Enhanced descriptions from Syndetics:

The authors have been beguiled and entranced by mathematics all of their lives, and both believe it is the highest expression of pure thought and an essential component-one might say the quintessence-of nature. How else can one ex­ plain the remarkable effectiveness of mathematics in describing and predicting the physical world? The projection of the mathematical method onto the subspace of human endeav­ 1 ors has long been a source of societal progress and commercial technology. The invention of the electronic digital computer (not the mechanical digital computer of Babbage) has made the role of mathematics in civilization even more central by making mathematics active in the operation of products. The writing of this book was intertwined with the development of a start-up company, Aware, Inc. Aware was founded in 1987 by one of the authors (H.L.R.), and the second author (R.O.W.) put his shoulder to the wheel as a consultant soon after.

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CHOICE Review

The construction of compactly supported wavelets by I. Daubechies just a bit more than a decade ago has precipitated an explosion of research, so far generating some 2000 journal articles and at least 80 books. Why all the fuss? Wavelets, as an alternative to classical Fourier analysis, offer technical advantages critical to certain timely high-profile applications, such as image compression. Though recently the subject has also attracted attention from pure mathematicians, mainly the peculiar difficulties arising in diverse application domains drive the literature's prodigious growth. The beginner can already choose from a shelf's worth of general introductions mostly covering similar ground, and another shelf of targeted introductions tied to specific applications. Among the general introductions, Blatter's primer counts as the minimalist approach. He assumes little background but makes short work of summarizing classical Fourier analysis. He explains the rationale and gets right down to work on the constructions, but concentrates on 1-dimensional functions and says virtually nothing about applications. He takes a rigorous approach, but shuns sophisticated tools like distributions and Sobelov spaces. Recommended for curious, casual readers, including undergraduates, looking for a fast way into the heart of the subject and few distractions. While no less inviting for the beginner, Resnikoff and Wells offer a much more ambitious treatment. Both their book's underlying philosophy and its scope reflect the authors' experience running Aware, Inc., a mathematical engineering company that focused on the development of mathematical tools for the efficient representation of data best suited to exploit state-of-the-art hardware and algorithms. The author devote just the final part 4, merely an eighth of the book, specifically to applications, so this book hardly counts as a targeted introduction. Nevertheless, even as they set forth basic theory the authors share the sort of savvy, about implementation details, about real-world performance, that only comes from practical experience. Upper-division undergraduates through faculty. Gasquet and Witomski's book amounts to a practical-minded introduction to real analysis especially well suited to students with an interest in digital signal processing. Despite the promise of the subtitle, only the final chapter, some 30 pages, introduces wavelets, but then the French version appeared in 1990, just as the subject began. Ryan's translation includes exercises from another work in French (by Delmasso and Witomski), but sadly, for the sake of space, omits the solutions found in the original. Upper-division undergraduates through faculty. D. V. Feldman University of New Hampshire