Riesz bases associated with regular representations of semidirect product groups

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