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

Abstract

A regular sampling theory in a multiply generated unitary invariant subspace of a separable Hilbert space H is proposed. This subspace is associated to a unitary representation of a countable discrete abelian group G on H. The samples are defined by means of a filtering process which generalizes the usual sampling settings. The multiply generated setting allows to consider some examples where the group G is non-abelian as, for instance, crystallographic groups. Finally, it is worth to mention that classical average or pointwise sampling in shift-invariant subspaces are particular examples included in the followed approach.

Sampling in $ \Lambda $-shift-invariant subspaces of Hilbert-Schmidt operators on $ L^2(\mathbb{R}^d) $

“…where λ min (respectively, λ max ) denotes the smallest (respectively, the largest) eigenvalue of the positive semidefinite matrix A(ξ) * A(ξ) (see Ref. [15]).…”

“…To be precise, the samples used along this paper will be expressed as the output of a discrete convolution system in the product Hilbert space ℓ 2 N (Λ) := ℓ 2 (Λ)ו • •×ℓ 2 (Λ) (N times), and then it will be used the close relationship between discrete convolution systems and sequences of translates in ℓ 2 N (Λ) (see, for instance, Ref. [15]). The other involved tools are the Kohn-Nirenberg transform or the Weyl transform for Hilbert-Schmidt operators: both are unitary operators from L 2 (R 2d ) onto HS(R d ) which respect the translations in the sense that, if we denote any of them by L, we have L(T z f ) = α z (Lf ) for f ∈ L 2 (R 2d ) and z ∈ R 2d .…”

“…The results for discrete convolution systems and their relationship with frames of translates in ℓ 2 N (Λ) can be found, for instance, in Ref. [15]. with convergence of the series in the operator norm.…”

Sampling in the range of the analysis operator of a continuous frame having unitary structure

“…The used mathematical techniques are similar to those in Ref. [14]. They lie in exploiting the relationship between discrete convolution systems and frames of translates in the product Hilbert space ℓ 2 N (H) := ℓ 2 (H) × • • • × ℓ 2 (H) (N times); this is an auxiliary space isomorphic to H U,φ,Φ .…”

Discrete convolution operators and Riesz systems generated by actions of abelian groups

“…The second part of this article, shows that by means of these convolution operators, an interesting generalization of some relevant results about shift-invariant systems can be obtained. In reference [11], it is showed that they are also useful in order to obtain a regular sampling theory in a very general context.…”