Approximation by Group Invariant Suspaces

BARBIERI, DAVIDE; CABRELLI, CARLOS; HERNÁNDEZ, EUGENIO; MOLTER, URSULA

JOURNAL DE MATHEMATIQUES PURES ET APPLIQUEES

GAUTHIER-VILLARS/EDITIONS ELSEVIER

Lugar: Paris; Año: 2020

0021-7824

In this article we study the structure of Γ-invariant spaces of L2(S). Here S is a second countable LCA group. The invariance is with re- spect to the action of Γ, 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 ob- tain 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 sub- space that best approximates a set of functional data in L2(S). 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.