Stein-Fourier analysis.pdf

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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 PRINCETON UNIVERSITY 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-Publication Data 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 brother Karim 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 of ideas 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: pre- senting the various sub-areas of analysis not as separate disciplines, but rather as highly interconnected. It is our view that 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. We have organized our exposition into four volumes, each reﬂecting the material covered in a semester. Their contents may be broadly sum- marized 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, distri- butions, and elements of probability theory. However, this listing 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 ele- ments 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 in a number of

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