Groups with frames of translates

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 $\phi \in L^2(G)$ such that $({\lambda }_G(x)\phi {)}_{x\in \Gamma }$ 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.