Linear combinations of generators in multiplicatively invariant spaces
Volume 226 / 2015
Abstract
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 $L^2(\mathbb {R}^d)$.
