Definition 2.2 (bases and Frames)

1. Background If ever there was a collection of articles that needed no introduction, this is it. Undaunted, I shall fulfill my charge as introducer by describing some of the intellectual background of wavelet theory and relating this background to the articles in this volume and to their expert introductions by Jelena Kovačević, Jean-Pierre Antoine, Hans Feichtinger, Yves Meyer, Guido Weiss, and Victor Wickerhauser. I was not a contributor to wavelet theory, but was close enough in the mid-1980s to hear the commotion. I was in the enviable position of having talented graduate students (including the editors of this volume), and so I felt obliged to make a serious judgement about whether to pursue wavelet theory. Timewise, this was before the wavelet stampede and sporadic tarantisms of hype and eventual catharsis. I recall saying to Heil and Walnut late in 1986 that I hoped I was not leading them down a primrose path by studying Meyer’s Séminaire Bourbaki article from 1985/1986 (which is translated into English in this volume) with them. I assured them that Meyer was brilliant and deep, and hoped that he was on target since the material seemed so compelling. Early in 1987 we also read the articles of Grossmann and Morlet and Grossmann, Morlet, and Paul (both reprinted in this volume). In April 1987 I attended the Zygmund lectures by Meyer (this volume) with Ray Johnson. Zygmund appeared and Meyer was dazzling. He gave an alternative proof of (the recently proved and not yet published) Daubechies’ theorem constructing smooth compactly supported wavelets; see the article by Daubechies reprinted in this volume. It was during the Zygmund lectures that Salem prize winner Dahlberg reminded me from high in the Hancock Center that the view was better than that in College Park. My wavelet excitement in the spring of 1987 was juxtaposed with my existing research interests, some of which seemed too inbred, a veritable “Glass Bead Game”. These interests included several significant problems which seemed out of reach, for instance, fathoming the arithmetic structure of some of the spectral synthesis problems originally formulated by Wiener and Beurling. On the other hand, and perhaps naively, many of us believed in the regenerative and centralizing power of harmonic analysis, and, then, voila! — les ondelettes arrived. I became hooked during those exhilarating nascent days of wavelets and made an effort to study wavelet theory: where it came from and where it was going. To the extent that