Scalar and Hermite subdivision schemes
type: Item
instance of: scholarly article
title: Scalar and Hermite subdivision schemes
zbMATH Open document ID: 1111.65020
author: Serge Dubuc
published in: Applied and Computational Harmonic Analysis
publication date: 2007-01-09T00:00:00Z
review text: A criterion of convergence for stationary nonuniform subdivision sche\-mes is provided. For periodic subdivision schemes, this criterion is optimal and can be applied to Hermite subdivision schemes which are not necessarily interpolatory. For the Merrien family of Hermite subdivision schemes which involve two parameters, the values of the parameters for which the Hermite subdivision scheme is convergent can be described explicitly. The paper is organized as follows. The basic terminology for subdivision schemes is described in Section 2. In Section 3, the authors obtain a difference subdivision scheme from an affine subdivision scheme. Section 4 contains a criterion of convergence for nonuniform subdivision schemes. In Section 5, the authors study the basic properties of \(C^1\)-convergence and the reproduction of linear functions for Hermite subdivision schemes. In Section 6, they associate a nonuniform scalar subdivision scheme to any interpolating Hermite subdivision scheme that reproduces constants, and obtain a necessary and sufficient condition for \(C^1\)-convergence for a given Hermite subdivision scheme that reproduces linear functions. Section 7 recalls the Merrien family of Hermite subdivision schemes. Section 8 contains necessary conditions for convergence in the Merrien family. These conditions are in fact sufficient, and in Section 9, authors obtain the precise definition of the convergence region, and provide examples in Section~10.
Mathematics Subject Classification ID: 65D18
zbMATH DE Number: 5083830
zbMATH Keywords: Hermite interpolation
zbMATH Keywords: numerical examples
zbMATH Keywords: convergence
reviewed by: Vladimir Yu. Rovenskij
MaRDI profile type: Publication
full work available at URL: j.acha.2006.04.007
OpenAlex ID: W2041216738
cites work: Commutation for irregular subdivision
cites work: Convergent vector and Hermite subdivision schemes
cites work: Analysis of uniform binary subdivision schemes for curve design
cites work: Q4026009
cites work: Q4856407
cites work: Q4259911
cites work: Approximation properties and construction of Hermite interpolants and biorthogonal multiwavelets
cites work: Noninterpolatory Hermite subdivision schemes
cites work: Polynomial generation and quasi-interpolation in stationary non-uniform subdivision
cites work: Using Laurent polynomial representation for the analysis of non-uniform binary subdivision schemes
cites work: C1 Interpolatory Subdivision with Shape Constraints for Curves
cites work: Subdivision schemes and irregular grids
cites work: A family of Hermite interpolants by bisection algorithms
cites work: Interpolants d'HermiteC2obtenus par subdivision
cites work: Multiple refinable Hermite interpolants
DOI: 10.1016/J.ACHA.2006.04.007
description: scientific article; zbMATH DE number 5083830