The Uncertainty Principle in Harmonic Analysis
The present book is a collection of variations on a theme which can be summed up as follows: It is impossible for a non-zero function and its Fourier transform to be simultaneously very small. In other words, the approximate equalities x :::::: y and x ::
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Editorial Board E. Bombieri, Princeton S. Feferman, Stanford M. Gromov, Bures-sur-Yvette H.W. Lenstra, Jr., Berkeley P.-L. Lions, Paris R. Remmert (Managing Editor), MUnster W. Schmid, Cambridge, Mass. J-P. Serre, Paris J.Tits, Paris
Victor Havin Burglind Joricke
The Uncertainty Principle in Harmonic Analysis With 45 Figures
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Victor Havin Department of Mathematics St. Petersburg State University 198904 St. Petersburg, Russia Burglind J6ricke Max-Planck-Gesellschaft zur F6rderung der Wissenschaften MohrenstraBe 39 D-10117 Berlin, Germany
Mathematics Subject Classification (1991): 41AlO, 41A28, 3IB35, 3IB36, 42A05, 42AlO, 42A55, 42A65, 42A70, 42A85
Library of Congress Cataloging-in-Publication Data. Khavin, Viktor Petrovich. The uncertainty principle in harmonic analysis 1 Victor Havin, Burglind Joricke. p. cm. - (Ergebnisse der Mathematik und ihrer Grenzgebiete; 3. Folge, Bd. 28) Includes bibliographical references and indexes. ISBN-13: 978-3-642-78379-1
001: 10.1007/978-3-642-78377-7
e-ISBN-13: 978-3-642-78377-7
1. Harmonic analysis. 2. Approximation theory. 3. Potential theory (Mathematics) I. Joricke, Burglind. II. Title. III. Series. QA403. K47 1994 515'.2433-dc20 93-40608 CIP This work is subject to copyright. All rights are reserved, whether the whole or part ofthe material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other ways, and storage in data banks. Duplication of this publication or parts thereof is permitted only under the provisions of the German Copyright Law of September 9, 1965, in its current version, and permission for use must always be obtained from Springer-Verlag. Violations are liable for prosecution under the German Copyright Law. © Springer-Verlag Berlin Heidelberg 1994 Softcover reprint of the hardcover I st edition 1994 Typesetting: Camera-ready copy produced from the authors' output file based on a Springer TEX macro package SPIN 10075837 41/3140 - 5 432 I 0 - Printed on acid-free paper
Foreword
The present book is a collection of variations on a theme which can be summed up as follows:
It is impossible for a non-zero function and its Fourier transform to be simultaneously very small. In other words, the approximate equalities x :::::: y and fj cannot hold, at the same time and with a high degree of accuracy, unless the functions x and yare identical. Any information gained about x (in the form of a good approximation y) has to be paid for by a corresponding loss of control on x, and vice versa. Such is, roughly speaking, the import of the Uncertainty Principle (or UP for short) referred to in the title ofthis book. That principle has an unmistakable kinship with its namesake in physics - Heisenberg's famous Uncertainty Principle - and may indeed be regarded as providing one of mathematical interpretations for the latter. But we mention these links with Quantum Mechanics and o
