Two results on polynomial interpolation in equally spaced points
type: Item
instance of: scholarly article
title: Two results on polynomial interpolation in equally spaced points
zbMATH Open document ID: 0736.41005
DOI: 10.1016/0021-9045(91)90090-W
author: J. A. C. Weideman
author: Lloyd N. Trefethen
published in: Journal of Approximation Theory
publication date: 1992-06-25T00:00:00Z
review text: The authors present two results on the behavior of polynomial interpolation in equally spaced points which, although not new, are, in the authors' opinion, generally unknown to approximation theorists. First, if \(p_ n(t)\) is the polynomial interpolant to \(e^{i\alpha x}\), \(\alpha\in\mathbb{R}\), in the points \(0,1,\dots,n\), then \(\| f-p_ n\|_{[0,n]}\to 0\) as \(n\to\infty\) if and only if \(|\alpha|\leq\pi/3\). Second, the Lebesgue constant \(\Lambda_ n\) (supremum norm of the \(n\)-the interpolation operator) satisfies \(\Lambda^{1/n}_ n\to 2\) as \(n\to\infty\). Both theorems are proved by the use of the Newton interpolation formula.
Mathematics Subject Classification ID: 41A05
Mathematics Subject Classification ID: 41A50
zbMATH DE Number: 12878
zbMATH Keywords: polynomial interpolation
zbMATH Keywords: Newton interpolation formula
reviewed by: Radu Precup
MaRDI profile type: Publication
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full work available at URL: 0021-9045(91)90090-w
OpenAlex ID: W2006770307
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description: scientific article; zbMATH DE number 12878