Recent Progress in Shearlet Theory: Systematic Construction of Shearlet Dilation Groups, Characterization of Wavefront Sets, and New Embeddings

TL;DR

This paper reviews recent advances in shearlet theory, focusing on systematic group construction, wavefront set characterization, and embedding into symplectic groups, unifying many previous results.

Contribution

It introduces a unified framework for generalized shearlet dilation groups, extending known results to infinitely many new cases and exploring their algebraic, geometric, and analytic properties.

Findings

Unified construction of shearlet groups from associative algebras

Characterization of wavefront sets using shearlet groups

Embeddings into symplectic groups linking different representations

Abstract

The class of generalized shearlet dilation groups has recently been developed to allow the unified treatment of various shearlet groups and associated shearlet transforms that had previously been studied on a case-by-case basis. We consider several aspects of these groups: First, their systematic construction from associative algebras, secondly, their suitability for the characterization of wavefront sets, and finally, the question of constructing embeddings into the symplectic group in a way that intertwines the quasi-regular representation with the metaplectic one. For all questions, it is possible to treat the full class of generalized shearlet groups in a comprehensive and unified way, thus generalizing known results to an infinity of new cases. Our presentation emphasizes the interplay between the algebraic structure underlying the construction of the shearlet dilation groups, the geometric properties of the dual action, and the analytic properties of the associated shearlet transforms.