A complex extremal problem of Chebyshev type
type: Item
instance of: scholarly article
title: A complex extremal problem of Chebyshev type
zbMATH Open document ID: 0923.30002
DOI: 10.1007/BF02791261
author: Peter Yuditskii
published in: Journal d'Analyse Mathématique
publication date: 1999-10-28T00:00:00Z
review text: Consider the class \({\mathcal P}_n\) of all complex polynomials of fixed degree \(n\) which are bounded by 1 on the interval \([-1,1]\). What can we say about \(\sup\{| p(z_0) |:p\in {\mathcal P}_n\}= :L(z_0)\)? For \(z_0\in \mathbb{R} \setminus]-1,1[\) it is a classical result that \(L(z_0)\) is given by the modulus of the \(n\)th degree Chebyshev polynomial. The case \(\mathfrak R\mathfrak z_0=0\) has been studied by \textit{R. Freund} and \textit{St. Ruscheweyh} [Numer. Math. 48, 525-542 (1986; Zbl 0611.65016)]. The main subject of this paper is to solve this extremal problem for the remaining points \(z_0\). An explicit expression in terms of elliptic functions is given.
reviewed by: Gerald Schmieder
Mathematics Subject Classification ID: 30C10
zbMATH DE Number: 1309904
zbMATH Keywords: Chebyshev methods
zbMATH Keywords: Pell's equation
zbMATH Keywords: extremal problem
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 1309904