Discrete Frames For $L^2({\mathbb{R}}^{n^2})$ Arising From Tiling Systems On ${\mathrm{GL}}_n(\mathbb{R})$

Abstract

A discrete frame for $L^2({\mathbb{R}}^d)$ is a countable sequence $\{e_j{\}}_{j\in J}$ in $L^2({\mathbb{R}}^d)$ together with real constants $0<A\le B<\infty$ such that $$A\Vert f{\Vert }_2^2\le \sum _{j\in J}|⟨f,e_j⟩{|}^2\le B\Vert f{\Vert }_2^2,$$ for all $f\in L^2({\mathbb{R}}^d)$. We present a method of sampling continuous frames, which arise from square-integrable representations of affine-type groups, to create discrete frames for high-dimensional signals. Our method relies on partitioning the ambient space by using a suitable "tiling system". We provide all relevant details for constructions in the case of ${\mathrm{M}}_n(\mathbb{R})⋊{\mathrm{GL}}_n(\mathbb{R})$, although the methods discussed here are general and could be adapted to many other settings. Finally, we prove significantly improved frame bounds over the previously known construction for the case of $n=2$.