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

近年来，越来越多的小波类型的系统，专门泰勒有效地近似显着的结构，在多元数据。突出的例子是张量积小波，曲波和剪切波。这些构造中的一些是基于局部紧群的表示理论;它们中的大多数实际上是基于某些矩阵群的仿射作用。 目前的项目的目的是发展一个统一的小波逼近理论的一大类群理论定义的小波系统。一个必要的工具，为这一目的将是一类Banach空间，共轨空间，和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申请人在最近的工作中开始的、由半直积群的自然作用引起的连续和离散小波系统的近似理论性质，主要挑战在于发展有关的、明确可验证的膨胀群标准，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，包括拟Banach空间，以及Triebel-Lizorkin型空间。此外，我们想发展一个系统的理解嵌入的结果，部分用于比较与不同的膨胀群相关的共轨空间，但也用于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，我们打算使用的分解空间语言作为一个通用的框架，广义和组相关的小波变换。部分的研究将关注的分解空间框架的扩展，其目的是允许一个更容易的过渡coorbit和分解空间理论。同时，我们打算制定标准的伸缩组的局部正则性分析的适用性。在这里，我们将集中在本地赫尔德正则性，并在波前集的表征。

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.