Extra-invariance of group actions

Given discrete groups $Γ\subset Δ$ we characterize $(Γ,σ)$-invariant spaces that are also invariant under $Δ$. 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 $(Γ,σ)$-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 $(Γ,σ)$-invariant space nearest - in the sense of least squares - to a given set of data.

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