Shift Invariant Spaces on LCA Groups

TL;DR

This paper generalizes the theory of shift-invariant spaces to locally compact abelian (LCA) groups, introducing H-invariant spaces and extending key concepts like range functions and fiberization to this broader setting.

Contribution

It extends the theory of shift-invariant spaces to LCA groups, including new characterizations of frames and Riesz bases in this context.

Findings

Established H-invariant spaces on LCA groups.

Extended range function and fiberization techniques.

Provided new characterizations of frames and Riesz bases.

Abstract

In this article we extend the theory of shift-invariant spaces to the context of LCA groups. We introduce the notion of H-invariant space for a countable discrete subgroup H of an LCA group G, and show that the concept of range function and the techniques of fiberization are valid in this context. As a consequence of this generalization we prove characterizations of frames and Riesz bases of these spaces extending previous results, that were known for Rd and the lattice Zd .