Gaussian versus optimal integration of analytic functions
type: Item
instance of: scholarly article
title: Gaussian versus optimal integration of analytic functions
zbMATH Open document ID: 0893.41019
published in: Constructive Approximation
publication date: 1998-08-03T00:00:00Z
review text: The author's abstract states: We consider error estimates for optimal and Gaussian quadrature formulas if the integrand is analytic and bounded in a certain complex region. First, a simple technique for the derivation of lower bounds for the optimal error constants is presented. This method is applied to Szegő-type weight functions and ellipses as regions of analyticity. In this situation, the error constants for the Gaussian formulas are close to the obtained lower bounds, which proves the quality of the Gaussian formulas and also the lower bounds. In the sequel, different regions of analyticity are investigated. It turns out that almost exclusively for ellipses, the Gaussian formulas are near-optimal. For classes of simply connected regions of analyticity, which are additionally symmetric about the real axis, the asymptotic of the worst ratio between the error constants of the Gaussian formulas and the optimal error constants is calculated. As a by-product, we prove explicit lower bounds for the Christoffel function for the constant weight function and for arguments outside the interval of integration.
Mathematics Subject Classification ID: 41A55
Mathematics Subject Classification ID: 65D30
Mathematics Subject Classification ID: 42C05
zbMATH DE Number: 1143071
zbMATH Keywords: Szegö weight
zbMATH Keywords: Gaussian quadrature
zbMATH Keywords: optimal error constants
zbMATH Keywords: Christoffel function
author: Knut Petras
reviewed by: Aldric L. Brown
MaRDI profile type: Publication
full work available at URL: s003659900072
OpenAlex ID: W2028953969
DOI: 10.1007/S003659900072
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description: scientific article; zbMATH DE number 1143071