A note on convergence of Newton interpolating polynomials
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 \([- 1,1]\).
scientific article; zbMATH DE number 641405
scientific article