On the minimum moduli of normalized polynomials with two prescribed values
type: Item
instance of: scholarly article
title: On the minimum moduli of normalized polynomials with two prescribed values
zbMATH Open document ID: 0602.30008
DOI: 10.1007/BF01893437
published in: Constructive Approximation
publication date: 1986-01-01T00:00:00Z
review text: Let \(P_ n(\mu)\) denote the set of complex polynomials of degree at most n which satisfy \(p(0)=1\) and \(p(1)=\mu\). Let \(S_ n(\mu)\) denote the supremum over P(\(\mu)\) of the quantity min\(\{| p(z)|:| z| \leq 1\}\). The authors determine the exact value of \(S_ n(\mu)\) for all \(n\geq 1\) and all complex \(\mu\). For \(\mu >0\), Corollary 1 states that \(S_ n(\mu)=\mu\) if \(0<\mu \leq 1\); \(S_ n(\mu)=0\) if \(\mu \geq 2^ n\); and if \(1<\mu <2^ n\) then \(S_ n(\mu)=\sigma\), the unique solution with \(0<\sigma <1\) of \(\sigma =T_{n+1}(\sigma^{-1/(n+1)})\), where \(T_{n+1}\) is the standard Chebyshev polynomial. For \(1<\mu <2^ n\), there is a unique extremal polynomial and it has positive coefficients. This latter result, for \(\mu =2\), has an application to the analysis of global descent methods for polynomials. The proof of the main theorem is not easy but is easy to follow since the paper is well-written and almost self-contained.
Mathematics Subject Classification ID: 30C10
Mathematics Subject Classification ID: 30C70
Mathematics Subject Classification ID: 65H05
zbMATH DE Number: 3971123
zbMATH Keywords: Chebyshev polynomial
zbMATH Keywords: extremal polynomial
author: Stephan Ruscheweyh
author: Richard S. Varga
reviewed by: David W. Boyd
MaRDI profile type: Publication
cites work: Q4072022
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cites work: A Unified Approach to Certain Problems of Approximation and Minimization
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full work available at URL: https://doi.org/10.1007/bf01893437
OpenAlex ID: W1965338029
description: scientific article; zbMATH DE number 3971123