INTRODUCTION 4. B. Ruskai. The success of the conference, which unexpectedly turned out to have many more participants than customary for CBMS conferences, was due in large part to its very efficient organization. As experienced conference organizer . M. James(1991)says, "every conference is mainly due to the efforts of a single individual who does almost all the work" for the 1990 Wavelet CBMs conference, this individual was Mary Beth Ruskai. I am especially grateful to her for proposing the conference in the first place, for organizing it in such a way that I had a minimal paperwork load, while keeping me posted about all the developments, and for generally being the organizational backbone, no small task. Prior to the conference I had the opportunity to teach much of this material as a graduate course in the Mathematics Department of the University of Michigan, in Ann Arbor. My one-term visit there was supported jointly by a Visiting Professorship for Women from the National Science Foundation, and by the University of Michigan. I would like to thank both institutions for their support. I would also like to thank all the faculty and students who sat in on the course and who provided feedback and useful suggestions. The manuscri was typeset by Martina Sharp, who I thank for her patience and diligence, and for doing a wonderful job. I wouldn't even have attempted to write this book without her. I am grateful to Jeff Lagarias for editorial comments. Several people helped me spot typos in the galley proofs, and i am grateful to all of them; I would like to thank especially Pascal Auscher, Gerry Kaiser, Ming-Jun Lai, and Martin Vetterli. All remaining mistakes are of course my responsiblity. I also would like to thank Jim Driscoll and Sharon Murrel for helping me prepare the author index. Finally, I want to thank my husband Robert Calderbank for being extremely supportive and committed to our two-career-track with family, even though it occasionally means that he as well as i prove a few theorems less ngri ATET Bell Laboratories Rutgers University In this second printing several minor mistakes and many typographical errors have been corrected. I am grateful to everybody who helped me to spot them I have also updated a few things: some of the previously unpublished references have appeared and some of the problems that were listed as open have been solved. I have made no attempt to include the many other interesting papers on wavelets that have appeared since the first printing; in any case, the list of references was not and is still not meant as a complete bibliography of the subject ngrid Daubechies, Sept. 1992