Deitmar, A., Universität
Tübingen,
Germany
Echterhoff, S., University of Münster, Germany
Written for:
graduate math students
Book category:
Graduate
Textbook
Publication
language: English
go here
to buy the book.
For Ken Ross's review of the book look here.
For a correction of known errors look here.
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Principles
of
Harmonic
Analysis
From the review of Kenneth Ross for the Mathematical
Association of America:
"Principles of Harmonic Analysis is an excellent and thorough
introduction to both commutative and non-commutative harmonic analysis.
"In summary, this is a superb book. It covers a great deal of important
material, but it is extremely readable and well organized. Graduate
students, and other newcomers to the field, will greatly appreciate the
authors’ clear and careful writing.
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 by Anton Deitmar, 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 \cite{HA1}. 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.
Contents:
Haar Integration - Banach Algebras - Duality for Abelian Groups -
Structure of LCA-Groups - Operators on Hilbert Space - Representations
- Compact Groups - The Selberg Trace Tormula - The Heisenberg Group -
SL(2,R) - Wavelets
Appendices: Topology - Measure and
Integration - Functional Analysis
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