Groups with frames of translates

TL;DR

This paper characterizes which locally compact groups admit frames of translates in their L^2 spaces, providing conditions and identifying broad classes of Lie groups that possess such frames.

Contribution

It offers a comprehensive characterization of groups with frames of translates and identifies large classes of Lie groups that admit these frames.

Findings

Derived necessary and sufficient conditions for frames of translates.

Identified broad classes of Lie groups admitting frames of translates.

Provided new insights into the structure of groups supporting such frames.

Abstract

Let $G$ be a locally compact group with left regular representation $\lambda_{G}.$ We say that $G$ admits a frame of translates if there exist a countable set $\Gamma\subset G$ and $\varphi\in L^{2}(G)$ such that $(\lambda_{G}(x) \varphi)_{x \in\Gamma}$ is a frame for $L^{2}(G).$ The present work aims to characterize locally compact groups having frames of translates, and to this end, we derive necessary and/or sufficient conditions for the existence of such frames. Additionally, we exhibit surprisingly large classes of Lie groups admitting frames of translates.