Approximation error for linear polynomial interpolation on \(n\)-simplices
type: Item
instance of: scholarly article
title: Approximation error for linear polynomial interpolation on \(n\)-simplices
zbMATH Open document ID: 0899.41002
DOI: 10.1007/BF02305420
author: Yu. A. Kilizhekov
published in: Mathematical Notes
publication date: 1998-11-11T00:00:00Z
review text: Let \(\Delta_n \subset \mathbb{R}^n\) be an \(n\)-simplex and let \(W^2_nM\) be the class of functions \(f:\Delta_n \to\mathbb{R}\) having continuous second derivatives not exceeding \(M> 0\) in absolute value. The author gives an (exact) pointwise bound for the error \(| f(x)-P_{1,n} (f,x) |\), where \(P_{1,n} (f,x)\) denotes the linear polynomial interpolating \(f\in W^2_nM\) at the vertices of the simplex \(\Delta_n\). More precisely, he shows that \[ \sup \biggl\{ \bigl| f(x)-P_{1,n} (f,x)\bigr|: f\in W^2_n M\biggr\} =\textstyle {1\over 2} M\bigl( R^2_n-d^2_n (x)\bigr), \] where \(R_n\) is the radius of the sphere circumscribing \(\Delta_n\) and \(d_n(x)\) denotes the distance between \(x\) and the center of this sphere. As a corollary he obtains a uniform bound in terms of the Chebyshev radius of the simplex \(\Delta_n\).
Mathematics Subject Classification ID: 41A05
zbMATH DE Number: 1158925
zbMATH Keywords: Chebyshev radius
reviewed by: Jürgen Müller
MaRDI profile type: Publication
cites work: The Hausdorff metric and Cebysev centres
full work available at URL: https://doi.org/10.1007/bf02305420
OpenAlex ID: W1997734280
description: scientific article; zbMATH DE number 1158925