Druck auf Anfrage Neuware - Printed after ordering - The tread of this book is formed by two fundamental principles of Harmonic Analysis: the Plancherel Formula and the Poisson S- mation Formula. We rst 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 S- berg Trace Formula, which we prove for arbitrary groups admitting uniform lattices. As examples for the application of the Trace F- mula we treat the Heisenberg group and the group SL (R). In the 2 2 former case the trace formula yields a decomposition of the L -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 nally include a chapter on the app- 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 indep- 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].

Bestandsnummer des Verk�ufers 9780387854687

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Bibliografische Details

Titel: Principles of Harmonic Analysis
Verlag: Springer New York
Erscheinungsjahr: 2008
Sprache: Englisch
ISBN-10: 0387854681
ISBN-13: 9780387854687
Einband: Taschenbuch
Zustand: Neu
Abmessungen: Breite: 235 mm, H�he: 156 mm, Tiefe: 16 mm
Gewicht: 501 Gramm

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The book is written for graduate students who have read the first book and like to see the proofs which were not given there and/or want to see the full extent of the theory. On the other hand it can be read independently from the first one, only a modest knowledge on Fourier series/tranform is required to understand the examples. This book fills a major gap in the textbook literature, as a full proof of Pontryagin Duality and Plancherel Theorem is hard to come by. It is usually given in books that focus on C*-algebras and thus carry too much technical overload for the reader who only wants these basic results of Harmonic Analysis. Other proofs use the structure theory which carries the reader away in a different direction. Here the authors consider the Banach-algebra approach more elegant and enlighting. They provide a streamlined approach that reaches the main results directly, and they also give the generalizations to the non-Abelian case. Another main pillar of Harmonic analysis is the Poisson Summation Formula. We give its generalization to LCA-groups. The Selberg Trace Formula is considered the generalization of the Poisson Formula to non-abelian groups. The authors give the first textbook approach to this deep and useful formula in full generality. The last two chapters are devoted to examples of applications of the Selberg Trace Formula.

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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 T�bingen, 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 M�nster, Germany.

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