Extra Invariance of Group Actions

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 well-known characterization of shift-invariant spaces in terms of the Fourier transform. 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.