For other titles in this series, go to Universitext

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1 For other titles in this series, go to Universitext

2 Anton Deitmar Siegfried Echterhoff Principles of Harmonic Analysis 123

3 Anton Deitmar Universität Tübingen Inst. Mathematik Auf der Morgenstelle Tübingen Germany Siegfried Echterhoff Universität Münster Mathematisches Institut Einsteinstr Münster Germany Editorial board: Sheldon Axler, San Francisco State University, San Francisco, CA, USA Vincenzo Capasso, University of Milan, Milan, Italy Carles Casacuberta, Universitat de Barcelona, Barcelona, Spain Angus MacIntyre, Queen Mary, University of London, London, UK Kenneth Ribet, University of California, Berkeley, CA, USA Claude Sabbah, Ecole Polytechnique, Palaiseau, France Endre Süli, Oxford University, Oxford, UK Wojbor Woyczynski, Case Western Reserve University, Cleveland, OH, USA ISBN: e-isbn: DOI: / Library of Congress Control Number: Mathematics Subject Classification (2000): 43-01, 42-01, 22Bxx c Springer Science+Business Media, LLC 2009 All rights reserved. This work may not be translated or copied in whole or in part without the written permission of the publisher (Springer Science+Business Media, LLC, 233 Spring Street, New York, NY 10013, USA), except for brief excerpts in connection with reviews or scholarly analysis. Use in connection with any form of information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed is forbidden. The use in this publication of trade names, trademarks, service marks, and similar terms, even if they are not identified as such, is not to be taken as an expression of opinion as to whether or not they are subject to proprietary rights. Printed on acid-free paper springer.com

4 Preface The tread of this book is formed by two fundamental principles of Harmonic Analysis: 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 of the Poisson Summation Formula to non-abelian groups is the Selberg Trace Formula, which we prove for arbitrary groups admitting uniform lattices. As examples for the application of the Trace Formula we treat the Heisenberg group and the group SL 2 (R). In the former case the trace formula yields a decomposition of the L 2 -space of the Heisenberg group modulo a lattice. In the case SL 2 (R), the 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 applications 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 independently, 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 duality, we also obtain various structure theorems for locally compact abelian groups. Knowledge of set theoretic topology, Lebesgue integration, and functional 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

5 Acknowledgments The authors thank the following people for corrections and comments 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

6 Chapter Dependency 1 2 3, 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 A for the characteristic function of A, i.e., 1 A (x) is 1 if x A and zero otherwise. The Kronecker-delta function is defined to be δ i,j def = { 1 if i = j, 0 otherwise. The word positive will always mean 0. For > 0, we use the words strictly positive. ix

7 Contents 1 Haar Integration Topological Groups Locally Compact Groups Haar Measure The Modular Function The Quotient Integral Formula Convolution The Fourier Transform Exercises Banach Algebras Banach Algebras The Spectrum σ A (a) Adjoining a Unit The Gelfand Map Maximal Ideals The Gelfand-Naimark Theorem The Continuous Functional Calculus Exercises and Notes Duality for Abelian Groups 63 xi

8 xii CONTENTS 3.1 The Dual Group The Fourier Transform The C -Algebra of an LCA-Group The Plancherel Theorem Pontryagin Duality The Poisson Summation Formula Exercises and Notes The Structure of LCA-Groups Connectedness The Structure Theorems Exercises Operators on Hilbert Spaces Functional Calculus Compact Operators Hilbert-Schmidt and Trace Class Exercises Representations Schur s Lemma Representations of L 1 (G) Exercises Compact Groups Finite Dimensional Representations The Peter-Weyl Theorem Isotypes Induced Representations...148

9 CONTENTS xiii 7.5 Representations of SU(2) Exercises Direct Integrals Von Neumann Algebras Weak and Strong Topologies Representations Hilbert Integrals The Plancherel Theorem Exercises The Selberg Trace Formula Cocompact Groups and Lattices Discreteness of the Spectrum The Trace Formula Locally Constant Functions Exercises and Notes The Heisenberg Group Definition The Unitary Dual The Plancherel Theorem for H The Standard Lattice Exercises and Notes SL 2 (R) The Upper Half Plane The Hecke Algebra An Explicit Plancherel Theorem...214

10 xiv CONTENTS 11.4 The Trace Formula Weyl s Asymptotic Law The Selberg Zeta Function Exercises and Notes Wavelets First Ideas Discrete Series Representations Examples of Wavelet Transforms Exercises and Notes A Topology 259 A.1 Generators and Countability A.2 Continuity A.3 Compact Spaces A.4 Hausdorff Spaces A.5 Initial- and Final-Topologies A.6 Nets A.7 Tychonov s Theorem A.8 The Lemma of Urysohn A.9 Baire Spaces A.10 The Stone-Weierstraß Theorem B Measure and Integration 281 B.1 Measurable Functions and Integration B.2 The Riesz Representation Theorem B.3 Fubini s Theorem B.4 L p -Spaces and the Riesz-Fischer Theorem...291

11 CONTENTS xv B.5 The Radon-Nikodym Theorem B.6 Vector-Valued Integrals C Functional Analysis 305 C.1 Basic Concepts C.2 Seminorms C.3 Hilbert Spaces C.4 Unbounded Operators Bibiliography 323 Index 329