Harmonic Analysis on Spaces of Homogeneous Type
The dramatic changes that came about in analysis during the twentieth century are truly amazing. In the thirties, complex methods and Fourier series played a seminal role. After many improvements, mostly achieved by the Calderón-Zygmund school, the action
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1966
Donggao Deng · Yongsheng Han
Harmonic Analysis on Spaces of Homogeneous Type With a Preface by Yves Meyer
ABC
Donggao Deng (deceased) Yongsheng Han Department of Mathematics and Statistics Auburn University Auburn, AL 36849-5310 USA [email protected]
ISBN: 978-3-540-88744-7 e-ISBN: 978-3-540-88745-4 DOI: 10.1007/978-3-540-88745-4 Lecture Notes in Mathematics ISSN print edition: 0075-8434 ISSN electronic edition: 1617-9692 Library of Congress Control Number: 2008938190 Mathematics Subject Classification (2000): 35J40, 41A30, 42B20, 42B30, 42C15, 42C40, 43A15, 46F12 c 2009 Springer-Verlag Berlin Heidelberg ° This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilm or in any other way, 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. Violations are liable to prosecution under the German Copyright Law. The use of general descriptive names, registered names, trademarks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. Cover design: SPi Publishing Services Printed on acid-free paper 987654321 springer.com
Preface
This book could have been entitled “Analysis and Geometry.” The authors are addressing the following issue: Is it possible to perform some harmonic analysis on a set? Harmonic analysis on groups has a long tradition. Here we are given a metric set X with a (positive) Borel measure μ and we would like to construct some algorithms which in the classical setting rely on the Fourier transformation. Needless to say, the Fourier transformation does not exist on an arbitrary metric set. This endeavor is not a revolution. It is a continuation of a line of research which was initiated, a century ago, with two fundamental papers that I would like to discuss briefly. The first paper is the doctoral dissertation of Alfred Haar, which was submitted at to University of G¨ ottingen in July 1907. At that time it was known that the Fourier series expansion of a continuous function may diverge at a given point. Haar wanted to know if this phenomenon happens for every orthonormal basis of L2 [0, 1]. He answered this question by constructing an orthonormal basis (today known as the Haar basis) with the property that the expansion (in this basis) of any continuous function uniformly converges to that function. Today we know that Haar was the grandfather of wavelets and we also know that wavelet bases offer a powerful and flexible alternative to Fourier analysis. Indeed wavelet bases are unconditional bases of most of the functional spaces we are using in analysis. In other words wavelet expan
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