INTRODUCTION transform, and the two cases are presented in parallel, with analogies and differ ences pointed out as we go along. The remaining chapters all focus on orthonor- mal bases of wavelets. multiresolution analysis and a first general strategy for the construction of orthonormal wavelet bases( Chapter 5), orthonormal bases of compactly supported wavelets and their link to subband coding(Chapter 6) sharp regularity estimates for these wavelet bases(Chapter 7), symmetry for compactly supported wavelet bases(Chapter 8)Chapter 9 shows that orthonor- mal bases are"good"bases for many functional spaces where Fourier methods are not well adapted. This chapter is the most mathematical of the whole book most of its material is not connected to the applications discussed in other chap- ters, so that it can be skipped by readers uninterested in this aspect of wavelet theory. I included it for several reasons: the kind of estimates used in the proof are very important for harmonic analysis, and similar(but more complicated) estimates in the proof of the"r(1)"-theorem of David and Journe have turned out to be the groundwork for the applications to numerical analysis in the work of Beylkin, Coifman, and Rokhlin(1991)Moreover, the Calderon-Zygmuin theorem, explained in this chapter, illustrates how techniques using different scales, one of the forerunners of wavelets, were used in harmonic analysis long before the advent of wavelets. Finally, Chapter 10 sketches several extensions of the constructions of orthonornal wavelet bases: to more than one dimension to dilation factors different from two(even noninteger), with the possibility of better frequency localization, and to wavelet bases on a finite interval instead of the whole line. Every chapter concludes with a section of numbered"note referred to in the text of the chapter by superscript numbers. These contai additional references, extra proofs excised to keep the text Rowing, remarks, etc This book is a mathematics book: it states and proves many theorems. It also presupposes some mathematical background. In particular, I assume that the reader is familiar with the basic properties of the fourier transform and Fourier series. I also use some basic theorems of measure and integration theory (Fatou' s lemma, dominated convergence theorem, Fubini's theorem; these cath be found in any good book on real analysis). In some chapters, familiarity with basic Hilbert space techniques is useful. A list of the basic notions and theorems used in the book is given in the preliminaries The reader who finds that he or she does not know all of these prerequisites should not be dismayed, however; most of the book can be followed with just the basic notions of Fourier analysis. Moreover, I have tried to keep a very pedes- trian pace in almost all the proofs, at the risk of boring some mathematically ophisticated readers. I hope therefore that these lecture notes will interest peo- ple other than mathematicians. For this reason I have often shied away from he "Definition-Lemma-Proposition-Theorem-Corollary"sequence, and I have tried to be intuitive in many places, even if this meant that the exposition be- came less succinct. I hope to succeed in sharing with my readers some of the excitement that this interdisciplinary subject has brought into my scientific life I want to take this opportunity to express my gratitude to the many people ho made the lowell conference happen: the cbms board, and the Mathematics Department of the University of Lowell, in particular Professors G. Kaiser and