An approximation problem in multiplicatively invariant spaces

Let H be Hilbert space and (Ω, m) a σ-finite measure space. Multiplicatively invariant (MI) spaces are closed subspaces of L2(Ω, H) that are invariant under point-wise multiplication by functions from a fixed subset of L∞(Ω). Given a finite set of data F ⊆ L2 (Ω, H), in this paper we prove the exist...

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Detalles Bibliográficos
Publicado: 2017
Materias:
Approximation
Extra invariance
Multiplicatively invariant spaces
Shift-invariant spaces
Acceso en línea:https://bibliotecadigital.exactas.uba.ar/collection/paper/document/paper_02714132_v693_n_p143_Cabrelli
http://hdl.handle.net/20.500.12110/paper_02714132_v693_n_p143_Cabrelli
Aporte de:
Biblioteca Digital - Facultad de Ciencias Exactas y Naturales (UBA) de Universidad de Buenos Aires
id paper:paper_02714132_v693_n_p143_Cabrelli
record_format dspace
spelling paper:paper_02714132_v693_n_p143_Cabrelli2023-06-08T15:24:58Z An approximation problem in multiplicatively invariant spaces Approximation Extra invariance Multiplicatively invariant spaces Shift-invariant spaces Let H be Hilbert space and (Ω, m) a σ-finite measure space. Multiplicatively invariant (MI) spaces are closed subspaces of L2(Ω, H) that are invariant under point-wise multiplication by functions from a fixed subset of L∞(Ω). Given a finite set of data F ⊆ L2 (Ω, H), in this paper we prove the existence and construct an MI space M that best fits F, in the least squares sense. MI spaces are related to shift-invariant (SI) spaces via a fiberization map, which allows us to solve an approximation problem for SI spaces in the context of locally compact abelian groups. On the other hand, we introduce the notion of decomposable MI spaces (MI spaces that can be decomposed into an orthogonal sum of MI subspaces) and solve the approximation problem for the class of these spaces. Since SI spaces having extra invariance are in one-to-one relation to decomposable MI spaces, we also solve our approximation problem for this class of SI spaces. Finally we prove that translation-invariant spaces are in correspondence with totally decomposable MI spaces. © 2017 American Mathematical Society. 2017 https://bibliotecadigital.exactas.uba.ar/collection/paper/document/paper_02714132_v693_n_p143_Cabrelli http://hdl.handle.net/20.500.12110/paper_02714132_v693_n_p143_Cabrelli
institution Universidad de Buenos Aires
institution_str I-28
repository_str R-134
collection Biblioteca Digital - Facultad de Ciencias Exactas y Naturales (UBA)
topic Approximation
Extra invariance
Multiplicatively invariant spaces
Shift-invariant spaces
spellingShingle Approximation
Extra invariance
Multiplicatively invariant spaces
Shift-invariant spaces
An approximation problem in multiplicatively invariant spaces
topic_facet Approximation
Extra invariance
Multiplicatively invariant spaces
Shift-invariant spaces
description Let H be Hilbert space and (Ω, m) a σ-finite measure space. Multiplicatively invariant (MI) spaces are closed subspaces of L2(Ω, H) that are invariant under point-wise multiplication by functions from a fixed subset of L∞(Ω). Given a finite set of data F ⊆ L2 (Ω, H), in this paper we prove the existence and construct an MI space M that best fits F, in the least squares sense. MI spaces are related to shift-invariant (SI) spaces via a fiberization map, which allows us to solve an approximation problem for SI spaces in the context of locally compact abelian groups. On the other hand, we introduce the notion of decomposable MI spaces (MI spaces that can be decomposed into an orthogonal sum of MI subspaces) and solve the approximation problem for the class of these spaces. Since SI spaces having extra invariance are in one-to-one relation to decomposable MI spaces, we also solve our approximation problem for this class of SI spaces. Finally we prove that translation-invariant spaces are in correspondence with totally decomposable MI spaces. © 2017 American Mathematical Society.
title An approximation problem in multiplicatively invariant spaces
title_short An approximation problem in multiplicatively invariant spaces
title_full An approximation problem in multiplicatively invariant spaces
title_fullStr An approximation problem in multiplicatively invariant spaces
title_full_unstemmed An approximation problem in multiplicatively invariant spaces
title_sort approximation problem in multiplicatively invariant spaces
publishDate 2017
url https://bibliotecadigital.exactas.uba.ar/collection/paper/document/paper_02714132_v693_n_p143_Cabrelli
http://hdl.handle.net/20.500.12110/paper_02714132_v693_n_p143_Cabrelli
_version_ 1768545281499987968

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