One-Dimensional Dyadic Wavelets - Luthy, Peter M.; Šikić, Hrvoje; Soria, Fernando; Weiss, Guido L.; Wilson, Edward N. (Memoirs of the American Mathematical Society, 2022)
The theory of wavelets has been thoroughly studied by many authors; standard references include books by I. Daubechies, by Y. Meyer, by R. Coifman and Y. Meyer, by C.K. Chui, and by M.V. Wickerhauser. In addition, the development of wavelets influenced the study of various other reproducing function systems. Interestingly enough, some open questions remained unsolved or only partially solved for more than twenty years even in the most basic case of dyadic orthonormal wavelets in a single dimension. These include issues related to the MRA structure (for example, a complete understanding of filters), the structure of the space of negative dilates (in particular, with respect to what is known as the Baggett problem), and the variety of resolution structures that may occur. In this article we offer a comprehensive, yet technically fairly elementary approach to these questions. On this path, we present a multitude of new results, resolve some of the old questions, and provide new advances for some problems that remain open for the future. In this study, we have been guided mostly by the philosophy presented some twenty years ago in a book by E. Hernandez and G. Weiss (one of us), in which the orthonormal wavelets are characterized by four basic equations, so that the interplay between wavelets and Fourier analysis provides a deeper insight into both fields of research. This book has influenced hundreds of researchers, and their effort has produced a variety of new techniques, many of them reaching far beyond the study of one-dimensional orthonormal wavelets. Here we are trying to close the circle in some sense by applying these new techniques to the original subject of one-dimensional wavelets. We are primarily interested in the quality of new results and their clear presentations; for this reason, we keep our study on the level of a single dimension, although we are aware that many of our results can be extended beyond that case. Given Given a principal shift invariant space The third and final chapter is devoted to the second case, i.e., when the space