Riesz bases associated with regular representations of semidirect product groups

TL;DR

This paper investigates Bessel and Riesz systems generated by the regular representation of semidirect product groups, linking their properties to matrix functions and applications in sampling and $C^*$-algebras.

Contribution

It introduces a matrix-valued function approach to analyze these systems and explores their applications in sampling theory and operator algebras.

Findings

Characterization of Bessel and Riesz systems via matrix functions

Application to sampling in $ ext{Gamma}$-invariant spaces

Connection established between these systems and $C^*$-algebras

Abstract

This work is devoted to the study of Bessel and Riesz systems of the type $\big\{L_{\gamma}\mathsf{f}\big\}_{\gamma\in \Gamma}$ 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 $\big\{L_{\gamma}\mathsf{f}\big\}_{\gamma\in \Gamma}$ are closely related to convolution operators, a connection with $C^*$-algebras is also established.