Anisotropic dilations of shift-invariant subspaces and approximation properties in L2(Rd)

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Title: Anisotropic dilations of shift-invariant subspaces and approximation properties in L2(Rd)
Authors: Cifuentes, Patricio | San Antolín Gil, Ángel | Soto-Bajo, Moisés
Research Group/s: Curvas Alpha-Densas. Análisis y Geometría Local
Center, Department or Service: Universidad de Alicante. Departamento de Matemáticas
Keywords: Anisotropic Sobolev spaces | Approximate continuity | Approximation order | Density order | Expansive linear maps | Shift-invariant spaces | Spectral function
Knowledge Area: Análisis Matemático
Issue Date: Apr-2015
Publisher: Wiley-VCH Verlag GmbH & Co. KGaA
Citation: Mathematische Nachrichten. 2015, 288(5-6): 525-539. doi:10.1002/mana.201300319
Abstract: Let A be an expansive linear map in Rd. Approximation properties of shift-invariant subspaces of L2(Rd) when they are dilated by integer powers of A are studied. Shift-invariant subspaces providing approximation order α or density order α associated to A are characterized. These characterizations impose certain restrictions on the behavior of the spectral function at the origin expressed in terms of the concept of point of approximate continuity. The notions of approximation order and density order associated to an isotropic dilation turn out to coincide with the classical ones introduced by de Boor, DeVore and Ron. This is no longer true when A is anisotropic. In this case the A-dilated shift-invariant subspaces approximate the anisotropic Sobolev space associated to A and α. Our main results are also new when S is generated by translates of a single function. The obtained results are illustrated by some examples
Sponsor: The first author is supported in part by MEC/MICINN grants MTM2010-16518 and MTM2013-40945-P (Spain). The second author was partially supported by MEC/MICINN grant MTM2011-27998 (Spain). The third author is supported in part by MEC/MlClNN grant MTM2013-40945-P (Spain).
URI: http://hdl.handle.net/10045/52993
ISSN: 0025-584X (Print) | 1522-2616 (Online)
DOI: 10.1002/mana.201300319
Language: eng
Type: info:eu-repo/semantics/article
Rights: © 2014 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim
Peer Review: si
Publisher version: http://dx.doi.org/10.1002/mana.201300319
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