Wavelets are a relatively recent development in applied mathematics. Their name itself was coined approximately a decade ago (Morlet, Arens, Fourgeau, and Giard (1982), Morlet (1983), Grossmann and Morlet (1984)); in the last ten years interest in them has grown at an explosive rate. There are several reasons for their present success. On the one hand, the concept of wavelets can be viewed as a synthesis of ideas which originated during the last twenty or thirty years in engineering (subband coding), physics (coherent states, renormalization group), and pure mathematics (study of Calder n-Zygmund operators). As a consequence of these interdisciplinary origins, wavelets appeal to scientists and engineers of many different backgrounds. On the other hand, wavelets are a fairly simple mathematical tool with a great variety of possible applications. Already they have led to exciting applications in signal analysis (sound, images) (some early references are Kronland-Martinet, Morlet and Grossmann (1987), Mallat (1989b), (1989c); more recent references are given later) and numerical analysis (fast algorithms for integral transforms in Beylkin, Coifman, and Rokhlin (1991)); many other applications are being studied. This wide applicability also contributes to the interest they generate.

This book contains ten lectures I delivered as the principal speaker at the CBMS conference on wavelets organized in June 1990 by the Mathematics Department at the University of Lowell, Massachusetts. According to the usual format of the CBMS conferences, other speakers (G. Battle, G. Beylkin, C. Chui, A. Cohen, R. Coifman, K. Gr chenig, J. Liandrat, S. Mallat, B. Torr sani, and A.