Continuous action of Lie groups on $\mathbb{R}^n$ and Frames

Abstract

Wavelet and frames have become a widely used tool in mathematics, physics, and applied science during the last decade. In this article we discuss the construction of frames for $L^2(\R^n)$ using the action of closed subgroups $H\subset \mathrm{GL}(n,\mathbb{R})$ such that $H$ has an open orbit $\cO$ in $\R^n$ under the action $(h,ω)\mapsto (h^{-1})^T(ω)$. If $H$ has the form $ANR$, where $A$ is simply connected and abelian, $N$ contains a co-compact discrete subgroup and $R$ is compact containing the stabilizer group of $ω\in\cO$ then we construct a frame for the space $L^2_{\cO}(\R^n)$ of $L^2$-functions whose Fourier transform is supported in $\cO$. We apply this to the case where $H^T=H$ and the stabilizer is a symmetric subgroup, a case discussed for the continuous wavelet transform in a paper by Fabec and Olafsson.