A First Course in Harmonic Analysis

Library of Congress Cataloging-in-Publication Data Deitmar, Anton.A first course in harmonie analysis / Anton Deitmar. p. cm. -(Universitext) Includes bibliographical references and index. This book is a primer in harmonic analysis on the undergraduate level. It gives a lean and streamlined introduction to the central concepts of this beautiful and utile theory. In contrast to other books on the topic, A First Course in Harmonic Analysis is entirely based on the Riemann integral and metric spaces instead of the more demanding Lebesgue integral and abstract topology. Nevertheless, almost all proofs are given in full and all central concepts are presented clearly. The first aim of this book is to provide an introduction to Fourier analysis, leading up to the Poisson Summation Formula. The second 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 reader interested in the central concepts and results of harmonic analysis will benefit from the streamlined and direct approach of this book. Professor Deitmar holds a Chair in Pure Mathematics at the University of Exeter, U.K. He is a former Heisenberg fellow and was awarded the main prize of the Japanese Association of Mathematical Sciences in 1998. In his leisure time he enjoys hiking in the mountains and practising Aikido. This book is intended as a primer in harmonic analysis at the un­ dergraduate level. All the central concepts of harmonic analysis are introduced without too much technical overload. For example, the book is based entirely on the Riemann integral instead of the more demanding Lebesgue integral. Furthermore, all topological questions are dealt with purely in the context of metric spaces. It is quite sur­ prising that this works. Indeed, it turns out that the central concepts theory can be explained using very little of this beautiful and useful 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. Front Matter....Pages i-xi Front Matter....Pages 1-1 Fourier Series....Pages 3-20 Hilbert Spaces....Pages 21-36 The Fourier Transform....Pages 37-53 Front Matter....Pages 55-55 Finite Abelian Groups....Pages 57-63 LCA Groups....Pages 65-78 The Dual Group....Pages 79-87 Plancherel Theorem....Pages 89-104 Front Matter....Pages 105-105 Matrix Groups....Pages 107-118 The Representations of SU(2)....Pages 119-125 The Peter-Weyl Theorem....Pages 127-134 Back Matter....Pages 135-152