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Linear combinations of generators in multiplicatively invariant spaces

Volume 226 / 2015

Victoria Paternostro Studia Mathematica 226 (2015), 1-16 MSC: Primary 42C40, 43A70; Secondary 42C15, 22B99. DOI: 10.4064/sm226-1-1

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 $L^2(\mathbb {R}^d)$.

Authors

  • Victoria PaternostroDepartamento 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
    e-mail

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