On the convergence of polynomial approximation of rational functions
type: Item
instance of: scholarly article
title: On the convergence of polynomial approximation of rational functions
zbMATH Open document ID: 0878.41004
author: T. W. Sederberg
author: Falai Chen
author: Guojin Wang
published in: Journal of Approximation Theory
publication date: 1998-01-13T00:00:00Z
review text: Based on the notion of hybrid polynomials, the authors derive necessary and sufficient convergence criteria for various polynomial approximations of rational functions and rational curves. This paper investigates the convergence condition for the polynomial approximation of rational functions and rational curves. The main result, based on a hybrid expression of rational functions (or curves), is that two-pointed Hermite interpolation converges if all eigenvalue moduli or a certain \(n \times n\) matrix are less than 2, where \(n\) is the degree of the rational function (or curve), and where the elements of the matrix are expressions involving only the denominator polynomial coefficients (weights) of the rational function (or curve). As a corollary for the special case of \(n=1\), a necessary and sufficient condition for convergence is also obtained which only involves the roots of the denominator of the rational function and which is shown to be superior to the condition obtained by the traditional remainder theory for polynomial interpolation. For the low degree case \((n= 1,2\), and 3), concrete conditions are derived. Application to rational Bernstein-Bézier curves is also discussed.
Mathematics Subject Classification ID: 41A10
zbMATH DE Number: 1033752
zbMATH Keywords: Bernstein-Bézier curves
reviewed by: R. S. Dahiya
MaRDI profile type: Publication
OpenAlex ID: W2025665036
cites work: Q3848301
cites work: Methods of matrix algebra
cites work: Q3943169
cites work: Q4026190
cites work: Q3996421
cites work: Q4041056
DOI: 10.1006/JATH.1996.3060
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description: scientific article; zbMATH DE number 1033752