A closed PP form of box splines via Green's function decomposition
type: Item
instance of: scholarly article
title: A closed PP form of box splines via Green's function decomposition
zbMATH Open document ID: 1392.41006
author: Joshua Horacsek
author: Usman Alim
published in: Journal of Approximation Theory
publication date: 2018-06-06T00:00:00Z
review text: A set construction scheme is presented for the class of non-degenerate box splines. This scheme reparably decomposes the Green's function of a box spline, yielding its explicit piecewise polynomial form. It is possible to use the well-known recursive formulation to obtain these polynomial pieces, but that procedure is quite expensive. The fast evaluation schemes using piecewise polynomial form of box splines are created. The spatial Fourier form of Green's function and the spatial form of the difference operator are used. It is proved that, under certain conditions, the proposed decomposition procedure is of asymptotically orders of magnitude lower than the recursive procedure. This allows to evaluate box splines with more direction vectors than what would be feasible under the recursive scheme. The exemplary constructions for some known box splines are provided.
Mathematics Subject Classification ID: 41A15
Mathematics Subject Classification ID: 41A10
zbMATH DE Number: 6881221
zbMATH Keywords: piecewise polynomial form of box splines
zbMATH Keywords: Fourier form of Green's function
zbMATH Keywords: recursive procedure
zbMATH Keywords: fast evaluation schemes
reviewed by: V. L. Leontiev
describes a project that uses: SageMath
MaRDI profile type: Publication
full work available at URL: j.jat.2018.04.002
OpenAlex ID: W2802280571
cites work: On the evaluation of box splines
cites work: Box splines
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cites work: Symmetric box-splines on the \(\mathcal A^*_n\) lattice
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cites work: On the Multivariate Horner Scheme
cites work: On the multivariate Horner scheme. II: Running error analysis
Wikidata QID: Q129912562
DOI: 10.1016/J.JAT.2018.04.002
description: scientific article; zbMATH DE number 6881221