没有合适的资源?快使用搜索试试~ 我知道了~
资源推荐
资源详情
资源评论
Universitext
Editorial Board
(North America):
S. Axler
F.W. Gehring
K.A. Ribet
Anton Deitmar
A First Course in
Harmonic Analysis
Second Edition
Anton Deitmar
Department of Mathematics
University of Exeter
Exeter, Devon EX4 4QE
UK
a.h.j.deitmar@exeter.ac.uk
Editorial Board
(North America):
S. Axler F.W. Gehring
Mathematics Department Mathematics Department
San Francisco State University East Hall
San Francisco, CA 94132 University of Michigan
USA Ann Arbor, MI 48109-1109
USA
K.A. Ribet
Mathematics Department
University of California, Berkeley
Berkeley, CA 94720-3840
USA
Mathematics Subject Classification (2000): 43-01, 42Axx, 22Bxx, 20Hxx
Library of Congress Cataloging-in-Publication Data
Deitmar, Anton.
A first course in harmonic analysis / Anton Deitmar. – 2nd ed.
p. cm. — (Universitext)
Includes bibliographical references and index.
ISBN 0-387-22837-3 (alk. paper)
1. Harmonic analysis. I. Title.
QA403 .D44 2004
515′.2433—dc22 2004056613
ISBN 0-387-22837-3 Printed on acid-free paper.
© 2005, 2002 Springer-Verlag New York, Inc.
All rights reserved. This work may not be translated or copied in whole or in part without the
written permission of the publisher (Springer Science+Business Media, Inc., 233 Spring Street, New
York, NY 10013, USA), except for brief excerpts in connection with reviews or scholarly analysis.
Use in connection with any form of information storage and retrieval, electronic adaptation, com-
puter software, or by similar or dissimilar methodology now known or hereafter developed is for-
bidden.
The use in this publication of trade names, trademarks, service marks, and similar terms, even if
they are not identified as such, is not to be taken as an expression of opinion as to whether or not
they are subject to proprietary rights.
Photocomposed copy prepared from the author’s
files.
Printed in the United States of America. (MP)
987654321 SPIN 11019138
springeronline.com
v
Preface to the second edition
This book is intended as a primer in harmonic analysis at the upper
undergraduate or early graduate level. All central concepts of har-
monic analysis are introduced without too much technical overload.
For example, the book is based entirely on the Riemann integral in-
stead of the more demanding Lebesgue integral. Furthermore, all
topological questions are dealt with purely in the context of metric
spaces. It is quite surprising that this works. Indeed, it turns out
that the central concepts of this beautiful and useful theory can be
explained using very little technical background.
The first aim of this book is to give a lean introduction to Fourier
analysis, leading up to the Poisson summation formula. The sec-
ond aim is to make the reader aware of the fact that both principal
incarnations of Fourier theory, the Fourier series and the Fourier
transform, are special cases of a more general theory arising in the
context of locally compact abelian groups. The third goal of this
book is to introduce the reader to the techniques used in harmonic
analysis of noncommutative groups. These techniques are explained
in the context of matrix groups as a principal example.
The first part of the book deals with Fourier analysis. Chapter 1
features a basic treatment of the theory of Fourier series, culminating
in L
2
-completeness. In the second chapter this result is reformulated
in terms of Hilbert spaces, the basic theory of which is presented
there. Chapter 3 deals with the Fourier transform, centering on
the inversion theorem and the Plancherel theorem, and combines
the theory of the Fourier series and the Fourier transform in the
most useful Poisson summation formula. Finally, distributions are
introduced in chapter 4. Modern analysis is unthinkable without this
concept that generalizes classical function spaces.
The second part of the book is devoted to the generalization of the
concepts of Fourier analysis in the context of locally compact abelian
groups, or LCA groups for short. In the introductory Chapter 5 the
entire theory is developed in the elementary model case of a finite
abelian group. The general setting is fixed in Chapter 6 by introduc-
ing the notion of LCA groups; a modest amount of topology enters
at this stage. Chapter 7 deals with Pontryagin duality; the dual is
shown to be an LCA group again, and the duality theorem is given.
化身
vi PREFACE
The second part of the book concludes with Plancherel’s theorem in
Chapter 8. This theorem is a generalization of the completeness of
the Fourier series, as well as of Plancherel’s theorem for the real line.
The third part of the book is intended to provide the reader with a
first impression of the world of non-commutative harmonic analysis.
Chapter 9 introduces methods that are used in the analysis of matrix
groups, such as the theory of the exponential series and Lie algebras.
These methods are then applied in Chapter 10 to arrive at a classi-
fication of the representations of the group SU(2). In Chapter 11 we
give the Peter-Weyl theorem, which generalizes the completeness of
the Fourier series in the context of compact non-commutative groups
and gives a decomposition of the regular representation as a direct
sum of irreducibles. The theory of non-compact non-commutative
groups is represented by the example of the Heisenberg group in
Chapter 12. The regular representation in general decomposes as a
direct integral rather than a direct sum. For the Heisenberg group
this decomposition is given explicitly.
Acknowledgements: I thank Robert Burckel and Alexander Schmidt
for their most useful comments on this book. I also thank Moshe
Adrian, Mark Pavey, Jose Carlos Santos, and Masamichi Takesaki
for pointing out errors in the first edition.
Exeter, June 2004 Anton Deitmar
剩余187页未读,继续阅读
资源评论
pulj26
- 粉丝: 1
- 资源: 1
上传资源 快速赚钱
- 我的内容管理 展开
- 我的资源 快来上传第一个资源
- 我的收益 登录查看自己的收益
- 我的积分 登录查看自己的积分
- 我的C币 登录后查看C币余额
- 我的收藏
- 我的下载
- 下载帮助
安全验证
文档复制为VIP权益,开通VIP直接复制
信息提交成功