Linear combinations of generators in multiplicatively invariant spaces

Victoria Paternostro. "Linear combinations of generators in multiplicatively invariant spaces." Studia Mathematica 226.1 (2015): 1-16. <http://eudml.org/doc/285693>.

@article{VictoriaPaternostro2015,
abstract = {Multiplicatively invariant (MI) spaces are closed subspaces of L²(Ω, ) that are invariant under multiplication by (some) functions in $L^\{∞\}(Ω)$; 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 $L²(ℝ^\{d\})$.},
author = {Victoria Paternostro},
journal = {Studia Mathematica},
keywords = {shift invariant space; fibers; range functions; multiplicatively invariant spaces; frame; Gramian; LCA groups},
language = {eng},
number = {1},
pages = {1-16},
title = {Linear combinations of generators in multiplicatively invariant spaces},
url = {http://eudml.org/doc/285693},
volume = {226},
year = {2015},
}

TY - JOUR
AU - Victoria Paternostro
TI - Linear combinations of generators in multiplicatively invariant spaces
JO - Studia Mathematica
PY - 2015
VL - 226
IS - 1
SP - 1
EP - 16
AB - Multiplicatively invariant (MI) spaces are closed subspaces of L²(Ω, ) that are invariant under multiplication by (some) functions in $L^{∞}(Ω)$; 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 $L²(ℝ^{d})$.
LA - eng
KW - shift invariant space; fibers; range functions; multiplicatively invariant spaces; frame; Gramian; LCA groups
UR - http://eudml.org/doc/285693
ER -