Analysis of subdivision schemes for nets of functions by proximity and controllability
type: Item
instance of: scholarly article
title: Analysis of subdivision schemes for nets of functions by proximity and controllability
zbMATH Open document ID: 1235.65020
published in: Journal of Computational and Applied Mathematics
publication date: 2011-12-08T00:00:00Z
review text: The authors develop tools for analyzing convergence, smoothness and approximation order of net subdivision schemes, that are schemes which recursively refine nets of bivariate continuous functions defined on grids of lines, and generate denser and denser nets. The two main ingredients of their analysis are (i) control on the size of the Lipschitz constants in grid intervals of the univariate net function and/or their derivatives, (ii) proximity to a point subdivision scheme. They derive sufficient conditions for the convergence of these sequence of refined nets, and for the smoothness of the limit function. They also derive approximation orders of net subdivision schemes which are in proximity with positive schemes refining points. The authors introduce an example of a family of blending spline-type net subdivision schemes and then analyze it with the tools developed in the paper. Although it is expected that the limit functions generated by the schemes analogous to higher order spline schemes are smoother, they can show only that the limit functions are \(C^1\). All the schemes in this family have approximation order 2.
Mathematics Subject Classification ID: 65D18
Mathematics Subject Classification ID: 65D07
zbMATH DE Number: 5987768
zbMATH Keywords: net subdivision schemes
zbMATH Keywords: point subdivision schemes
zbMATH Keywords: proximity
zbMATH Keywords: controllability
zbMATH Keywords: convergence
zbMATH Keywords: smoothness
zbMATH Keywords: blending spline-type
author: Costanza Conti
author: Nira Dyn
MaRDI profile type: Publication
full work available at URL: j.cam.2011.06.004
OpenAlex ID: W2060729466
cites work: Q5702279
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cites work: Q4026009
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cites work: A Theoretical Development for the Computer Generation and Display of Piecewise Polynomial Surfaces
cites work: Interpolatory blending net subdivision schemes of Dubuc-Deslauriers type
DOI: 10.1016/J.CAM.2011.06.004
description: scientific article; zbMATH DE number 5987768