 200
A.4 SolutionsinH0 ofasingularsecondorderdifferentialequation. . 205
(cid:2)
A.5 ExpressionofMb.(cid:2)/intermsofBesselfunctions . . . . . . . . . 208
f
Contents ix
B TheAveragingMappingontheSpaceX 210
B.1 SpecialcaseofatheoremofHarish-Chandra. . . . . . . . . . . . 210
B.2 AnalogofMéthée’sresults . . . . . . . . . . . . . . . . . . . . . 210
B.3 Tengstrand’sresultsforX . . . . . . . . . . . . . . . . . . . . . 212
B.4 SolutionsinH0 ofasingularsecondorderdifferentialequation. . 213
(cid:2)
Bibliography 217
Index 221