On an extremal problem of Fejér
type: Item
instance of: scholarly article
title: On an extremal problem of Fejér
zbMATH Open document ID: 0663.41007
DOI: 10.1016/0021-9045(88)90065-2
author: Tai Shing Lau
author: William J. Studden
published in: Journal of Approximation Theory
publication date: 1988-01-01T00:00:00Z
review text: Let \(-\infty \leq a<b\leq \infty\) and let \({\mathcal P}_{n+1}\) denote the class of \((n+1)\)-triples \((x_ 1,x_ 2,...,x_{n+1})\) with \(a\leq x_ 1<...<x_{n+1}\leq b\) \((x_ 1\) and \(x_{n+1}\) are finite). Let w(x) be a non-negative function on [a,b] and let \(\ell_ i(x)\) denote the fundamental polynomials of Lagrange on the \((n+1)\)-triples. The authors consider the extremal problem of finding the value \[ N=\inf_{P_{n+1}}\sup_{a\leq x\leq b}\{w(x)\sum^{n+1}_{j=1}\ell^ 2_ j(x)/w(x_ j)\}=\inf_{P_{n+1}}\tilde N(x_ 1,...,x_{n+1}) \] when w(x) is a non-negative function on [a,b]. They also determine the set of points \(\{x_ j\}_ 1^{n+1}\) which minimizes \(\tilde N\) for a fixed w(x). This problem when \(w(x)=1\) goes back to Fejer. For other relevant results see the book on Chebyshev systems by S. Karlin and W. J. Studden and a paper by \textit{C. Balazs} [Acta Math. Acad. Sci. Hungar. 34, 307-315 (1979; Zbl 0434.41004)].
Mathematics Subject Classification ID: 41A05
Mathematics Subject Classification ID: 41A10
zbMATH DE Number: 4084243
zbMATH Keywords: extremal problem
zbMATH Keywords: Chebyshev systems
reviewed by: A. Sharma
MaRDI profile type: Publication
full work available at URL: 0021-9045(88)90065-2
OpenAlex ID: W1985092179
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description: scientific article; zbMATH DE number 4084243