Principles of Harmonic Analysis - Brossura

Deitmar, Anton; Echterhoff, Siegfried

Brossura

ISBN 10: 038785519X ISBN 13: 9780387855196

Casa editrice: Springer, 2009

Riassunto

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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Review:

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)

From the Back Cover:

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.

Su questa edizione

• Editore: Springer
• Data di pubblicazione: 2009
• ISBN 10: 038785519X
• ISBN 13: 9780387855196
• Rilegatura: Paperback
• Numero di pagine: 352