On wavelet coorbit spaces associated to different dilation groups

TL;DR

This paper introduces a unified framework using coarse geometry to compare wavelet coorbit spaces generated by different dilation groups, including reducible representations, to encompass a broad class of function spaces.

Contribution

It develops criteria for comparing coorbit spaces across different dilation groups, emphasizing the importance of reducible representations for a comprehensive analysis.

Findings

Criteria for when different dilation groups produce the same coorbit spaces

Characterization of subgroups yielding identical coorbit spaces

Clarification of when anisotropic Besov spaces can be described as coorbit spaces

Abstract

This paper develops methods based on coarse geometry for the comparison of wavelet coorbit spaces defined by different dilation groups, with emphasis on establishing a unified approach to both irreducible and reducible quasi-regular representations. We show that the use of reducible representations is essential to include a variety of examples, such as anisotropic Besov spaces defined by general expansive matrices, in a common framework. The obtained criteria yield, among others, a simple characterization of subgroups of a dilation group yielding the same coorbit spaces. They also allow to clarify which anisotropic Besov spaces have an alternative description as coorbit spaces associated to irreducible quasi-regular representations.