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.
Categories: math.RT