Subspaces with extra invariance nearest to observed data
Given an arbitrary finite set of data F={f1,…,fm}⊂L2(Rd) we prove the existence and show how to construct a “small shift invariant space” that is “closest” to the data F over certain class of closed subspaces of L2(Rd). The approximating subspace is required to have extra-invariance properties, that...
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| Acceso en línea: | https://bibliotecadigital.exactas.uba.ar/collection/paper/document/paper_10635203_v41_n2_p660_Cabrelli http://hdl.handle.net/20.500.12110/paper_10635203_v41_n2_p660_Cabrelli |
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paper:paper_10635203_v41_n2_p660_Cabrelli2023-06-08T16:03:31Z Subspaces with extra invariance nearest to observed data Cabrelli, Carlos Alberto Extra-invariance Paley–Wiener spaces Sampling Shift invariant spaces Harmonic analysis Sampling Approximation problems Closed subspace Parseval frames Primary Secondary Shift-invariant space Translation invariants Wiener spaces Functional analysis Given an arbitrary finite set of data F={f1,…,fm}⊂L2(Rd) we prove the existence and show how to construct a “small shift invariant space” that is “closest” to the data F over certain class of closed subspaces of L2(Rd). The approximating subspace is required to have extra-invariance properties, that is to be invariant under translations by a prefixed additive subgroup of Rd containing Zd. This is important for example in situations where we need to deal with jitter error of the data. Here small means that our solution subspace should be generated by the integer translates of a small number of generators. An expression for the error in terms of the data is provided and we construct a Parseval frame for the optimal space. We also consider the problem of approximating F from generalized Paley–Wiener spaces of Rd that are generated by the integer translates of a finite number of functions. That is finitely generated shift invariant spaces that are translation invariant. We characterize these spaces in terms of multi-tile sets of Rd, and show the connections with recent results on Riesz basis of exponentials on bounded sets of Rd. Finally we study the discrete case for our approximation problem. © 2015 Elsevier Inc. Fil:Cabrelli, C. Universidad de Buenos Aires. Facultad de Ciencias Exactas y Naturales; Argentina. 2016 https://bibliotecadigital.exactas.uba.ar/collection/paper/document/paper_10635203_v41_n2_p660_Cabrelli http://hdl.handle.net/20.500.12110/paper_10635203_v41_n2_p660_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 |
Extra-invariance Paley–Wiener spaces Sampling Shift invariant spaces Harmonic analysis Sampling Approximation problems Closed subspace Parseval frames Primary Secondary Shift-invariant space Translation invariants Wiener spaces Functional analysis |
| spellingShingle |
Extra-invariance Paley–Wiener spaces Sampling Shift invariant spaces Harmonic analysis Sampling Approximation problems Closed subspace Parseval frames Primary Secondary Shift-invariant space Translation invariants Wiener spaces Functional analysis Cabrelli, Carlos Alberto Subspaces with extra invariance nearest to observed data |
| topic_facet |
Extra-invariance Paley–Wiener spaces Sampling Shift invariant spaces Harmonic analysis Sampling Approximation problems Closed subspace Parseval frames Primary Secondary Shift-invariant space Translation invariants Wiener spaces Functional analysis |
| description |
Given an arbitrary finite set of data F={f1,…,fm}⊂L2(Rd) we prove the existence and show how to construct a “small shift invariant space” that is “closest” to the data F over certain class of closed subspaces of L2(Rd). The approximating subspace is required to have extra-invariance properties, that is to be invariant under translations by a prefixed additive subgroup of Rd containing Zd. This is important for example in situations where we need to deal with jitter error of the data. Here small means that our solution subspace should be generated by the integer translates of a small number of generators. An expression for the error in terms of the data is provided and we construct a Parseval frame for the optimal space. We also consider the problem of approximating F from generalized Paley–Wiener spaces of Rd that are generated by the integer translates of a finite number of functions. That is finitely generated shift invariant spaces that are translation invariant. We characterize these spaces in terms of multi-tile sets of Rd, and show the connections with recent results on Riesz basis of exponentials on bounded sets of Rd. Finally we study the discrete case for our approximation problem. © 2015 Elsevier Inc. |
| author |
Cabrelli, Carlos Alberto |
| author_facet |
Cabrelli, Carlos Alberto |
| author_sort |
Cabrelli, Carlos Alberto |
| title |
Subspaces with extra invariance nearest to observed data |
| title_short |
Subspaces with extra invariance nearest to observed data |
| title_full |
Subspaces with extra invariance nearest to observed data |
| title_fullStr |
Subspaces with extra invariance nearest to observed data |
| title_full_unstemmed |
Subspaces with extra invariance nearest to observed data |
| title_sort |
subspaces with extra invariance nearest to observed data |
| publishDate |
2016 |
| url |
https://bibliotecadigital.exactas.uba.ar/collection/paper/document/paper_10635203_v41_n2_p660_Cabrelli http://hdl.handle.net/20.500.12110/paper_10635203_v41_n2_p660_Cabrelli |
| work_keys_str_mv |
AT cabrellicarlosalberto subspaceswithextrainvariancenearesttoobserveddata |
| _version_ |
1768544831226773504 |