Approximation by group invariant subspaces

In this article we study the structure of Gamma-invariant spaces of L-2 (S). Here S is a second countable LCA group. The invariance is with respect to the action of Gamma, a non commutative group in the form of a semidirect product of a discrete cocompact subgroup of S and a group of automorphisms. This class includes in particular most of the crystallographic groups. We obtain a complete characterization of Gamma-invariant subspaces in terms of range functions associated to shift-invariant spaces. We also define a new notion of range function adapted to the Gamma-invariance and construct Parseval frames of orbits of some elements in the subspace, under the group action. These results are then applied to prove the existence and construction of a Gamma-invariant subspace that best approximates a set of functional data in L-2 (S). This is very relevant in applications since in the euclidean case, Gamma-invariant subspaces are invariant under rigid movements, a very sought feature in models for signal processing. (C) 2020 Elsevier Masson SAS. All rights reserved.