Quadrature formulas for rational functions
type: Item
instance of: scholarly article
title: Quadrature formulas for rational functions
zbMATH Open document ID: 0956.41017
publication date: 2000-07-24T00:00:00Z
full work available at URL: https://eudml.org/doc/120827
full work available at URL: pp39-52.html
review text: In this interesting paper, the authors study convergence of quadrature formulas that exactly integrate rational functions with given denominators. Let us suppose that we are given monic polynomials \(\pi_n\), \(n\geq 1\), whose zeros lie at least a distance \(\delta\) away from \([-1,1]\) for \(n\geq 1\). Of course \(\delta >0\) should be independent of \(n\). Let \(\omega: [-1,1]\to [0,\infty)\) be a non-negative function, and consider the following quadrature problem: find weights \(\{\lambda_{j,n}\}^n_{j=1}\) and points \(\{x_{jn}\}^n_{j =1}\) such that \[ \int^1_{-1} Rw=\sum^n_{j=1} \lambda_{j,n} R(x_{j,n}) \] for every rational function \(R\) of the form \(R={P \over|\pi_n|^2}\), where \(P\) is a polynomial of degree \(\leq 2n-1\). The authors relate such quadrature formulae to Gauss quadratures (with varying weights) and also present the relationship to multipoint Padé-type approximation. They give conditions for positivity of the weights \(\lambda_{j,n}\) and show how that positivity leads to convergence of the rules as \(n\to \infty\), for bounded Riemann integrable \(f\). More general convergence aspects are explored.
Mathematics Subject Classification ID: 41A55
Mathematics Subject Classification ID: 41A21
Mathematics Subject Classification ID: 65D30
Mathematics Subject Classification ID: 42C05
Mathematics Subject Classification ID: 30E10
zbMATH DE Number: 1476910
zbMATH Keywords: numerical quadrature
zbMATH Keywords: rational functions
zbMATH Keywords: multipoint Padé-type approximation
author: Francisco Cala Rodríguez
author: Pablo González-Vera
author: M. Jiménez Paiz
reviewed by: Doron S. Lubinsky
MaRDI profile type: Publication
published in: ETNA - Electronic Transactions on Numerical Analysis
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description: scientific article; zbMATH DE number 1476910