On the convergence of the rational interpolation approximant of Carathéodory functions
type: Item
instance of: scholarly article
title: On the convergence of the rational interpolation approximant of Carathéodory functions
zbMATH Open document ID: 0818.41011
DOI: 10.1016/0377-0427(94)90257-7
author: V. Pereyra
published in: Journal of Computational and Applied Mathematics
publication date: 1995-08-21T00:00:00Z
review text: Let \(d\mu\) denote a finite positive Borel measure on the unit circle with associated Carathéodory function \(F(z)= \int [u/(u- z)] d\mu(u)\). Given a triangular table of interpolation nodes \(\{z_{n, k}: | z_{n, k}|< 1\}_{k= 1}^ n\), \(n= 0,1,\dots\), the rational interpolants \(p_{n- 1}/q_ n\) of type \((n- 1,n)\) for \(F\) in the points \(\{z_{n, k}, 1/\overline z_{n, k}\}_{k= 1}^ n\) are considered. First it is shown that \(q_ n\) can be chosen to be the \(n\)th orthogonal polynomial with respect to the varying measure \(d\mu_ n(u)= d\mu(u)/ | w_ n(u)|^ 2\), where \(w_ n(u)= \prod_{k= 1}^ n (1- \overline z_{n, k}u)\). From this, expressions for \(p_{n- 1}\) and the error \(e_ n= q_ n F- p_{n- 1}\) can be obtained. Then it is shown that the interpolants converge to \(F\) locally uniformly outside the unit disk when \[ \lim_{n\to\infty} \sum_{i= 1}^ n (1-| z_{n, i}|^ 2)= \infty. \] Finally, it is shown that some knowledge about the distribution of the nodes \(z_{n, k}\) leads to corresponding knowledge of the rate of convergence.
reviewed by: Adhemar Bultheel
Mathematics Subject Classification ID: 41A20
Mathematics Subject Classification ID: 30E05
zbMATH DE Number: 739565
zbMATH Keywords: complex approximation
zbMATH Keywords: orthogonal rational functions
zbMATH Keywords: orthogonal polynomials for varying measure
zbMATH Keywords: rational interpolation
zbMATH Keywords: Carathéodory function
MaRDI profile type: Publication
full work available at URL: 0377-0427(94)90257-7
OpenAlex ID: W2094267724
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description: scientific article; zbMATH DE number 739565