Wavelet approximation theory in higher dimensions: Foundations for a systematic comparison of diverse wavelet systems

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Recent years have witnessed an increasing amount of wavelet-type systems specifically taylored to efficiently approximate salient structures in multivariate data. Prominent examples are tensor product wavelets, curvelets and shearlets. Some of these constructions are based on the representation theory of locally compact groups; most of them are in fact based on the affine actions of certain matrix groups. The aim of the current project is to develop a unified wavelet approximation theory for a large class of group-theoretically defined wavelet systems. An essential tool for this purpose will be a class of Banach spaces, the coorbit spaces, and the closely related decomposition spaces. The chief objective of the project is to continue the systematic study of approximation-theoretic properties of continuous and discrete wavelet systems arising from the natural action of a semidirect product group, initiated in recent work of the applicant.The main challenge consists in the development of pertinent, explicitly verifiable criteria on the dilation group that should serve as a basis of such an analysis; here we expect the dual action of the dilation group to be of central importance. The chief objectives related to coorbit spaces are the following: We would like to extend the scale of coorbit spaces under considerations, e.g., to include quasi-Banach spaces, as well as Triebel-Lizorkin-type spaces. Furthermore, we would like to develop a systematic understanding of embedding results, in part for the comparison of coorbit spaces associated to different dilation groups, but also for the comparison to classical smoothness spaces (e.g. Besov spaces) and more recently defined, anisotropic analogs. Here we intend to use the decomposition space language as a common framework for generalized and group-related wavelet transforms. Part of the research will be concerned with the extension of the decomposition space framework, with the aim of allowing an easier transition between coorbit and decomposition space theory. Concurrently, we intend to develop criteria for the suitability of a dilation group for the analysis of local regularity. Here we will focus on local Hölder regularity, and on the characterization of the wavefront set.