Pointwise error estimates for interpolation
type: Item
instance of: scholarly article
title: Pointwise error estimates for interpolation
zbMATH Open document ID: 0653.41008
DOI: 10.1016/0377-0427(87)90207-X
author: Thomas E. jun. Price
published in: Journal of Computational and Applied Mathematics
publication date: 1987-01-01T00:00:00Z
review text: Let f have an uniformly and absolutely convergent expansion of the form \(f(x)=d_ 0T_ 0(x)+\sum^{\infty}_{k=1}d_ kT_ k(x),\)-1\(\leq x\leq 1\), where \(T_ k(x)\) denotes the kth Chebyshev polynomial of the first kind. Let \(L_{n-1}(x;f)\) denote the unique Lagrange polynomial interpolant of f in the roots of \(T_ n(x)\). For the error \(R_{n- 1}(x;f):=f(x)-L_{n-1}(x;f)\) the author proves that \[ R_{n- 1}(x;f)=2T_ n(x)\sum^{\infty}_{m=1}(-1)^{m-1}[D_{(2m-1)n}T_ 0(x)+\sum^{\infty}_{k=1}d_{(2m-1)n+k}T_ k(x)], \] \[ | R_{n-1}(x;f)| \leq 2| T_ n(x)| (D_ 0+\sum^{\infty}_{k=1}D_ k), \] where \(D_ k:=| \sum^{\infty}_{m=1}(-1)^{m-1}d_{(2m-1)n+k}|\), \(k=0,1,... \). The author obtains a similar error representation and bound for polynomial interpolants in the extrema of \(T_ k(x)\).
Mathematics Subject Classification ID: 41A05
Mathematics Subject Classification ID: 41A25
zbMATH DE Number: 4065577
zbMATH Keywords: error bound
zbMATH Keywords: Chebyshev polynomial
zbMATH Keywords: Lagrange polynomial
zbMATH Keywords: error representation
MaRDI profile type: Publication
cites work: Ableitungsfreie Abschätzungen bei trigonometrischer Interpolation und Konjugierten-Bestimmung
cites work: Pointwise error estimates for trigonometric interpolation and conjugation
cites work: Q5588103
cites work: Some new characterizations of the Chebyshev polynomials
description: scientific article; zbMATH DE number 4065577