Principles of harmonic analysis

336 Pages·2009·1.32 MB·English

by  Anton Deitmar

Universitext
Forothertitlesinthisseries,goto
www.springer.com/series/223
·
Anton Deitmar Siegfried Echterhoff
Principles of Harmonic
Analysis
123
AntonDeitmar SiegfriedEchterhoff
UniversitätTübingen UniversitätMünster
Inst.Mathematik MathematischesInstitut
AufderMorgenstelle10 Einsteinstr.62
72076Tübingen 48149Münster
Germany Germany
[email protected] [email protected]
Editorialboard:
SheldonAxler,SanFranciscoStateUniversity,SanFrancisco,CA,USA
VincenzoCapasso,UniversityofMilan,Milan,Italy
CarlesCasacuberta,UniversitatdeBarcelona,Barcelona,Spain
AngusMacIntyre,QueenMary,UniversityofLondon,London,UK
KennethRibet,UniversityofCalifornia,Berkeley,CA,USA
ClaudeSabbah,EcolePolytechnique,Palaiseau,France
EndreSüli,OxfordUniversity,Oxford,UK
WojborWoyczynski,CaseWesternReserveUniversity,Cleveland,OH,USA
ISBN:978-0-387-85468-7 e-ISBN:978-0-387-85469-4
DOI:10.1007/978-0-387-85469-4
LibraryofCongressControlNumber:2008938333
MathematicsSubjectClassification(2000):43-01,42-01,22Bxx
(cid:1)c SpringerScience+BusinessMedia,LLC2009
Allrightsreserved.Thisworkmaynotbetranslatedorcopiedinwholeorinpartwithoutthewritten
permissionofthepublisher(SpringerScience+BusinessMedia,LLC,233SpringStreet,NewYork,
NY 10013, USA), except for brief excerpts inconnection with reviews orscholarly analysis. Use
in connection with any form of information storage and retrieval, electronic adaptation, computer
software,orbysimilarordissimilarmethodologynowknownorhereafterdevelopedisforbidden.
Theuseinthispublicationoftradenames,trademarks,servicemarks,andsimilarterms,evenifthey
arenotidentifiedassuch,isnottobetakenasanexpressionofopinionastowhetherornottheyare
subjecttoproprietaryrights.
Printedonacid-freepaper
springer.com
Preface
The tread of this book is formed by two fundamental principles of
Harmonic Analysis: the Plancherel Formula and the Poisson Sum-
mation 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 of
the Poisson Summation Formula to non-abelian groups is the Sel-
berg Trace Formula, which we prove for arbitrary groups admitting
uniform lattices. As examples for the application of the Trace For-
mula we treat the Heisenberg group and the group SL (R). In the
2
former case the trace formula yields a decomposition of the L2-space
of the Heisenberg group modulo a lattice. In the case SL (R), the
2
trace formula is used to derive results like the Weil asymptotic law
for hyperbolic surfaces and to provide the analytic continuation of
the Selberg zeta function. We finally include a chapter on the appli-
cations of abstract Harmonic Analysis on the theory of wavelets.
The present book is a text book for a graduate course on abstract
harmonic analysis and its applications. The book can be used as a
follow up of the First Course in Harmonic Analysis, [9], or indepen-
dently, if the students have required a modest knowledge of Fourier
Analysis already. In this book, among other things, proofs are given
of Pontryagin Duality and the Plancherel Theorem for LCA-groups,
which were mentioned but not proved in [9]. Using Pontryagin du-
ality, we also obtain various structure theorems for locally compact
abelian groups.
Knowledge of set theoretic topology, Lebesgue integration, and func-
tional analysis on an introductory level will be required in the body
of the book. For the convenience of the reader we have included all
necessary ingredients from these areas in the appendices.
v
Acknowledgments
The authors thank the following people for corrections and com-
ments on the book: Ralf Beckmann, Wolfgang Bertram, Alexander
Schmidt, Dana Williams.
Chapters 3 and 4 are partly based on written notes of a course given
by Professor Eberhard Kaniuth on duality theory for abelian locally
compact groups. The authors are grateful to Professor Kaniuth for
allowing us to use this material.
vii
Chapter Dependency
1 2
(cid:2)(cid:1)
(cid:2)(cid:2)(cid:2)(cid:1)(cid:2)(cid:1)(cid:2)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1) (cid:3)(cid:3)(cid:3)(cid:3)(cid:3)(cid:3)(cid:3)
(cid:2) (cid:1)(cid:1) (cid:3)(cid:3)(cid:3) (cid:4)(cid:4)
(cid:2)
(cid:2)
(cid:2) 3,4 5
(cid:2)(cid:2)(cid:2) (cid:3)(cid:3)(cid:3)
(cid:2) (cid:3)
(cid:2)(cid:2) (cid:3)(cid:3)(cid:3)
(cid:2)(cid:2) (cid:3)(cid:3)(cid:3)
6
(cid:1)
(cid:3) (cid:1)
(cid:3) (cid:1)
(cid:3) (cid:1)
(cid:3) (cid:1)
(cid:3) (cid:1)
(cid:3) (cid:1)
(cid:3) (cid:1)
(cid:3)(cid:3)(cid:3) (cid:4)(cid:4) (cid:1)(cid:1)
7 8 9
(cid:1) (cid:1)
(cid:3) (cid:1) (cid:3) (cid:1)
(cid:3) (cid:1) (cid:3) (cid:1)
(cid:3) (cid:1) (cid:3) (cid:1)
(cid:3) (cid:1) (cid:3) (cid:1)
(cid:3) (cid:1) (cid:3) (cid:1)
(cid:3) (cid:1) (cid:3) (cid:1)
(cid:3) (cid:1) (cid:3) (cid:1)
(cid:3)(cid:3)(cid:3) (cid:1)(cid:1) (cid:3)(cid:3)(cid:3) (cid:1)(cid:1)
12 11 10
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 the characteristic function of A, i.e., 1 (x)
A A
is 1 if x ∈ A and zero otherwise. The Kronecker-delta function is
defined to be (cid:1)
1 if i = j,
δ d=ef
i,j
0 otherwise.
The word positive will always mean ≥ 0. For > 0, we use the words
strictly positive.
ix
Contents
1 Haar Integration 1
1.1 Topological Groups . . . . . . . . . . . . . . . . . . . . 1
1.2 Locally Compact Groups. . . . . . . . . . . . . . . . . 6
1.3 Haar Measure . . . . . . . . . . . . . . . . . . . . . . . 8
1.4 The Modular Function . . . . . . . . . . . . . . . . . . 17
1.5 The Quotient Integral Formula . . . . . . . . . . . . . 21
1.6 Convolution . . . . . . . . . . . . . . . . . . . . . . . . 25
1.7 The Fourier Transform . . . . . . . . . . . . . . . . . . 29
1.8 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . 30
2 Banach Algebras 33
2.1 Banach Algebras . . . . . . . . . . . . . . . . . . . . . 33
2.2 The Spectrum σA(a) . . . . . . . . . . . . . . . . . . . 37
2.3 Adjoining a Unit . . . . . . . . . . . . . . . . . . . . . 41
2.4 The Gelfand Map . . . . . . . . . . . . . . . . . . . . . 43
2.5 Maximal Ideals . . . . . . . . . . . . . . . . . . . . . . 47
2.6 The Gelfand-Naimark Theorem . . . . . . . . . . . . . 49
2.7 The Continuous Functional Calculus . . . . . . . . . . 54
2.8 Exercises and Notes . . . . . . . . . . . . . . . . . . . 58
3 Duality for Abelian Groups 63
xi
xii CONTENTS
3.1 The Dual Group . . . . . . . . . . . . . . . . . . . . . 63
3.2 The Fourier Transform . . . . . . . . . . . . . . . . . . 67
3.3 The C∗-Algebra of an LCA-Group . . . . . . . . . . . 69
3.4 The Plancherel Theorem . . . . . . . . . . . . . . . . . 72
3.5 Pontryagin Duality . . . . . . . . . . . . . . . . . . . . 78
3.6 The Poisson Summation Formula . . . . . . . . . . . . 83
3.7 Exercises and Notes . . . . . . . . . . . . . . . . . . . 86
4 The Structure of LCA-Groups 91
4.1 Connectedness . . . . . . . . . . . . . . . . . . . . . . 91
4.2 The Structure Theorems . . . . . . . . . . . . . . . . . 95
4.3 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . 107
5 Operators on Hilbert Spaces 109
5.1 Functional Calculus . . . . . . . . . . . . . . . . . . . 109
5.2 Compact Operators . . . . . . . . . . . . . . . . . . . 115
5.3 Hilbert-Schmidt and Trace Class . . . . . . . . . . . . 119
5.4 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . 124
6 Representations 127
6.1 Schur’s Lemma . . . . . . . . . . . . . . . . . . . . . . 127
6.2 Representations of L1(G) . . . . . . . . . . . . . . . . 131
6.3 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . 135
7 Compact Groups 139
7.1 Finite Dimensional Representations . . . . . . . . . . . 139
7.2 The Peter-Weyl Theorem . . . . . . . . . . . . . . . . 141
7.3 Isotypes . . . . . . . . . . . . . . . . . . . . . . . . . . 145
7.4 Induced Representations . . . . . . . . . . . . . . . . . 148
CONTENTS xiii
7.5 Representations of SU(2) . . . . . . . . . . . . . . . . 150
7.6 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . 156
8 Direct Integrals 159
8.1 Von Neumann Algebras . . . . . . . . . . . . . . . . . 159
8.2 Weak and Strong Topologies . . . . . . . . . . . . . . 160
8.3 Representations . . . . . . . . . . . . . . . . . . . . . . 162
8.4 Hilbert Integrals . . . . . . . . . . . . . . . . . . . . . 165
8.5 The Plancherel Theorem . . . . . . . . . . . . . . . . . 167
8.6 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . 169
9 The Selberg Trace Formula 171
9.1 Cocompact Groups and Lattices . . . . . . . . . . . . 171
9.2 Discreteness of the Spectrum . . . . . . . . . . . . . . 174
9.3 The Trace Formula . . . . . . . . . . . . . . . . . . . . 179
9.4 Locally Constant Functions . . . . . . . . . . . . . . . 185
9.5 Exercises and Notes . . . . . . . . . . . . . . . . . . . 186
10 The Heisenberg Group 189
10.1 Definition . . . . . . . . . . . . . . . . . . . . . . . . . 189
10.2 The Unitary Dual . . . . . . . . . . . . . . . . . . . . 190
10.3 The Plancherel Theorem for H . . . . . . . . . . . . . 195
10.4 The Standard Lattice . . . . . . . . . . . . . . . . . . 196
10.5 Exercises and Notes . . . . . . . . . . . . . . . . . . . 199
11 SL (R) 201
2
11.1 The Upper Half Plane . . . . . . . . . . . . . . . . . . 201
11.2 The Hecke Algebra . . . . . . . . . . . . . . . . . . . . 205
11.3 An Explicit Plancherel Theorem . . . . . . . . . . . . 214