Simplified vanishing moment criteria for wavelets over general dilation groups, with applications to abelian and shearlet dilation groups

TL;DR

This paper simplifies the vanishing moment criteria for wavelet analysis over general dilation groups, verifies these for broad classes including abelian and shearlet groups, and establishes the existence of Banach frames with compactly supported wavelets.

Contribution

It introduces simplified vanishing moment criteria for wavelets over general dilation groups and systematically constructs and classifies shearlet dilation groups, including new examples.

Findings

Criteria depend on technical assumptions, which are now simplified.

Verification of assumptions for abelian and generalized shearlet groups.

Existence of Banach frames with compactly supported wavelets for these groups.

Abstract

We consider the coorbit theory associated to general continuous wavelet transforms arising from a square-integrable, irreducible quasi-regular representation of a semidirect product group $G = \mathbb{R}^d \rtimes H$. The existence of coorbit spaces for this very general setting has been recently established, together with concrete vanishing moment criteria for analyzing vectors and atoms that can be used in the coorbit scheme. These criteria depend on fairly technical assumptions on the dual action of the dilation group, and it is one of the chief purposes of this paper to considerably simplify these assumptions. We then proceed to verify the assumptions for large classes of dilation groups, in particular for all abelian dilation groups, as well as a class called {\em generalized shearlet dilation groups}, containing and extending all known examples of shearlet dilation groups employed in dimensions two and higher. We explain how these groups can be systematically constructed from certain commutative associative algebras of the same dimension, and give a full list, up to conjugacy, of shearing groups in dimensions three and four. In the latter case, three previously unknown groups arise. As a result the existence of Banach frames consisting of compactly supported wavelets, with simultaneous convergence in a whole range of coorbit spaces, is established for all groups involved.