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 γ f γ∈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 f ∈ ℓ 2 (Γ). The main features about these systems can be conveniently studied by means of a simple matrix-valued function 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 L γ f γ∈Γ are closely related to convolution operators, a connection with C *algebras is also established.

Convolution Systems on Discrete Abelian Groups as a Unifying Strategy in Sampling Theory

“…holds, then the subspace V Φ is a reproducing kernel Hilbert space of continuous bounded functions in L 2 ( G). In fact, it is a necessary and sufficient condition as the following result shows; its proof is analogous to that in [14,Lemma 4.2].…”

“…If the generator ϕ is continuous in R d and the function t → p∈Z d |ϕ(t−p)| 2 is bounded on R d , a standard argument shows that V ϕ is a RKHS of bounded continuous functions in L 2 (R d ) (see, for instance, Ref. [14]). As a consequence, convergence in L 2 (R d )-norm implies pointwise convergence which is absolute and uniform on R d .…”