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

TL;DR

This paper develops a regular sampling theory for the analysis operator of continuous frames with a unitary structure linked to locally compact abelian groups, generalizing traditional sampling methods.

Contribution

It introduces a novel sampling framework for continuous frames with unitary group actions, extending classical sampling theory to more general group-based settings.

Findings

Established a regular sampling theory for the analysis operator of continuous frames.

Connected the unitary structure with group representations to facilitate sampling.

Generalized traditional sampling methods through discrete convolution systems.

Abstract

We establish a regular sampling theory in the range of the analysis operator of a continuous frame having a unitary structure. The unitary structure is related with a unitary representation of a locally compact abelian group on a separable Hilbert space. The samples are defined by means of suitable discrete convolution systems which generalize some usual sampling settings; here regular sampling means that the samples are taken at a countable discrete subgroup.