A weak-star convergence result for orthogonal rational functions
type: Item
instance of: scholarly article
title: A weak-star convergence result for orthogonal rational functions
zbMATH Open document ID: 1079.42019
author: Adhemar Bultheel
author: Joris Van Deun
published in: Journal of Computational and Applied Mathematics
publication date: 2005-04-18T00:00:00Z
review text: Let \(\lambda\) be a finite positive Borel measure on \(I=[-1,1]\) satisfying \(\lambda'>0\) a.e. and \(B=\{\beta_1,\beta_2,\ldots\}\) be a sequence of poles such that \(B\subset{\mathbb R}\cup\{\infty\}\backslash I\) and \(| \beta_k| \geq \rho>1\) for all \(k.\) The space of rational functions with poles in \(\{\beta_1,\ldots,\beta_n\}\) is defined as \({\mathcal L}_n=\)span\(\{b_0,\ldots,b_n\},\) where \(b_0=1,b_k(x)=\prod_{k=1}^n\frac{x}{1-x/\beta_k}.\) Rational functions \(\{\varphi_0,\ldots,\varphi_n\}\) are the orthonormalization of the basis of \(\mathcal L_n\) with respect to \(\lambda.\) The main result: \[ \frac{1-x/\beta_n}{\sqrt{1-1/\beta_n^2}}\varphi_n^2(x)d\lambda(x)\longrightarrow\frac{1}{\pi}\frac{dx}{\sqrt{1-x^2}} \] in the sense of weak-star convergence of measures. Applications to calculations of quadrature error for formulas based on the orthogonal rational functions are discussed
Mathematics Subject Classification ID: 42C05
Mathematics Subject Classification ID: 65D30
zbMATH DE Number: 2157686
zbMATH Keywords: orthogonal rational functions
zbMATH Keywords: asymptotic properties
reviewed by: Alexei Lukashov
MaRDI profile type: Publication
full work available at URL: j.cam.2004.08.011
OpenAlex ID: W2089409279
cites work: Q4223056
cites work: A generalized orthonormal basis for linear dynamical systems
cites work: Strong and Weak Convergence of Rational Functions Orthogonal on the Circle
cites work: ON THE ASYMPTOTICS OF THE RATIO OF ORTHOGONAL POLYNOMIALS AND CONVERGENCE OF MULTIPOINT PADÉ APPROXIMANTS
cites work: Asymptotics of polynomials orthogonal with respect to varying measures
cites work: Rational basis functions for robust identification from frequency and time-domain measurements
cites work: A unifying construction of orthonormal bases for system identification
cites work: ON THE ASYMPTOTICS OF THE RATIO OF ORTHOGONAL POLYNOMIALS
cites work: Identification of rational approximate models in H/sup /spl infin// using generalized orthonormal basis
cites work: System identification with generalized orthonormal basis functions
cites work: The computation of orthogonal rational functions on an interval
cites work: Orthogonal rational functions for system identification: numerical aspects
cites work: State space representation for arbitrary orthogonal rational functions
cites work: System identification using Laguerre models
cites work: System identification using Kautz models
DOI: 10.1016/J.CAM.2004.08.011
description: scientific article; zbMATH DE number 2157686