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

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 $L^2({\mathbb{R}}^n)$ using the action of closed subgroups $H\subset \mathrm{G}\mathrm{L}(n,\mathbb{R})$ such that $H$ has an open orbit $\text{cO}$ in ${\mathbb{R}}^n$ under the action $(h,\omega )\mapsto (h^{-1}{)}^T(\omega )$. 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 $\omega \in \text{cO}$ then we construct a frame for the space $L_{\text{cO}}^2({\mathbb{R}}^n)$ of $L^2$-functions whose Fourier transform is supported in $\text{cO}$. 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.