Some quadrature formulae with nonstandard weights
type: Item
instance of: scholarly article
title: Some quadrature formulae with nonstandard weights
zbMATH Open document ID: 1210.41008
published in: Journal of Computational and Applied Mathematics
publication date: 2010-10-11T00:00:00Z
review text: The paper deals with the problem of computing integrals of the form \[ \int_0^\infty f(x)w(x)\,dx, \quad \int_0^\infty f(x)K(x,y)u(x)\,dx, \] where \(f\) is continuous. Besides, \(u(x)=x^\alpha e^{-x^\beta/2}/(1+x)^\lambda\), and \(w(x)=x^\alpha e^{-x^\beta}\), with \(\alpha>-1\), \(\beta>1/2\), \(\gamma > -1\), \( \lambda\geq 0\). When \(f\) has exponential growth at infinity, the authors propose the use of truncated quadrature rules to evaluate these integrals. The main results are summarized in two theorems. When evaluating the first integral, an upper estimate for the quadrature error is obtained given in terms of the Sobolev norm of \(f\). Such estimate holds true when it is applied to a Gauss quadrature formula conveniently truncated. The other result deals with the stability and the convergence of a quadrature rule, which is a modification of the previous one, when it is applied to the second integral. The construction of the latter formula is more complicated than the former because of the presence of the kernel \(K\). Hence the end of the article, there is an appendix where the authors discuss the calculation of coefficients for the special case \(\lambda=0\), \(\beta=1\). Some numerical tests are shown to illustrate the performance of the proposed method.
Mathematics Subject Classification ID: 41A55
Mathematics Subject Classification ID: 65D32
Mathematics Subject Classification ID: 41A10
zbMATH DE Number: 5798184
zbMATH Keywords: quadrature formulae
zbMATH Keywords: orthogonal polynomials
zbMATH Keywords: approximation by polynomials
zbMATH Keywords: Gauss-type quadrature formulae
author: Giuseppe Mastroianni
author: Donatella Occorsio
reviewed by: J. Illán
describes a project that uses: OrthogonalPolynomials
MaRDI profile type: Publication
full work available at URL: j.cam.2010.06.011
OpenAlex ID: W2030258152
cites work: Truncated Quadrature Rules Over $(0,\infty)$ and Nyström-Type Methods
cites work: Q3629290
cites work: Q4431906
cites work: \(L^p\)-convergence of Lagrange interpolation on the semiaxis
cites work: Q4801527
cites work: Gaussian rules on unbounded intervals
cites work: Numerical approximation of weakly singular integrals on the half line
cites work: Markov-Sonin Gaussian rule for singular functions
cites work: Q5714904
cites work: Half-Range Generalized Hermite Polynomials and the Related Gaussian Quadratures
cites work: Orthonormal polynomials with generalized Freud-type weights.
cites work: Q3621865
cites work: Q3567752
cites work: Q5607484
cites work: Q3838552
DOI: 10.1016/J.CAM.2010.06.011
description: scientific article; zbMATH DE number 5798184