Ten Lectures on Wavelets

Wavelet theory is an attempt to address the pervasive problem of describing the frequency content of a function locally in time. The wavelet approach is to analyze a function using an appropriate family of dilates and translates of one or more wavelets. Although this term is relatively new, wavelet-like techniques have been independently invented over the past 30 years in harmonic analysis, quantum mechanics, signal processing, etc. Some of wavelet theory has therefore been the rediscovery of those ideas, but the introduction of a single logical framework has allowed new insights and the formulation of new problems not possible before. Moreover, the applicability of wavelets to such diverse fields has led to an unusual—and highly productive—feedback between mathematicians, physicists, and engineers. This book is a greatly expanded and updated version of ten lectures given by Daubechies at a CBMS conference at the University of Lowell in June, 1990. It is an introductory-level mathematics text, intended to introduce mathematicians and other scientists to wavelets. In this respect it is superficially similar to Chui's book [C1]; however, whereas Chui is exclusively mathematical and concentrates on only a few aspects of wavelet theory, Daubechies effectively surveys most of the current theory and specifically discusses the links with other areas, especially signal processing. To be sure, the topics which are discussed in the greatest detail are the ones in which Daubechies has herself worked—but this is not much of a limitation since she has been involved in most of the major advances, from the first " mathematical " wavelet paper [DGM] onward. All this is packaged into a well-written, organized format which should be accessible to first-year graduate students who have some knowledge of Fourier transforms and basic real analysis (a short chapter reviewing the essentials of Fourier analysis, integration theory, and Hilbert space techniques is included, but serves more as a quick-reference guide than a primer). Proofs and calculations are given in considerable detail, yet are carefully organized so that the essential ideas are clear. When details are omitted the reader is directed to the appropriate literature. There are a large number of minor typographical errors; most of these (hopefully all) will be corrected in the second printing. The ten lectures begin with a short chapter which neatly surveys the two major incarnations of the wavelet transform, and compares these to analogous forms of a standard time-frequency transform known as the windowed Fourier …