The Bernstein constant and polynomial interpolation at the Chebyshev nodes.
type: Item
instance of: scholarly article
title: The Bernstein constant and polynomial interpolation at the Chebyshev nodes.
zbMATH Open document ID: 1035.41015
author: Michael I. Ganzburg
published in: Journal of Approximation Theory
publication date: 2003-04-02T00:00:00Z
review text: By giving explicit upper bounds, the author shows that the Bernstein constants \[ B_{\lambda,p} := \lim_{n\to\infty} n^{\lambda+1/p} \inf_{c_k} \Biggl\| | x| ^\lambda - \sum^n_{k=0} c_k x^k\Biggl\|_{L_p[-1,1]} \] are finite for all \(\lambda > 0\) and \(p\in (1/3,\infty)\). For \(p = 1\), the upper bounds turn out to be sharp. The main result follows from asymptotic relations for the error of approximation of \(| x| ^\lambda\) in \(L_p [-1,1]\) by polynomials of even degree given as the sum of the Lagrange interpolation polynomial to \(| x| ^\lambda\) at the Chebyshev nodes of 1st and 2nd kind and certain scalar multiples of the Chebyshev polynomials of 1st and 2nd kind (with scalars depending on \(n\) and \(\lambda\)). Such asymptotic relations are also given for the case \(p=\infty\), that is, for uniform approximation.
Mathematics Subject Classification ID: 41A44
Mathematics Subject Classification ID: 41A05
Mathematics Subject Classification ID: 41A10
zbMATH DE Number: 1891513
zbMATH Keywords: Lagrange interpolation
zbMATH Keywords: Bernstein constant
zbMATH Keywords: Chebyshev nodes
reviewed by: Jürgen Müller
MaRDI profile type: Publication
OpenAlex ID: W2004859972
cites work: On the divergence of Lagrange interpolation to \(| x|\)
cites work: On Lagrange interpolation with equidistant nodes
cites work: Limit theorems for the best polynomial approximations in the \(L_ \infty\) metric
cites work: Q4486636
cites work: Limit theorems and best constants in approximation theory
cites work: Limit theorems for the best polynomial approximations in the L? metric
cites work: Local convergence of Lagrange interpolation associated with equidistant nodes
cites work: Q5783394
cites work: On the approximation of certain functions by interpolating polynomials
cites work: On Lagrange interpolation with equally spaced nodes
cites work: The divergence of Lagrange interpolation for \(|x|^\alpha\) at equidistant nodes
cites work: Approximation constants in equidistant Lagrange interpolation
cites work: On Lagrange interpolatory parabolas to \(|x|^{\alpha}\) at equally spaced nodes
cites work: Q5727762
cites work: On the Bernstein conjecture in approximation theory
cites work: Q5762274
DOI: 10.1006/JATH.2002.3729
Recommended article: Asymptotics of polynomial interpolation and the Bernstein constants
Recommended article: Q4368080
Recommended article: A note on polynomial interpolation at the Chebyshev extrema nodes
Recommended article: Generalization of Polynomial Interpolation at Chebyshev Nodes
Recommended article: On the asymptotics of polynomial interpolation to \(|x|^{\alpha}\) at the Chebyshev nodes
Recommended article: Bernshtein's interpolational polynomials
Recommended article: Q4897327
Recommended article: Approximation by interpolation: the Chebyshev nodes
Recommended article: On the Bernstein constants of polynomial approximation
Recommended article: On the filtered polynomial interpolation at Chebyshev nodes
description: scientific article; zbMATH DE number 1891513