A remark to the extended Hermite-Fejér type interpolation of higher order
type: Item
instance of: scholarly article
title: A remark to the extended Hermite-Fejér type interpolation of higher order
zbMATH Open document ID: 0607.41003
DOI: 10.1007/BF01848102
author: D. L. Berman
published in: Periodica Mathematica Hungarica
publication date: 1987-01-01T00:00:00Z
review text: For \(f\in C[-1,1]\) and any natural n we consider the algebraic polynomials \(\delta _{n,r}(f,x)\), \(r=0,1,2,3\). \(\delta _{u,r}(f,x)\) is the polynomial of degree \(4n+2r+1\) which is uniquely determined by the conditions: \(\delta _{n,r}(f,x_{kn})=f(x_{kn}),\) \(S^{(j)}_{n,r}(f,x_{kn})=0,\) \(j=1,2,3\), \(k=1,2,...,n\). \(\delta _{n,r}(f,\pm 1)=f(\pm 1)\), \(S^{(i)}_{n,r}(f,\pm 1)=0\), \(i=1,...,r\), \(x_{kn}=\cos (2k-1)\pi /2n.\) The author states the theorem: the interpolation process \(\{\delta _{n,r}(f,x)\}^{\infty}_{n=1}\), \((r=1,2,3)\), \(n=1,2,...\), constructed with the nodes \(\{\pm 1\}\cup \{x_{kn}\}^ n_{k=1}\), \(n=1,2,..\). for a function f converges to f uniformly in [-1,1] provided \(| f^{(r+1)}(x)| \leq C_ 1,\) \(| x| \leq 1\), and \(f'(\pm 1)=f''(\pm 1)=f^{(r)}(\pm 1)=0\) \((C_ 1\) is an absolute constant).
Mathematics Subject Classification ID: 41A05
Mathematics Subject Classification ID: 41A10
zbMATH DE Number: 3981731
zbMATH Keywords: algebraic polynomials
zbMATH Keywords: interpolation process
MaRDI profile type: Publication
cites work: Q5509106
cites work: Q5590170
cites work: On Berman's phenomenon in interpolation theory
cites work: Q4178010
cites work: Quasi-Hermite-Fejer type interpolation of higher order
full work available at URL: https://doi.org/10.1007/bf01848102
OpenAlex ID: W2363755828
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description: scientific article; zbMATH DE number 3981731