# Convolution systems on discrete abelian groups as a unifying strategy in sampling theory

## TL;DR

This paper develops a unified sampling theory framework using convolution systems on discrete abelian groups, extending classical and average sampling methods to more general group-invariant subspaces in Hilbert spaces.

## Contribution

It introduces a general sampling approach based on group representations, encompassing classical and non-abelian group cases like crystallographic groups.

## Key findings

- Unified sampling framework for abelian and non-abelian groups
- Generalizes classical average and pointwise sampling methods
- Applicable to shift-invariant and group-invariant subspaces

## 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.