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

近年来，越来越多的小波类系统被用来有效地逼近多变量数据中的显著结构。突出的例子是张量积小波、曲线小波和剪切波。其中一些构造是基于局部紧群的表示理论，其中大多数实际上是基于某些矩阵群的仿射作用。本项目的目的是为一大类群理论定义的小波系统发展一个统一的小波逼近理论。为了达到这个目的，一个基本的工具将是一类Banach空间，共轨空间，以及密切相关的分解空间。该项目的主要目标是继续系统地研究由半直积群的自然作用产生的连续和离散小波系统的逼近理论性质。主要的挑战在于关于伸缩群的相关的、明确可验证的准则的发展，它应该作为这种分析的基础；在这里，我们期望伸缩群的对偶作用是至关重要的。与共轨空间有关的主要目标如下：我们希望扩大正在考虑的共轨空间的规模，例如，包括准Banach空间以及Triebel-Lizorkin型空间。此外，我们希望对嵌入结果有一个系统的理解，部分是为了比较与不同膨胀群有关的共轨空间，也是为了比较经典的光滑空间(例如Besov空间)和最近定义的各向异性类似物。在这里，我们打算使用分解空间语言作为广义和群相关小波变换的公共框架。部分研究将涉及分解空间框架的扩展，目的是允许在共轨和分解空间理论之间更容易地过渡。同时，我们打算为扩张组的适宜性制定标准，以分析局部正则性。在这里，我们将集中讨论局部Hölder正则性，以及波阵面集合的特征。

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.