Riesz and frame systems generated by unitary actions of discrete groups

Abstract

Esta es la versión de autor del artículo publicado en: This is an author produced version of a paper published in: El acceso a la versión del editor puede requerir la suscripción del recurso Access to the published version may require subscriptionRiesz and frame systems generated by unitary actions of discrete groupsOctober 6, 2014 AbstractWe characterize orthonormal bases, Riesz bases and frames which arise from the action of a countable discrete group Γ on a single element ψ of a given Hilbert space H. As Γ might not be abelian, this is done in terms of a bracket map taking values in the L 1 -space associated to the group von Neumann algebra of Γ. Our result generalizes recent work for LCA groups in [26]. In many cases, the bracket map can be computed in terms of a noncommutative form of the Zak transform.

Riesz bases associated with regular representations of semidirect product groups

“…By taking the N -Fourier transform in the second term of expression (1) we obtain the so called transfer matrix of the MIMO system (2). This motivates the following definition: Definition 1.…”

“…To illustrate the results in this section we consider group Γ = D ∞ , a unidimensional crystallographic group, and a real continuous generator ϕ ∈ L 2 (R) supported in the interval [0,2]. Notice that we can check if a system {U (γ)ϕ(t)} γ∈D∞ = {ϕ(t−n)} n∈Z ∪{ϕ(n−t)} n∈Z is a Riesz basis for A ϕ = n∈Z a(n)ϕ(t − n) + b(n)ϕ(n − t) : a, b ∈ ℓ 2 (Z) by using the Gramian condition (see, for instance, Refs.…”

Knit Product of Finite Groups and Sampling

“…[6,7,8,9,10,11,12,21]. For non Abelian groups a classical Fourier transform is not available and consequently other techniques should be considered as in [1]; for the finite case, see, for instance, Part II of Ref. [23].…”

On Regular Generalized Sampling in T-Invariant Subspaces of a Hilbert Space: An Overview

“…4. In case the operator T = U is unitary, a necessary and sufficient condition for {T (h)a} h∈H to be a Riesz sequence in H can be found in [5].…”