Extra Invariance of a Shift-Invariant Space in LCA Groups

MAGALÍ ANASTASIO; CARLOS CABRELLI; VICTORIA PATERNOSTRO

JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS

ACADEMIC PRESS INC ELSEVIER SCIENCE

Lugar: Amsterdan; Año: 2010 vol. 370 p. 530 - 537

0022-247X

This article generalizes recent results in the extra invariance for shift-invariant spaces to the context of LCA groups.Let G be a locally compact abelian (LCA) group and K a closed subgroup of G.A closed subspace of L^2(G) is called K-invariant if  it is invariant under translations by elements of K.Assume now that H is a countable uniform lattice in G and M is any closed subgroup of G containing H.In this article we study  necessary and sufficient conditions for an H-invariant space to be M-invariant.As a consequence of our results we prove that for each closed subgroup Mof G containing the lattice H, there exists an H-invariant space S that is exactly M-invariant.That is, S is not invariant under any other subgroup M´ containing M. We also obtain estimates on the support of the  Fourier transform  of the generators of the H-invariant space, related to  its  M-invariance.