Uncertainty Principles in Finitely generated Shift-Invariant Spaces with additional invariance

TL;DR

This paper explores the limitations of generators in finitely generated shift-invariant spaces with extra invariance, showing they must meet strict conditions at infinity if they form a frame, extending previous results.

Contribution

It generalizes existing results on principal shift-invariant spaces to broader cases with additional invariance, revealing new restrictions on generators and their translates.

Findings

Generators must satisfy restrictions at infinity when forming a frame

Generalization of prior results from principal to finitely generated spaces

Establishment of conditions linking invariance, frames, and generator behavior

Abstract

We consider finitely generated shift-invariant spaces (SIS) with additional invariance in $L^2(\R^d)$. We prove that if the generators and their translates form a frame, then they must satisfy some stringent restrictions on their behavior at infinity. Part of this work (non-trivially) generalizes recent results obtained in the special case of a principal shift-invariant spaces in $L^2(\R)$ whose generator and its translates form a Riesz basis.