Wavelets and wavelet sets.

摘要

In this dissertation, we investigate some basic properties of wavelets. All the new results contained in the dissertation were obtained during the past two years of research under the supervision and guidance of my thesis advisor Dr. Larson. The research consists of several closely related topics in the theory of wavelets. These topics are initiated mainly by a recent AMS Memoir paper of Dai and Larson ( (5)) and the subsequent paper of Dai, Larson, and Speegle ( (7)). In the first paper mentioned above, the authors introduced the notions of wavelet set, interpolation family of wavelet sets and wavelet multiplier, among other things. These concepts proved to be important in the understanding of the topological and algebraic structures of the collection of all wavelets. In the second paper, the authors proved the existence of wavelets in higher dimensional setting using a constructive argument. In an attempt to answer several open problems raised in (5) we prove the existence of some interpolation families of wavelet sets, characterize all functional wavelet multipliers, and explore the connection between functional wavelet multipliers and other aspects of wavelets. Following the lead of (5) and (7), we also discuss the construction of wavelet sets, in particular those with finitely many intervals in one dimensional case and those that are associated with multiresolution analysis in higher dimensional case. In the latter case, we give a necessary and sufficient condition under which there exist higher dimensional wavelets which are associated with multiresolution analysis. Other questions concerning wavelet sets, interpolation families of wavelet sets and wavelet multipliers are also studied and some other aspects of higher dimensional wavelets are discussed. It is worth pointing out that the construction of wavelet sets plays a role in the investigation of almost all of these topics, implicitly or explicitly.