Extremal polynomials for weighted Markov inequalities
type: Item
instance of: scholarly article
title: Extremal polynomials for weighted Markov inequalities
zbMATH Open document ID: 0641.41014
DOI: 10.1016/0021-9045(87)90039-6
author: R. S. Rodriguez
author: Ram N. Mohapatra
author: P. O'Hara
published in: Journal of Approximation Theory
publication date: 1987-01-01T00:00:00Z
review text: Let \(\Pi^*_ n\) denote the class of all real monic polynomials of degree \(\leq n\) and let w be a positive weight function defined on (- \(\infty,\infty)\) with continuous derivative there. For each \(n=1,2,...\), define \(\lambda_ n=\sup \{\| wp'\| /\| wp\|:p\in \Pi^*_ n\}\) and \(\mu_ n=\sup \{\| (wp)'\| /\| wp\|:p\in \Pi^*_ n\}\) where the prime denotes differentiation and \(\| \cdot \|\) denotes the sup-norm on (-\(\infty,\infty)\). In this setting, alternation information for \(wp^*\) is developed where \(p^*\) is an extremal polynomial for \(\lambda_ n\) or \(\mu_ n\). In addition, for the special cases that \(w(x)=\exp (-x^ 2)\) and \(p^*\in \Pi^*_ n\) is an extremal for \(\mu_ n\), it is shown that \(wp^*\) is extremal.
Mathematics Subject Classification ID: 41A17
Mathematics Subject Classification ID: 41A10
zbMATH DE Number: 4044403
zbMATH Keywords: monic polynomials
zbMATH Keywords: weight function
zbMATH Keywords: extremal polynomial
MaRDI profile type: Publication
full work available at URL: 0021-9045(87)90039-6
OpenAlex ID: W2046933422
cites work: Q5543516
cites work: Q3848301
cites work: Q5625674
cites work: Extremal Problems for Polynomials with Exponential Weights
cites work: Q5343522
cites work: Inequalities for weighted polynomials
cites work: Some weighted polynomial inequalities
description: scientific article; zbMATH DE number 4044403