On oscillating polynomials
type: Item
instance of: scholarly article
title: On oscillating polynomials
zbMATH Open document ID: 1206.33013
author: Nikola Naidenov
author: Borislav D. Bojanov
published in: Journal of Approximation Theory
publication date: 2010-11-18T00:00:00Z
review text: An algebraic polynomial that has all its zeros in a finite interval \([a,b]\) is called oscillating on \([a,b]\). In the present paper the authors prove the following Theorem. Let \(P\) be an oscillating on \([-1,1]\) polynomial of degree \(n\) with symmetric zeros, and let \(P\) attain its maximal absolute value at the endpoints. Then \(P'\) attains its maximal absolute value at the endpoints too. In fact, the authors prove more than that: for every oscillating polynomial \(P\) on \([-1,1]\) such that \(|P(-1)|=\|P\|\), each local extremum of \(P'\) from the first half is smaller in absolute values than \(|P'(-1)|\). As an application of the method developed in the paper the authors show that the classic Zolotarev polynomials have maximal derivative at one of the endpoints.
Mathematics Subject Classification ID: 33C52
Mathematics Subject Classification ID: 33C45
Mathematics Subject Classification ID: 41A17
Mathematics Subject Classification ID: 41A55
zbMATH DE Number: 5817259
zbMATH Keywords: Markov's inequality
zbMATH Keywords: maximal derivative
zbMATH Keywords: critical points
zbMATH Keywords: Tikhonov polynomials
zbMATH Keywords: Zolotarev polynomials
zbMATH Keywords: orthogonal polynomials
reviewed by: Leonid Golinskii
MaRDI profile type: Publication
full work available at URL: j.jat.2010.04.009
OpenAlex ID: W1989051326
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DOI: 10.1016/J.JAT.2010.04.009
description: scientific article; zbMATH DE number 5817259