Groups with frames of translates
Volume 167 / 2022
Colloquium Mathematicum 167 (2022), 73-91
MSC: Primary 22D10; Secondary 42C15.
DOI: 10.4064/cm7864-10-2020
Published online: 22 April 2021
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.
