Invariance of a Shift-Invariant Space in Several Variables

MAGALÍ ANASTASIO; CARLOS CABRELLI; VICTORIA PATERNOSTRO

COMPLEX ANALYSIS AND OPERATOR THEORY

BIRKHAUSER VERLAG AG

Lugar: Basel; Año: 2011 vol. 5 p. 1031 - 1050

1661-8254

In this article we study invariance properties of shift-invariant spaces in higher dimensions.We state and prove several necessary and sufficient conditions for a shift-invariant space to be invariant under a given closed subgroup of $R^d$, and prove the existence of shift-invariantspaces that are exactly invariant  for  each given subgroup. As an application we  relate the extra invariance to the size of support of the Fourier transform of the generators of the shift-invariant space.This work extends recent results obtained for the case of one variable to several variables.