Accuracy of several multidimensional refinable distributions

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Compactly supported distributions f1,..., fr on 9d are refinable if each fi is a finite linear combination of the reseated and translated distributions fj (Ax -k), where the translates k are taken along a lattice Γ ⊂ Rd and A is a dilation matrix that expansively maps Γ into itself. Refinable distri...

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Autores principales:	Cabrelli, C., Heil, C., Molter, U.
Formato:	JOUR
Materias:	Accuracy Dilation equation Dilation matrix Multidimensional wavelets Multiwavelets Refinable distributions Refinable functions Refinement equation Shift invariant spaces Wavelets
Acceso en línea:	http://hdl.handle.net/20.500.12110/paper_10695869_v6_n5_p482_Cabrelli
Aporte de:	Biblioteca Digital - Facultad de Ciencias Exactas y Naturales (UBA) de Universidad de Buenos Aires

id	todo:paper_10695869_v6_n5_p482_Cabrelli
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spelling	todo:paper_10695869_v6_n5_p482_Cabrelli2023-10-03T16:02:15Z Accuracy of several multidimensional refinable distributions Cabrelli, C. Heil, C. Molter, U. Accuracy Dilation equation Dilation matrix Multidimensional wavelets Multiwavelets Refinable distributions Refinable functions Refinement equation Shift invariant spaces Wavelets Compactly supported distributions f1,..., fr on 9d are refinable if each fi is a finite linear combination of the reseated and translated distributions fj (Ax -k), where the translates k are taken along a lattice Γ ⊂ Rd and A is a dilation matrix that expansively maps Γ into itself. Refinable distributions satisfy a refinement equation f(x) = ∑k∈Λ ck f(Ax-k), where Λ is a finite subset of Γ, the ck are r × r matrices, and f = (f1,..., fr)T. The accuracy of f is the highest degree p such that all multivariate polynomials q with degree(q) < p are exactly reproduced from linear combinations of translates of f1,..., fr along the lattice Γ. We determine the accuracy p from the matrices ck. Moreover, we determine explicitly the coefficients yα,i(k) such that xα = ∑i=1 r ∑k∈Gamma; yα,i fi(x + k). These coefficients are multivariate polynomials yα,i(x) of degree \|α\| evaluated at lattice points k ∈ Γ. © 2000 Birkhäuser Boston. All rights reserved. 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. JOUR info:eu-repo/semantics/openAccess http://creativecommons.org/licenses/by/2.5/ar http://hdl.handle.net/20.500.12110/paper_10695869_v6_n5_p482_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	Accuracy Dilation equation Dilation matrix Multidimensional wavelets Multiwavelets Refinable distributions Refinable functions Refinement equation Shift invariant spaces Wavelets
spellingShingle	Accuracy Dilation equation Dilation matrix Multidimensional wavelets Multiwavelets Refinable distributions Refinable functions Refinement equation Shift invariant spaces Wavelets Cabrelli, C. Heil, C. Molter, U. Accuracy of several multidimensional refinable distributions
topic_facet	Accuracy Dilation equation Dilation matrix Multidimensional wavelets Multiwavelets Refinable distributions Refinable functions Refinement equation Shift invariant spaces Wavelets
description	Compactly supported distributions f1,..., fr on 9d are refinable if each fi is a finite linear combination of the reseated and translated distributions fj (Ax -k), where the translates k are taken along a lattice Γ ⊂ Rd and A is a dilation matrix that expansively maps Γ into itself. Refinable distributions satisfy a refinement equation f(x) = ∑k∈Λ ck f(Ax-k), where Λ is a finite subset of Γ, the ck are r × r matrices, and f = (f1,..., fr)T. The accuracy of f is the highest degree p such that all multivariate polynomials q with degree(q) < p are exactly reproduced from linear combinations of translates of f1,..., fr along the lattice Γ. We determine the accuracy p from the matrices ck. Moreover, we determine explicitly the coefficients yα,i(k) such that xα = ∑i=1 r ∑k∈Gamma; yα,i fi(x + k). These coefficients are multivariate polynomials yα,i(x) of degree \|α\| evaluated at lattice points k ∈ Γ. © 2000 Birkhäuser Boston. All rights reserved.
format	JOUR
author	Cabrelli, C. Heil, C. Molter, U.
author_facet	Cabrelli, C. Heil, C. Molter, U.
author_sort	Cabrelli, C.
title	Accuracy of several multidimensional refinable distributions
title_short	Accuracy of several multidimensional refinable distributions
title_full	Accuracy of several multidimensional refinable distributions
title_fullStr	Accuracy of several multidimensional refinable distributions
title_full_unstemmed	Accuracy of several multidimensional refinable distributions
title_sort	accuracy of several multidimensional refinable distributions
url	http://hdl.handle.net/20.500.12110/paper_10695869_v6_n5_p482_Cabrelli
work_keys_str_mv	AT cabrellic accuracyofseveralmultidimensionalrefinabledistributions AT heilc accuracyofseveralmultidimensionalrefinabledistributions AT molteru accuracyofseveralmultidimensionalrefinabledistributions
_version_	1782028150527492096

Accuracy of several multidimensional refinable distributions

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