Accuracy of Several Multidimensional Refinable Distributions

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Compactly supported distributions f1, . . . , fr on Rd are refinable if each fi is a finite linear combination of the rescaled 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 dist...

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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_02182165_v6_n5_p482_Cabrelli
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Biblioteca Digital - Facultad de Ciencias Exactas y Naturales (UBA) de Universidad de Buenos Aires
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spelling todo:paper_02182165_v6_n5_p482_Cabrelli2023-10-03T15:10:51Z 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 Rd are refinable if each fi is a finite linear combination of the rescaled 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) = ΣkεΛ 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 = Σi=1 r ΣkεΓ yα,i(k) fi(x + k). These coefficients are multivariate polynomials yα,i (x) of degree |α| evaluated at lattice points k ε Γ. 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_02182165_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 Rd are refinable if each fi is a finite linear combination of the rescaled 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) = ΣkεΛ 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 = Σi=1 r ΣkεΓ yα,i(k) fi(x + k). These coefficients are multivariate polynomials yα,i (x) of degree |α| evaluated at lattice points k ε Γ.
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_02182165_v6_n5_p482_Cabrelli
work_keys_str_mv AT cabrellic accuracyofseveralmultidimensionalrefinabledistributions
AT heilc accuracyofseveralmultidimensionalrefinabledistributions
AT molteru accuracyofseveralmultidimensionalrefinabledistributions
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