Accuracy of Lattice Translates of Several Multidimensional Refinable Functions
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...
| Authors: | , , |
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| Format: | article |
| Status: | Published version |
| Publication Date: | 1998 |
| Country: | Argentina |
| Institution: | Universidad Nacional de Buenos Aires. Facultad de Ciencias Exactas y Naturales |
| Repository: | Biblioteca Digital (UBA-FCEN) |
| Language: | English |
| OAI Identifier: | paperaa:paper_00219045_v95_n1_p5_Cabrelli |
| Online Access: | http://hdl.handle.net/20.500.12110/paper_00219045_v95_n1_p5_Cabrelli |
| Access Level: | Open access |
| Keyword: | Accuracy; approximation by translates; dilation equations; dilation matrix; multidimensional refinable functions; multidimensional wavelets; multiwavelets; refinement equations; refinable functions; shift invariant spaces; wavelets |
| Summary: | 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. |
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