Riesz bases associated with regular representations of semidirect product groups

Abstract

This work is devoted to the study of Bessel and Riesz systems of the type $\{L_{\gamma }\mathsf{f}{\}}_{\gamma \in \Gamma }$ obtained from the action of the left regular representation $L_{\gamma }$ of a discrete non abelian group $\Gamma$ which is a semidirect product, on a function $\mathsf{f}\in {\ell }^2(\Gamma )$. The main features about these systems can be conveniently studied by means of a simple matrix-valued function $\mathbf{F}(\xi )$. These systems allow to derive sampling results in principal $\Gamma$-invariant spaces, i.e., spaces obtained from the action of the group $\Gamma$ on a element of a Hilbert space. Since the systems $\{L_{\gamma }\mathsf{f}{\}}_{\gamma \in \Gamma }$ are closely related to convolution operators, a connection with $C^{\ast }$-algebras is also established.