Accuracy of Lattice Translates of Several Multidimensional Refinable Functions

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Complex-valued functionsf1,...,fronRdarerefinableif they are linear combinations of finitely many of the rescaled and translated functionsfi(Ax-k), where the translateskare taken along a latticeΓ⊂RdandAis adilation matrixthat expansively mapsΓinto itself. Refinable functions satisfy arefinement equa...

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Autores principales: Cabrelli, C., Heil, C., Molter, U.
Formato: Artículo publishedVersion
Lenguaje:Inglés
Publicado: 1998
Materias:
Accuracy; approximation by translates; dilation equations; dilation matrix; multidimensional refinable functions; multidimensional wavelets; multiwavelets; refinement equations; refinable functions; shift invariant spaces; wavelets
Acceso en línea:http://hdl.handle.net/20.500.12110/paper_00219045_v95_n1_p5_Cabrelli
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Biblioteca Digital - Facultad de Ciencias Exactas y Naturales (UBA) de Universidad de Buenos Aires
id paperaa:paper_00219045_v95_n1_p5_Cabrelli
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spelling paperaa:paper_00219045_v95_n1_p5_Cabrelli2023-06-12T16:42:40Z Accuracy of Lattice Translates of Several Multidimensional Refinable Functions J. Approx. Theory 1998;95(1):5-52 Cabrelli, C. Heil, C. Molter, U. Accuracy; approximation by translates; dilation equations; dilation matrix; multidimensional refinable functions; multidimensional wavelets; multiwavelets; refinement equations; refinable functions; shift invariant spaces; wavelets Complex-valued functionsf1,...,fronRdarerefinableif they are linear combinations of finitely many of the rescaled and translated functionsfi(Ax-k), where the translateskare taken along a latticeΓ⊂RdandAis adilation matrixthat expansively mapsΓinto itself. Refinable functions satisfy arefinement equationf(x)=∑k∈Λckf(Ax-k), whereΛis a finite subset ofΓ, theckarer×rmatrices, andf(x)=(f1(x),...,fr(x))T. Theaccuracyoffis the highest degreepsuch that all multivariate polynomialsqwith degree(q)<pare exactly reproduced from linear combinations of translates off1,...,fralong the latticeΓ. In this paper, we determine the accuracypfrom the matricesck. Moreover, we determine explicitly the coefficientsyα,i(k) such thatxα=∑ri=1∑ k∈Γyα,i(k)fi(x+k). These coefficients are multivariate polynomialsyα,i(x) of degree α evaluated at lattice pointsk∈1. © 1998 Academic Press. Fil:Cabrelli, C. Universidad de Buenos Aires. Facultad de Ciencias Exactas y Naturales; Argentina. Fil:Molter, U. Universidad de Buenos Aires. Facultad de Ciencias Exactas y Naturales; Argentina. 1998 info:eu-repo/semantics/article info:ar-repo/semantics/artículo info:eu-repo/semantics/publishedVersion application/pdf eng info:eu-repo/semantics/openAccess http://creativecommons.org/licenses/by/2.5/ar http://hdl.handle.net/20.500.12110/paper_00219045_v95_n1_p5_Cabrelli
institution Universidad de Buenos Aires
institution_str I-28
repository_str R-134
collection Biblioteca Digital - Facultad de Ciencias Exactas y Naturales (UBA)
language Inglés
orig_language_str_mv eng
topic Accuracy; approximation by translates; dilation equations; dilation matrix; multidimensional refinable functions; multidimensional wavelets; multiwavelets; refinement equations; refinable functions; shift invariant spaces; wavelets
spellingShingle Accuracy; approximation by translates; dilation equations; dilation matrix; multidimensional refinable functions; multidimensional wavelets; multiwavelets; refinement equations; refinable functions; shift invariant spaces; wavelets
Cabrelli, C.
Heil, C.
Molter, U.
Accuracy of Lattice Translates of Several Multidimensional Refinable Functions
topic_facet Accuracy; approximation by translates; dilation equations; dilation matrix; multidimensional refinable functions; multidimensional wavelets; multiwavelets; refinement equations; refinable functions; shift invariant spaces; wavelets
description Complex-valued functionsf1,...,fronRdarerefinableif they are linear combinations of finitely many of the rescaled and translated functionsfi(Ax-k), where the translateskare taken along a latticeΓ⊂RdandAis adilation matrixthat expansively mapsΓinto itself. Refinable functions satisfy arefinement equationf(x)=∑k∈Λckf(Ax-k), whereΛis a finite subset ofΓ, theckarer×rmatrices, andf(x)=(f1(x),...,fr(x))T. Theaccuracyoffis the highest degreepsuch that all multivariate polynomialsqwith degree(q)<pare exactly reproduced from linear combinations of translates off1,...,fralong the latticeΓ. In this paper, we determine the accuracypfrom the matricesck. Moreover, we determine explicitly the coefficientsyα,i(k) such thatxα=∑ri=1∑ k∈Γyα,i(k)fi(x+k). These coefficients are multivariate polynomialsyα,i(x) of degree α evaluated at lattice pointsk∈1. © 1998 Academic Press.
format Artículo
Artículo
publishedVersion
author Cabrelli, C.
Heil, C.
Molter, U.
author_facet Cabrelli, C.
Heil, C.
Molter, U.
author_sort Cabrelli, C.
title Accuracy of Lattice Translates of Several Multidimensional Refinable Functions
title_short Accuracy of Lattice Translates of Several Multidimensional Refinable Functions
title_full Accuracy of Lattice Translates of Several Multidimensional Refinable Functions
title_fullStr Accuracy of Lattice Translates of Several Multidimensional Refinable Functions
title_full_unstemmed Accuracy of Lattice Translates of Several Multidimensional Refinable Functions
title_sort accuracy of lattice translates of several multidimensional refinable functions
publishDate 1998
url http://hdl.handle.net/20.500.12110/paper_00219045_v95_n1_p5_Cabrelli
work_keys_str_mv AT cabrellic accuracyoflatticetranslatesofseveralmultidimensionalrefinablefunctions
AT heilc accuracyoflatticetranslatesofseveralmultidimensionalrefinablefunctions
AT molteru accuracyoflatticetranslatesofseveralmultidimensionalrefinablefunctions
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