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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todo:paper_02714132_v693_n_p143_Cabrelli2023-10-03T15:14:52Z An approximation problem in multiplicatively invariant spaces Cabrelli, C. Mosquera, C.A. Paternostro, V. 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. JOUR info:eu-repo/semantics/openAccess http://creativecommons.org/licenses/by/2.5/ar 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 Cabrelli, C. Mosquera, C.A. Paternostro, V. 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. |
| format |
JOUR |
| author |
Cabrelli, C. Mosquera, C.A. Paternostro, V. |
| author_facet |
Cabrelli, C. Mosquera, C.A. Paternostro, V. |
| author_sort |
Cabrelli, C. |
| 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 |
| url |
http://hdl.handle.net/20.500.12110/paper_02714132_v693_n_p143_Cabrelli |
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