Extra-invariance of group actions

TL;DR

This paper characterizes when certain group-invariant spaces remain invariant under larger groups using a Zak transform, providing new insights and solutions for invariant space problems.

Contribution

It introduces a new Zak transform-based characterization of invariant spaces and solves the problem of finding the nearest invariant space to given data.

Findings

Characterization of $( ext{Gamma}, ext{sigma})$-invariant spaces using Zak transform

New description of principal invariant spaces via Zak transform of generators

Solution for the least squares problem of finding nearest invariant space

Abstract

Given discrete groups $\Gamma \subset \Delta$ we characterize $(\Gamma,\sigma)$-invariant spaces that are also invariant under $\Delta$. This will be done in terms of subspaces that we define using an appropriate Zak transform and a particular partition of the underlying group. On the way, we obtain a new characterization of principal $(\Gamma,\sigma)$-invariant spaces in terms of the Zak transform of its generator. This result is in the spirit of the analogous in the context of shift-invariant spaces in terms of the Fourier transform, which is very well-known. As a consequence of our results, we give a solution for the problem of finding the $(\Gamma,\sigma)$-invariant space nearest - in the sense of least squares - to a given set of data.