Quizás Quisiste Decir: Fourier Analysis And Partial Differential Equations

Ibookroot October 20, 2007

FOURIER ANALYSIS

Ibookroot October 20, 2007

Princeton Lectures in Analysis
I Fourier Analysis: An Introduction II Complex Analysis III Real Analysis: Measure Theory, Integration, and Hilbert Spaces

Ibookroot October 20, 2007

Princeton Lectures in Analysis

I

FOURIER ANALYSIS
an introduction

Elias M. Stein & Rami Shakarchi

PRINCETONUNIVERSITY PRESS PRINCETON AND OXFORD

Copyright © 2003 by Princeton University Press Published by Princeton University Press, 41 William Street, Princeton, New Jersey 08540 In the United Kingdom: Princeton University Press, 6 Oxford Street, Woodstock, Oxfordshire OX20 1TW All Rights Reserved Library of Congress Control Number 2003103688 ISBN 978-0-691-11384-5 British Library Cataloging-in-PublicationData is available The publisher would like to acknowledge the authors of this volume for providing the camera-ready copy from which this book was printed Printed on acid-free paper. ∞ press.princeton.edu Printed in the United States of America 5 7 9 10 8 6

Ibookroot October 20, 2007

To my grandchildren

Carolyn, Alison, Jason
E.M.S.

To my parents

Mohamed & Mireille
and my brotherKarim
R.S.

Ibookroot October 20, 2007

Foreword

Beginning in the spring of 2000, a series of four one-semester courses were taught at Princeton University whose purpose was to present, in an integrated manner, the core areas of analysis. The objective was to make plain the organic unity that exists between the various parts of the subject, and to illustrate the wide applicability ofideas of analysis to other fields of mathematics and science. The present series of books is an elaboration of the lectures that were given. While there are a number of excellent texts dealing with individual parts of what we cover, our exposition aims at a different goal: presenting the various sub-areas of analysis not as separate disciplines, but rather as highly interconnected. It is our viewthat seeing these relations and their resulting synergies will motivate the reader to attain a better understanding of the subject as a whole. With this outcome in mind, we have concentrated on the main ideas and theorems that have shaped the field (sometimes sacrificing a more systematic approach), and we have been sensitive to the historical order in which the logic of the subject developed. Wehave organized our exposition into four volumes, each reflecting the material covered in a semester. Their contents may be broadly summarized as follows: I. Fourier series and integrals. II. Complex analysis. III. Measure theory, Lebesgue integration, and Hilbert spaces. IV. A selection of further topics, including functional analysis, distributions, and elements of probability theory. However, thislisting does not by itself give a complete picture of the many interconnections that are presented, nor of the applications to other branches that are highlighted. To give a few examples: the elements of (finite) Fourier series studied in Book I, which lead to Dirichlet characters, and from there to the infinitude of primes in an arithmetic progression; the X-ray and Radon transforms, which arise ina number of

Ibookroot October 20, 2007

viii

FOREWORD

problems in Book I, and reappear in Book III to play an important role in understanding Besicovitch-like sets in two and three dimensions; Fatou’s theorem, which guarantees the existence of boundary values of bounded holomorphic functions in the disc, and whose proof relies on ideas developed in each of the first three books; and thetheta function, which first occurs in Book I in the solution of the heat equation, and is then used in Book II to find the number of ways an integer can be represented as the sum of two or four squares, and in the analytic continuation of the zeta function. A few further words about the books and the courses on which they were based. These courses where given at a rather intensive pace, with 48...