Principles of Harmonic Analysis

Paperback | November 21, 2008

byAnton Deitmar, Siegfried Echterhoff

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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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From the Publisher

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 ...

From the Jacket

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 thin...

Format:PaperbackDimensions:348 pages, 9.25 × 6.1 × 0.27 inPublished:November 21, 2008Publisher:Springer New YorkLanguage:English

The following ISBNs are associated with this title:

ISBN - 10:0387854681

ISBN - 13:9780387854687

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Table of Contents

Preface.- Haar Integration.- Banach Algebras.- Duality for Abelian Groups.- The Structure of LCA-Groups.- Operators on Hilbert Spaces.- Representations.- Compact Groups.- Direct Integrals.- The Selberg Trace Formula.- The Heisenberg Group.- SL2(R).- Wavelets.- Topology.- Measure and Integration.- Functional Analysis.- Bibliography.- Index.

Editorial Reviews

From the reviews:"Principles of Harmonic Analysis is an excellent and thorough introduction to both commutative and non-commutative harmonic analysis. It is suitable for any graduates student with the appropriate background . . In summary, this is a superb book. . it is extremely readable and well organized. Graduate students, and other newcomers to the field, will greatly appreciate the author's clear and careful writing." (Kenneth A. Ross, MAA Online, February, 2009)"The book under review is a nice presentation of all the standard, basic material on abstract harmonic analysis. . The most welcome aspect of the book under review is the inclusion of a discussion of the trace formula, a rather unusual feature in an introductory book on harmonic analysis. . This is a nice addition to the literature on the subject." (Krishnan Parthasarathy, Mathematical Reviews, Issue 2010 g)