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Linear combinations of generators in multiplicatively invariant spaces

100%

Paternostro V.

Studia Mathematica

2015

tom 226

nr 1

1-16

Multiplicatively invariant (MI) spaces are closed subspaces of L²(Ω,𝓗 ) 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 $L²(ℝ^{d})$.

Shift-modulation invariant spaces on LCA groups

63%

Cabrelli C., Paternostro V.

Studia Mathematica

2012

tom 211

nr 1

1-19

A (K,Λ) shift-modulation invariant space is a subspace of L²(G) that is invariant under translations along elements in K and modulations by elements in Λ. Here G is a locally compact abelian group, and K and Λ are closed subgroups of G and the dual group Ĝ, respectively. We provide a characterization of shift-modulation invariant spaces when K and Λ are uniform lattices. This extends previous results known for $L²(ℝ^{d})$. We develop fiberization techniques and suitable range functions adapted to LCA groups needed to provide the desired characterization.

Ograniczanie wyników

2 Studia Mathematica

2 Paternostro V.

1 Cabrelli C.

1 2015

1 2012

Wyniki wyszukiwania

Linear combinations of generators in multiplicatively invariant spaces

Shift-modulation invariant spaces on LCA groups