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WAVELETS RONALD A. DeVORE and BRADLEY J. LUCIER 1. Introduction The subject of "wavelets" is expanding at such a tremendous rate that it is impossible to give, within these few pages, a complete introduction to all aspects of its theory. We hope, however, to allow the reader to become sufficiently acquainted with the subject to understand, in part, the enthusiasm of its proponents toward its potential application to various numerical problems. Furthermore, we hope that our exposition can guide the reader who wishes to make more serious excursions into the subject. Our viewpoint is biased by our experience in approximation theory and data compression; we warn the reader that there are other viewpoints that are either not represented here or discussed only briefly. For example, orthogonal wavelets were developed primarily in the context of signal processing, an application which we touch on only indirectly. However, there are several good expositions (e.g., [Da1] and [RV]) of this application. A discussion of wavelet decompositions in the context of Littlewood-Paley theory can be found in the monograph of Frazier, Jawerth, and Weiss [FJW]. We shall also not attempt to give a complete discussion of the history of wavelets. Historical accounts can be found in the book of Meyer [Me] and the introduction of the article of Daubechies [Da1]. We shall try to give enough historical commentary in the course of our presentation to provide some feeling for the subject's development. The term "wavelet" (originally called wavelet of constant shape) was introduced by J. Morlet. It denotes a univariate function (multivariate wavelets exist as well and will be discussed subsequently), defined on R, which, when subjected to the fundamental operations of shifts (i.e., translation by integers) and dyadic dilation, yields an orthogonal basis of L2(R). That is, the functions j,k := 2k/2(2k* - j), j, k 2 Z, form a complete orthonormal system for L2(R). In this work, we shall call such a function an orthogonal wavelet, since there are many generalizations of wavelets that drop the requirement of orthogonality. The Haar function H := O/[0,1/2) - O/[1/2,1), which will be discussed in more detail in the section that follows, is the simplest example of an orthogonal wavelet. Orthogonal wavelets with higher smoothness (and even compact support) can also be constructed. But before A version of this paper appeared in Acta Numerica, A. Iserles, ed., Cambridge University Press, v. 1 (1992), pp. 1-56. This work was supported in part by the National Science Foundation (grants DMS-8922154 and DMS-9006219), the Air Force Office of Scientific Research (contract 89-0455-DEF), the Office of Naval Research (contracts N00014-90-1343, N00014-91-J-1152, and N00014-91-J-1076), the Defense Advanced Research Projects Agency (AFOSR contract 90-0323), and the Army High Performance Computing Research Center at the University of Minnesota. 1 %%[ Page: 1 ]%% 2 RONALD A. DEVORE AND BRADLEY J. LUCIER -1 0 1 OE 2k j 2k j + 1 2k OE(2k* - j) Figure 1. An example of functions OE and OE(2k* - j). considering that and other questions, we wish first to motivate the desire for such wavelets. We view a wavelet as a "bump" (and think of it as having compact support, though it need not). Dilation squeezes or expands the bump and translation shifts it (see Figure 1). Thus, j,k is a scaled version of centered at the dyadic integer j2-k. If k is large positive, then j,k is a bump with small support; if k is large negative, the support of j,k is large. The requirement that the set {j,k}j,k2Z forms an orthonormal system means that any function f 2 L2(R) can be represented as a series (1.1) f = X |
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