Approximation by group invariant subspaces

BARBIERI, DAVIDE; CABRELLI, CARLOS A.; HERNÁNDEZ, EUGENIO; MOLTER, URSULA M.

JOURNAL DE MATHEMATIQUES PURES ET APPLIQUEES

GAUTHIER-VILLARS/EDITIONS ELSEVIER

Lugar: Paris; Año: 2020 vol. 142 p. 72 - 100

0021-7824

In this article we study the structure of Γ-invariant spaces of L^2(G) . Here G is a second countable LCA group. The invariance is with respect to the action of Γ, a non commutative group in the form of a semidirect product of a discrete cocompact subgroup of G and a group of automorphisms. This class includes in particular most of the crystallographic groups. We obtain a complete characterization of Γ-invariant subspaces in terms of range functions associated to shift-invariant spaces. We also define a new notion of range function adapted to the Γ-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 Γ-invariant subspace that best approximates a set of functional data in L^2(G) . This is very relevant in applications since in the euclidean case, Γ-invariant subspaces are invariant under rigid movements, a very sought feature in models for signal processing.