Convergence of Gaussian quadrature formulas
type: Item
instance of: scholarly article
title: Convergence of Gaussian quadrature formulas
zbMATH Open document ID: 0966.41019
author: Yingguang Shi
published in: Journal of Approximation Theory
publication date: 2001-08-06T00:00:00Z
review text: Classical Gaussian formulas are well known. They construct a polynomial interpolating in the zeros of a polynomial orthogonal with respect to a positive measure \(\alpha\) and the integral of this polynomial is a quadrature formula for \(\int f(x) d\alpha(x)\) with maximal polynomial degree of exactness. The generalized Gaussian formulas considered in this paper allow the interpolation points to coincide and a Hermite interpolating polynomial is used instead. The knots are generalized Chebyshev points in the sense that the the monic polynomial with these zeros (multiplicity imposed) has minimal \(L_1(d\alpha)\) norm in \([-1,1]\). Under appropriate conditions the convergence of these generalized Gaussian quadrature formulas is proved and error estimates are derived.
reviewed by: Adhemar Bultheel
Mathematics Subject Classification ID: 41A55
zbMATH DE Number: 1523566
zbMATH Keywords: Gaussian quadrature
zbMATH Keywords: convergence
zbMATH Keywords: rate of convergence
MaRDI profile type: Publication
full work available at URL: jath.2000.3470
OpenAlex ID: W1968563543
cites work: Generalized Gaussian quadrature formulas
cites work: A class of orthogonal polynomials
cites work: Q5596362
cites work: The generalized Markov-Stieltjes inequality for Birkhoff quadrature formulas
cites work: On Hermite interpolation
DOI: 10.1006/JATH.2000.3470
description: scientific article; zbMATH DE number 1523566