Anton Deitmar, Siegfried Echterhoff (auth.)0387854681, 9780387854687, 038785469X, 9780387854694
The present book is intended as a text for a graduate course on abstract harmonic analysis and its applications. The book can be used as a follow up to Anton Deitmer’s previous book, A First Course in Harmonic Analysis, or independently, if the students already have a modest knowledge of Fourier Analysis. 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 A First Course in Harmonic Analysis. Using Pontryagin duality, the authors also obtain various structure theorems for locally compact abelian groups. The book then proceeds with Harmonic Analysis on non-abelian groups and its applications to theory in number theory and the theory of wavelets.
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, all necessary ingredients from these areas have been included in the appendices.
Professor Deitmar is Professor of Mathematics at the University of Tübingen, Germany. He is a former Heisenberg fellow and has taught in the U.K. for some years. Professor Echterhoff is Professor of Mathematics and Computer Science at the University of Münster, Germany.
Table of contents :
Front Matter….Pages i-xv
Haar Integration….Pages 1-32
Banach Algebras….Pages 33-61
Duality for Abelian Groups….Pages 63-90
The Structure of LCA-Groups….Pages 91-108
Operators on Hilbert Spaces….Pages 109-125
Representations….Pages 127-137
Compact Groups….Pages 139-158
Direct Integrals….Pages 159-170
The Selberg Trace Formula….Pages 171-187
The Heisenberg Group….Pages 189-200
SL 2 (ℝ)….Pages 201-231
Wavelets….Pages 233-257
Back Matter….Pages 259-333
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