A note on convergence of Newton interpolating polynomials
type: Item
instance of: scholarly article
title: A note on convergence of Newton interpolating polynomials
zbMATH Open document ID: 0810.41008
DOI: 10.1016/0377-0427(94)00057-3
author: George M. Phillips
author: Dimitar K. Dimitrov
published in: Journal of Computational and Applied Mathematics
publication date: 1994-09-25T00:00:00Z
review text: The first part of the paper deals with rational functions \(r\) of the form \(r(z)= {{p(z)} \over {az-b}}\) where \(p\) is a polynomial and \(a\), \(b\) are complex numbers with \(a\neq 0\) and \({b\over a}\not\in \{0,1,2,\dots\}\). It is shown that the corresponding sequence \((N_ k)_{k=0}^ \infty\) of Newton interpolating polynomials, based on the data \(r(l)\), \(l=0,1,2,\dots\), converges (pointwise) to \(r\) on the set \(\{r\in\mathbb{C}\mid \text{Re}(z) >\text{Re} ({b\over a})\}\). In the second part of the paper this result is used to prove that for suitable real parameters \(\alpha\) and \(\beta\) there exists a natural number \(n\) and a point \(x\in [-1,1]\) such that \[ \sum_{k=0}^ n {{P_ k^{(\alpha, \beta)} (x)} \over {P_ k^{(\beta, \alpha)} (1)}} <0. \] Here \(P_ k^{(\alpha, \beta)} (x)\) denotes the Jacobi polynomial of order \(k\) with respect to the weight function \((1- x)^ \alpha (1+ x)^ \beta\) on the interval \([- 1,1]\).
Mathematics Subject Classification ID: 41A10
Mathematics Subject Classification ID: 33C20
Mathematics Subject Classification ID: 33C45
zbMATH DE Number: 641405
zbMATH Keywords: positive sums
zbMATH Keywords: Newton interpolating polynomials
zbMATH Keywords: Jacobi polynomial
reviewed by: J. Saurer
MaRDI profile type: Publication
full work available at URL: 0377-0427(94)00057-3
OpenAlex ID: W2012036798
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description: scientific article; zbMATH DE number 641405