A note on the extended Hermite-Fejér interpolation
type: Item
instance of: scholarly article
title: A note on the extended Hermite-Fejér interpolation
zbMATH Open document ID: 0631.41002
DOI: 10.1007/BF01849500
author: D. L. Berman
published in: Periodica Mathematica Hungarica
publication date: 1989-01-01T00:00:00Z
review text: Let \(\{x_ k^{(n)}\}^ n_{k=1}\), \(n=1,2,..\). be a given matrix of nodes \(-1<x_ k^{(n)}<...<x_ 1^{(n)}<1\), \(n=1,2,... \). Let us denote \[ \ell _ k^{(n)}(x)=\omega _ n(x)[(x-x_ n^{(n)})\omega _ n'(x_ k^{(n)})]^{-1},\quad h_ k^{(n)}(x)=\nu _ k^{(n)}(x)[\ell _ k^{(n)}(x)]^ 2, \] \[ \nu _ k^{(n)}(x)=1- (x-x_ k^{(n)})\omega _ n''(x_ k^{(n)})(\omega _ n'(x_ k^{(n)}))^{-1},\quad \omega _ n(x)=\prod ^{n}_{k=1}(x-x_ k^{(n)}). \] We prove the following theorem. Let the matrix of nodes satisfy the conditions: a) \(h_ k^{(n)}(x)\geq 0\), \(x\in [-1,1]\), \(k=1,2,...,n\), \(n=1,2,...\); \(b)\quad \lim _{n\to \infty}\sum ^{n}_{k=1}(x_ k^{(n)})^ ih_ k^{(n)}(x)=x^ i,\) \(i=1,2\); \(c)\quad \sum ^{n}_{k=1}[\ell _ k^{(n)}(1)]^ 2\leq C,\) \(n=1,2,...\), where \(C>0\) is any constant; d) \(| \omega _ n(x)/\omega _ n(1)| =O(1)\), \(| x| \leq 1\); e) \(\omega _ n(1)/\omega _ n'(1)=O(1/n^ 2)\); f) \(\omega _ n(1)\omega _ n''(1)/(\omega _ n'(1))^ 2=O(1)\). Let f''(x)\(\in C[-1,1]\) and \(f'(1)=f''(1)=0\). Then \(N_ n(f,x)\to f(x),\) \(n\to \infty\), uniformly in [-1,1], where the polynomial \(N_ n(f,x)\) of degree \(2n+1\) is uniquely determined by the conditions: \(N_ n(f,x_ k^{(n)}=f(x_ k^{(n)}),\) \(N_ n'(f,x_ k^{(n)})=0\), \(k=1,2,...,n\), \(N_ n'(f,1)=N_ n''(f,1)=0\).
Mathematics Subject Classification ID: 41A05
zbMATH DE Number: 4026034
zbMATH Keywords: Hermite-Fejér interpolation
MaRDI profile type: Publication
cites work: Q5590170
full work available at URL: https://doi.org/10.1007/bf01849500
OpenAlex ID: W2077795814
description: scientific article; zbMATH DE number 4026034