Some notes on the zeros of power orthogonal polynomials
type: Item
instance of: scholarly article
title: Some notes on the zeros of power orthogonal polynomials
zbMATH Open document ID: 1156.42006
author: Yingguang Shi
published in: Journal of Approximation Theory
publication date: 2008-12-02T00:00:00Z
review text: One section of the paper is devoted to estimation of the largest zeros of power orthogonal polynomials. Here \(n\)th power orthogonal polynomial with respect to \(d\mu\) is a polynomial \(\omega_n(d\mu,{\mathbf m};x)=(x-x_{1n})\ldots(x-x_{nn})\) such that \[ \Phi_n(x_{1n},\ldots,x_{nn})=\inf_{{\mathbf y}\in\bar Y}\Phi_n(y_1,\ldots,y_n), \] where \(\Phi_n(y_1,\ldots,y_n)=\int_I\Omega_n({\mathbf y};x)d\mu(x),\) \[ \Omega_n({\mathbf y};x)=\prod_{k=1}^n|x-y_k|^{m_k},\quad Y=\{{\mathbf y}=(y_1,\ldots,y_n):a<y_1<y_2<\ldots<y_n<b\}, \] \({\mathbf m}=(m_1,\ldots,m_n), I=(a,b),-\infty\leq a<b\leq\infty.\) Another section contains results about usual orthogonal polynomials including inequalities involving the largext zeros and the Turán inequality.
Mathematics Subject Classification ID: 42C05
zbMATH DE Number: 5376867
zbMATH Keywords: power orthogonal polynomials
zbMATH Keywords: Turán inequality
zbMATH Keywords: extreme zeros of orthogonal polynomials
reviewed by: Alexei Lukashov
MaRDI profile type: Publication
full work available at URL: j.jat.2007.08.003
OpenAlex ID: W1986750327
cites work: Q4054149
cites work: On estimations of the greatest zeroes of orthogonal polynomials
cites work: Bounds for orthogonal polynomials for exponential weights
cites work: Strong and Weak Convergence of Orthogonal Polynomials
cites work: Bounds and inequalities for arbitrary orthogonal polynomials on finite intervals
cites work: Q5309312
cites work: Q3264312
DOI: 10.1016/J.JAT.2007.08.003
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description: scientific article; zbMATH DE number 5376867