On some problems of P. Turán concerning \(L_m\) extremal polynomials and quadrature formulas
type: Item
instance of: scholarly article
title: On some problems of P. Turán concerning \(L_m\) extremal polynomials and quadrature formulas
zbMATH Open document ID: 0937.41014
author: Yingguang Shi
published in: Journal of Approximation Theory
publication date: 2000-06-07T00:00:00Z
review text: Let be \(w(x)\) a weight function on \([-1,1]\). It is known that for each \(n\in\mathbb{N}\) there exists a unique extremal polynomial \(P_n(w,m;x)= x^n+\dots\) for which \[ \int^1_{-1}P_n(w,m;x)^m w(x)dx= \min_{P=x^n+ \cdots} \int^1_{-1} P(x)^mw(x)dx. \] The zeros of the polynomial \(P_n(w,m;x)\) are real, distinct and located in the interior of the interval \([-1,1]\). This paper deals with the explicit form of the extremal polynomials with respect to the weights \((1-x)^{-1/2} (1+x)^{m-1/2}\) and \((1-x)^{m-1/2} (1+x)^{-1/2}\) for even \(m\). The author also gives an explicit representation for the Cotes numbers of the corresponding Turán quadrature formulas and their asymptotic behavior.
Mathematics Subject Classification ID: 41A55
zbMATH DE Number: 1375221
zbMATH Keywords: extremal polynomial
zbMATH Keywords: Cotes numbers
zbMATH Keywords: Turán quadrature
reviewed by: Luigi Gatteschi
MaRDI profile type: Publication
full work available at URL: jath.1999.3342
OpenAlex ID: W1985271888
cites work: On a quadrature formula of Micchelli and Rivlin
cites work: Generalized Gaussian quadrature formulas
cites work: Q3671491
cites work: Turán Formulae and Highest Precision Quadrature Rules for Chebyshev Coefficients
cites work: A solution of problem 26 of P. Turán
cites work: An analogue of problem 26 of P. Turán
cites work: On some open problems of approximation theory
DOI: 10.1006/JATH.1999.3342
description: scientific article; zbMATH DE number 1375221