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Book review

Author(s) Daubechies, Ingrid
Title Ten lectures on wavelets
Publisher SIAM
Year of publication 1992
Reviewed by Frantisek Vcelar,

Wavelets, like the Fourier transform, provide a tool for cutting up a function into different frequency components; unlike the ordinary Fourier transform, however, they can be localized in space, and analyzed with a resolution matched to a given scale. These properties render them suitable for a wide range of practical applications both in mathematics and in applied sciences. Although the wavelets are a relatively recent branch of mathematics (some fifteen years old), their study seems to proceed at a really explosive rate.

The present book may serve as a tutorial of wavelets, written by one of the world's leading experts in the field. It consists of ten chapters, starting from the bare fundamentals, then dealing with both the continuous and the discrete wavelet transform, the frames and orthogonal bases of wavelets, and the multiresolution analysis, and finally proceeding to the discussion of regularity and symmetry properties of wavelets, of expansions in other spaces then L^2 (L^p for p != 2, Sobolev spaces) and similar slightly more advanced topics.

The exposition is concise and nicely written and requires but the minimal prerequisites (basic real analysis and Hilbert space theory, and some familiarity with the Fourier transform). Full proofs are almost always given, and the whole book is "mathematical" in spirit. On the other hand, motivations from and/or applications to other sciences are indicated whenever possible, including a lot of graphs and charts demonstrating

the practical utilization of the theory, which makes the book attractive also to the "engineering" audience. The volume is excellently typeset and supplied with both name and subject indexes and a good bibliography.

Many books (even many good ones) on the subject of wavelets have appeared recently: Chui's An introduction to wavelets (Academic Press, 1992) and Wavelets: A tutorial in theory and applications (Academic Press, 1992), Lemarié's Les ondolettes en 1989 (Lecture Notes in Mathematics 1438, Springer, 1990, in French), R. K. Young's Wavelet theory and its applications (Kluwer, 1993), Koornwinder's Wavelets: An elementary treatment through theory and applications (World Scientific, 1993), and, of course, Yves Meyer's Ondolettes et opérateurs (in 3 volumes, Hermann, 1990), to cite just a few of them. The present one ranks well among them; besides, this wealth of literature most clearly indicates the interest and the intensity of current research in this field. The book under review will appeal to anyone from graduate students to mature scientists who wants to learn something about wavelets, and it is a must for any library which is interested in applied mathematics.