The Continuous Wavelet Transform and Symmetric Spaces

TL;DR

This paper explores generalizations of the continuous wavelet transform through group actions, proposing a novel inversion method for cases involving symmetric subgroup stabilizers, expanding its applicability.

Contribution

It introduces a new approach to invert the wavelet transform when the stabilizer subgroup is symmetric, a case not previously addressed.

Findings

Develops a framework for wavelet transform generalizations

Proposes an inversion method for symmetric subgroup stabilizers

Extends wavelet analysis to new group action scenarios

Abstract

The continuous wavelet transform has become a widely used tool in applied science during the last decade. In this article we discuss some generalizations coming from actions of closed subgroups of $\mathrm{GL}(n,\mathbb{R})$ acting on $\mathbb{R}^n$. In particular, we propose a way to invert the wavelet transform in the case where the stabilizer of a generic point in $\mathbb{R}^n$ is not compact, but a symmetric subgroup, a case that has not previously been discussed in the literature.