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A friendly guide to wavelets. (English) Zbl 0839.42011

Boston: Birkhäuser. xiv, 300 p. (1994).
The opening-up of a new area of mathematics, especially one that impinges deeply on the ways mathematics can be applied, quickly leads to different types of expositions. Some fields, of course, require considerable background, while others can be included in advanced undergraduate or early graduate work. “Wavelets” form one of the latter fields, and the book under review is a fine attempt at lowering the ante for newcomers to the field. While it is possible to insist upon considerable functional-analytic and Fourier-analytic background, the author has decided to develop the analytical machinery ‘ab initio’ as much as possible, even including some basic linear algebra in Chapter 1 (along with other preliminaries from functional analysis).
The book is divided into two parts. The first develops basic wavelet theory with the guiding theme being applicability to the analysis of signals. Thus, the author first discusses windowed Fourier transforms and the appropriate localization and reconstruction theorems. Then he turns to (continuous) wavelet transforms, showing their relation to the windowed transforms. Next, he develops the theory of generalized frames as a general method for the analysis and synthesis of signals; such generalized frames combine the notion of resolution of unity (i.e., of the identity) with the usual (discrete) notion of frames. Generalized frames work in both the discrete and continuous setting and the results on windowed and continuous wavelet transforms are shown to be consequences of results on generalized frames. Then the resolutions of the identity from the continuous theory are discretized to obtain various sampling theorems in the time-frequency and time-scale domains. The last two chapters of this first part give algebraic treatments of multiresolution analysis and Daubechies’ orthonormal wavelet bases. An apparently new result here is the author’s use of the (statistical) theory of cumulants to obtain a (new) algorithm for the construction of scaling functions and wavelets from a given filter sequence.
Part II contains the author’s own results on “physical” wavelets. The main tool in the construction is the analytic-signal transform of a vector field \(F: \mathbb{R}^n\to \mathbb{C}^m\), which is the function \(\mathbb{F}: \mathbb{C}^n\to \mathbb{C}^m\) defined formally by \(\mathbb{F}(x+ iy)= (\pi i)^{- 1} \int^\infty_{- \infty} (\tau- i)^{- 1} F(x+ \tau y)d\tau\). In practice the vector field \(F\) will be an electromagnetic or acoustic wave (satisfying Maxwell’s equations or the wave equation). This transform extends the physical waves (defined on domains in \(\mathbb{R}^n)\) to a complex space-time domain. By applying it, the information content of the wavelets is displayed in a transparent way. The construction for electromagnetic waves is applied to problems in radar and electromagnetic scattering. The interesting part mathematically is that the physical wavelets \(\Psi_z\), which are solutions of the homogeneous Maxwell or wave equations, split into an incoming wavelet \(\Psi_z^-\) that is absorbed (or detected) and an outgoing wavelet \(\Psi^+_z\) that is emitted precisely when the incoming wavelet is absorbed. These incoming and outgoing wavelets are not global solutions of the homogeneous equation, but rather they solve the corresponding inhomogeneous equation.
The book contains numerous exercises, ranging from trivial to challenging. It should be very useful for its intended audience.
Reviewer: J.S.Joel (Kelly)

MSC:

42C40 Nontrigonometric harmonic analysis involving wavelets and other special systems
42-01 Introductory exposition (textbooks, tutorial papers, etc.) pertaining to harmonic analysis on Euclidean spaces
94A12 Signal theory (characterization, reconstruction, filtering, etc.)
78A45 Diffraction, scattering
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