Linear combinations of generators in multiplicatively invariant spaces

Victoria Paternostro¹

¹ Departamento de Matemática Facultad de Ciencias Exactas y Naturales Universidad de Buenos Aires Ciudad Universitaria, Pabellón I 1428 Buenos Aires, Argentina and IMAS-CONICET Consejo Nacional de Investigaciones Científicas y Técnicas Argentina

Studia Mathematica, Tome 226 (2015) no. 1, pp. 1-16

Résumé

Multiplicatively invariant (MI) spaces are closed subspaces of $L^2(\Omega ,\mathcal {H})$ that are invariant under multiplication by (some) functions in $L^{\infty }(\Omega )$; they were first introduced by Bownik and Ross (2014). In this paper we work with MI spaces that are finitely generated. We prove that almost every set of functions constructed by taking linear combinations of the generators of a finitely generated MI space is a new set of generators for the same space, and we give necessary and sufficient conditions on the linear combinations to preserve frame properties. We then apply our results on MI spaces to systems of translates in the context of locally compact abelian groups and we extend some results previously proven for systems of integer translates in $L^2(\mathbb {R}^d)$.

Détail

DOI : 10.4064/sm226-1-1

Keywords: multiplicatively invariant spaces closed subspaces omega mathcal invariant under multiplication functions infty omega first introduced bownik ross paper work spaces finitely generated prove almost every set functions constructed taking linear combinations generators finitely generated space set generators space necessary sufficient conditions linear combinations preserve frame properties apply results spaces systems translates context locally compact abelian groups extend results previously proven systems integer translates mathbb