On best approximation of the monomials on the unit ball of \(\mathbb{R}^ r\)
type: Item
instance of: scholarly article
title: On best approximation of the monomials on the unit ball of \(\mathbb{R}^ r\)
zbMATH Open document ID: 0890.41016
author: Ulrike Maier
published in: Journal of Approximation Theory
publication date: 1998-07-08T00:00:00Z
review text: The Kergin interpolant \({\mathcal K} (f)\) of a function \(f\in C^\mu (\mathbb{R}^r)\) with respect to \(\mu+1\) distinguished nodes is an unique polynomial in \(r\) variables of degree at most \(\mu\). [\textit{P. Kergin}, J. Approximation Theory 29, 278-293 (1980; Zbl 0492.41008)]. If \(f\) is a monomial \(x^m\), with \(|m|=\mu+1\), where \(m=(m_1, \dots, m_r) \in N^r\), then \({\mathcal K} (x^m)\) can be explicitly given, although for a general function \(f\in C^\mu (R^r)\) this fact is not possible. Bos and Liang have separately proved in the case \(r\geq 2\), that for a particular choice of the nodes, the Kergin interpolant \({\mathcal K} (x^n)\) is a best approximation of \(x^m\) in the uniform norm on the unit disk of \(\mathbb{R}^r\). In the present paper, sufficient conditions for \(\mu+1\) nodes are given in order that, in the case \(r=2\), the corresponding interpolant \({\mathcal K} (x^m)\) is a best approximation of \(x^m\) on the unit ball in the infinity norm and also, in the general case \(r\geq 3\), the corresponding interpolant \({\mathcal K} (x^m)\) is a best approximation of \(x^m\) on \(S^{r-1}\) with respect to the \(L_2\) norm. The given conditions make possible to get all the systems of nodes with the desired property.
reviewed by: Radu Paltanea
Mathematics Subject Classification ID: 41A50
Mathematics Subject Classification ID: 41A05
Mathematics Subject Classification ID: 41A63
zbMATH DE Number: 1121221
zbMATH Keywords: best approximation
zbMATH Keywords: Kergin interpolant
MaRDI profile type: Publication
full work available at URL: jath.1996.3110
OpenAlex ID: W2065974109
cites work: On Kergin interpolation in the disk
cites work: A natural interpolation of \(C^ k \)functions
cites work: Introduction to algebra. Transl. from the Russian by Neal Koblitz
cites work: Q3779158
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cites work: On multivariate polynomials of least deviation from zero on the unit ball
cites work: Q3854090
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DOI: 10.1006/JATH.1996.3110
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description: scientific article; zbMATH DE number 1121221