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 $${\mathcal {H}}$$ is proposed. This subspace is associated to a unitary representation of a countable discrete abelian group G on $${\mathcal {H}}$$. 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.