The Continuous Wavelet Transform and Symmetric Spaces

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.