Invariance of a Shift-Invariant Space

TL;DR

This paper characterizes shift-invariant spaces that remain invariant under non-integer translations, revealing that finitely generated spaces have specific invariance properties and that compactly supported generators cannot be invariant under non-integer shifts.

Contribution

It provides a complete characterization of shift-invariant spaces invariant under additional translations, especially for finitely generated spaces, and shows limitations for compactly supported generators.

Findings

Finitely generated shift-invariant spaces have specific invariance properties.

Principal shift-invariant spaces with compactly supported generators are not invariant under non-integer translations.

The paper offers a characterization in terms of generators for these spaces.

Abstract

A shift-invariant space is a space of functions that is invariant under integer translations. Such spaces are often used as models for spaces of signals and images in mathematical and engineering applications. This paper characterizes those shift-invariant subspaces S that are also invariant under additional (non-integer) translations. For the case of finitely generated spaces, these spaces are characterized in terms of the generators of the space. As a consequence, it is shown that principal shift-invariant spaces with a compactly supported generator cannot be invariant under any non-integer translations.