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Summary: The mathematics of wavelets
Review: The literature on wavelets has been growing at an increasing pace, as testified by 129 titles listed in Amazon.com, the great majority which were first published in the last five years. Many of these books are either (i)popular accounts for the general reader, (ii) user's guides for the practitioner or (iii) advanced mathematical treatises for experts. Thus we find very few textbooks where the serious student can obtain an honest introduction to the subject from a recognized authority. This is even more surprising, given that the origins of the subject can be traced back to the mathematics literature in the early part of the 20th century under the names of Alfred Haar, Phillip Franklin and Sir Edmund Whittaker---and later to further work by Alberto Calderon in the 1960's. With the flurry of activity in the 1980's in the French school, the subject came alive and is now a permanent part of the literature in mathematical analysis, growing by leaps and bounds.The book of Wojtaszczyk is a welcome addition to the literature on wavelets. Without the benefit of glossy pictures or computer output, the author has been extremely successful in presenting a clear and correct approach to the subject for readers who have a minimal acquaintance with mathematical analysis at the level of integration theory and elementary Fourier analysis. Going beyond other introductory works, the book contains a systematic sets of exercises at the end of each chapter, as a sort of "reality check" for the student, to test his/her understanding of the theory. The first four chapters of the book deal with wavelets in one dimensions, continued in chapter 9. Following the examples of the classical Haar system and the Stromberg spline wavelet, done in great detail, we are introduced to MRA systems as a systematic method for generating wavelet bases of the space of square-integrable functions on the line. An MRA (multi-reolution analysis) is defined by a "scaling function", which satisfies an orthonormality condition, a scaling condition and a smoothness condition in the Fourier domain. Any such function generates an MRA, which in turn generates a wavelet basis. In particular one can generate in this framework the smooth wavelets of Meyer by this method, followed by the compactly supported wavelets of Daubechies. Wavelet theory is first formulated for square integrable functions, but can be extended to other Banach spaces, where it often provides an "unconditional basis", which is not true of the classical Fourier series. Chapters 5-8 deal with some of the multi-dimensional theory, where several wavelets are necesary to generate the MRA, suitably defined. Chapter 6 contains a self-contained treatment of some important topics in analysis: the Hardy-Littlewood maximal inequality, the Banach spaces H^1 and BMO, the John-Nirenberg inequalty. These are used to develop the property of unconditional basis for wavelets in the spaces L^p and H^1. The author suceceds admirably in carrying out his stated goal beyond any reasonable expectation: in the preface we read "This is a purely mathematical book, although I constantly try to make the calculations as explicit as possible and I concentrate on theoretical questions that should have relevance in appplications, but regrettably discuss no real applications". With the flurry of literature on the uses of wavelets, these applications are best left to other works. One can expect that this book will be in print for many years to come.
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