Deitmar, A., Universität
Tübingen, Germany
Echterhoff, S., University of Münster,
Germany
Written
for: graduate math students
Book
category: Graduate Textbook
Publication
language:
English
For corrections go here.
We also have corrections for the first edition here.
Principles
of Harmonic Analysis (2nd Edition)
Springer New York 2014
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 - Adeles and Ideles