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Abstract:

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.

Registro:

Documento: Artículo
Título:Subspaces with extra invariance nearest to observed data
Autor:Cabrelli, C.; Mosquera, C.A.
Filiación:Departamento de Matemática, Facultad de Ciencias Exactas y Naturales, Universidad de Buenos Aires, Ciudad Universitaria, Pabellón I, Buenos Aires, 1428, Argentina
IMAS-CONICET, Consejo Nacional de Investigaciones Científicas y Técnicas, Argentina
Palabras clave: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
Año:2016
Volumen:41
Número:2
Página de inicio:660
Página de fin:676
DOI: http://dx.doi.org/10.1016/j.acha.2015.12.001
Título revista:Applied and Computational Harmonic Analysis
Título revista abreviado:Appl Comput Harmonic Anal
ISSN:10635203
CODEN:ACOHE
Registro:https://bibliotecadigital.exactas.uba.ar/collection/paper/document/paper_10635203_v41_n2_p660_Cabrelli

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Citas:

---------- APA ----------
Cabrelli, C. & Mosquera, C.A. (2016) . Subspaces with extra invariance nearest to observed data. Applied and Computational Harmonic Analysis, 41(2), 660-676.
http://dx.doi.org/10.1016/j.acha.2015.12.001
---------- CHICAGO ----------
Cabrelli, C., Mosquera, C.A. "Subspaces with extra invariance nearest to observed data" . Applied and Computational Harmonic Analysis 41, no. 2 (2016) : 660-676.
http://dx.doi.org/10.1016/j.acha.2015.12.001
---------- MLA ----------
Cabrelli, C., Mosquera, C.A. "Subspaces with extra invariance nearest to observed data" . Applied and Computational Harmonic Analysis, vol. 41, no. 2, 2016, pp. 660-676.
http://dx.doi.org/10.1016/j.acha.2015.12.001
---------- VANCOUVER ----------
Cabrelli, C., Mosquera, C.A. Subspaces with extra invariance nearest to observed data. Appl Comput Harmonic Anal. 2016;41(2):660-676.
http://dx.doi.org/10.1016/j.acha.2015.12.001