Book Review: Wavelets and operators
Charles K. Chui · Bulletin of the American Mathematical Society · 1996
Although wavelet analysis is a relatively young mathematical subject, it has already drawn a great deal of attention, not only among mathematicians themselves, but from various other disciplines as well.In fact, it is fair to attribute the main driving force of the rapid development of this field to the "users" rather than to the "inventors" of mathematics.To the mathematicians, Fourier analysis has been and still is a very important research area.Its theory is beautiful, its techniques powerful, and its impact on science and technology most profound.However, even as early as the decade of the 1940s, those who used the Fourier approach to analyze natural behaviors were already frustrated with the limitation of the Fourier transform and Fourier series in the investigation of physical phenomena with nonperiodic behavior and local variations.The need for simultaneous time-frequency analysis led to the introduction of Gabor's short-time Fourier transform in 1946 and the so-called Wigner-Ville transform in 1947.But the common ingredient of these two transforms is the sinusoidal kernel in the core of their definitions, so that both highand low-frequency behaviors are investigated in the same manner and any signal under investigation is matched by the same rigid sinusoidal waveform.In place of the sinusoidal kernel as modulation (for phase shift), a French geophysicist, J. Morlet, introduced in 1982 the operation of dilation, while keeping the translation operation, and developed an algorithm for the recovery of the signals under investigation from this "wavelet transform".It was the mathematical physics group in Marseille, led by A. Grossmann, in cooperation with I. Daubechies, T. Paul, etc., that extended Morlet's discrete version of wavelet transform to the continuous version, by relating it to the theory of coherent states in quantum physics.This was how the notion of the integral (or continuous) wavelet transform was introduced.The development of the mathematical analysis of the wavelet transform had really not begun, until a year later, in 1985, when the author of the first book under review learnt about the work of Morlet and the Marseille group and immediately recognized the connection of Morlet's algorithm to the notion of resolution of identity in harmonic analysis due to A. Calderón in 1964.He then applied the Littlewood-Paley theory to the study of "wavelet decomposition".In this regard, Yves Meyer may be considered as the founder of this mathematical subject, which we call wavelet analysis.Of course, Meyer's profound contribution to wavelet analysis is much more than being a pioneer of this new mathematical field.For the past ten years, he has been totally committed to its development, not only by building the mathematical foundation, but also by actively promoting the field as an interdisciplinary area of research.The first book under review is the English translation of his first monograph on this subject.In addition to the two subsequent volumes in this three-volume series (the last jointly with R. Coifman), he wrote at least two other shorter monographs on the theory, algorithms, and applications of this