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 $\big\{L_γ\mathsf{f}\big\}_{γ\in Γ}$ obtained from the action of the left regular representation $L_γ$ of a discrete non abelian group $Γ$ which is a semidirect product, on a function $\mathsf{f}\in \ell^2(Γ)$. The main features about these systems can be conveniently studied by means of a simple matrix-valued function $\mathbf{F}(ξ)$. These systems allow to derive sampling results in principal $Γ$-invariant spaces, i.e., spaces obtained from the action of the group $Γ$ on a element of a Hilbert space. Since the systems $\big\{L_γ\mathsf{f}\big\}_{γ\in Γ}$ are closely related to convolution operators, a connection with $C^*$-algebras is also established.
math.FA↗