Orthogonal rational functions and quadrature on an interval
type: Item
instance of: scholarly article
title: Orthogonal rational functions and quadrature on an interval
zbMATH Open document ID: 1016.41015
DOI: 10.1016/S0377-0427(02)00598-8
author: Adhemar Bultheel
author: Joris Van Deun
published in: Journal of Computational and Applied Mathematics
publication date: 2003-05-22T00:00:00Z
review text: This paper deals with quadrature formulas which are exact in certain spaces of rational functions, with poles in a prescribed set of complex numbers, orthogonal on the real line. The case of the formulas, similar to the Gauss ones for orthogonal polynomials, which are exact for rational functions with complex poles in the lower half-plane has been studied by \textit{A. Bultheel, P. González-Vera, E. Hendriksen} and \textit{O. Njastad} [in ``Orthogonal rational functions'' (1999; Zbl 0923.42017)]. This result is generalized to the case of orthonormal functions \(\phi _n\) with arbitrary complex poles. The orthogonality on a finite interval is studied. When the poles are real the functions \(\phi _n\) satisfy a three-term recurrence relation and their zeros are the nodes in the quadrature formulas. The zeros satisfy too a quadratic eigenvalue problem.
Mathematics Subject Classification ID: 41A55
Mathematics Subject Classification ID: 42C05
zbMATH DE Number: 1915259
zbMATH Keywords: orthogonal rational functions
zbMATH Keywords: quadrature
zbMATH Keywords: quadratic eigenvalue problem
reviewed by: Luigi Gatteschi
MaRDI profile type: Publication
cites work: Orthogonal rational functions and tridiagonal matrices
cites work: Numerical analysis of a quadratic matrix equation
cites work: The quadratic eigenvalue problem
full work available at URL: s0377-0427(02)00598-8
OpenAlex ID: W1984907140
description: scientific article; zbMATH DE number 1915259