ITRODUCTION the other third being devoted to more recent and unpublished work. This divi- sion is refected in the present write-up as well. As a result, I believe that this book will be useful as an introduction to the subject, to be used either for indi- vidual reading, or for a seminar or graduate course. None of the other lectures or, workshop papers presented at the CBMs conference have been incorporated here, As a result, this presentation is biased more toward my own work than the CRMS conference was. In many instances I have included pointers to references trther reading or a detailed exposition of particular applications, comple- ting the present text. Other books on wavelets published include Wavelets Tame Frequency Methods(Combes, Grossmann, and Tchamitchian(1987)) contains the proceedings of the International Wavelet Conference held in lille, France, in December 1987, Ondelettes, by Y Meyer(1990)(in French lish translation expected soon), which contains a mathematically more ex ded treatment than the present lectures, with fewer forays into other fields " wever, Les Ondelettes en 1989, edited by P G. Lemarie(1990), a collection of cks given at the Universite Paris XI in the spring of 1989, and An Introduction Wavelets, by C. K. Chui(1992b), an introduction from the approximation theory viewpoint. The proceedings of the International Wavelet Conference In May 1989, held again in Marseille, are due to come out soon(Meyer(1992)) Moreover, many of the other contributors to the CBMS conference, as well as some wavelet researchers who could not attend, were invited to write an essay on their wavelet work, the result is the essay collection Wavelets and their Ap- placations(Ruskai et al. (1992), which can be considered a companion book to thIs one. Another wavelet essay book is Wavelets: A Tutoral in Theory and Applacations, edited by C. K. Chui(1992c); in addition, I know of several other wavelet essay books in preparation(edited by J. Benedetto and M Frazier, an- other by M. Barlaud ) as well as a monograph by M. Holschneider; thele was a, special wavelet issue of IEEE Trans. Inform. Theory in March of will be another one, later in 1992, of ConstructIve Approzzmation Theory, and gne in 1993, of IEEE Trans. Sign. Prmc. In addition, several recent books in- clude chapters on wavelets. Examples are Multraie Systems and Falter Banks by P. P. Vaidyanathan(1992)and Quantum Fhysacs, Relatunty and Compler Spacetime: Towards a New Synthesus by G. Kaiser(1990 ). Readers interested in the present lectures will find these books and special issues useful for many details and other aspects not fully presented here. It is moreover clear that the subject is still developing rapidly This book more or less follows the path of my lectures: each of the ten chap- ters stands for one of the ten lectures, presented in the order in which they were delivered. The first chapter presents a quick overview of different aspects of the wavelet transform. It sketches the outlines of a big fresco; subsequent chapters then fill in more detail. From there on, we proceed to the continu ous wavelet transform( Chapter 2; with a short review of bandlimited functions and Shannon, s theorem), to discrete but redundant wavelet transforms(frames Chapter 3)and to a general discussion of time-frequiency density and the possible existence of orthonormal basis(Chapter 4). Many of the results in Chapters 2-4 can be formulated for the windowed Fourier transform as well as the wavelet