Orthogonal rational functions and tridiagonal matrices
Orthogonal rational functions and tridiagonal matrices
Adhemar Bultheel
Pablo González-Vera
Erik Hendriksen
Olav Njåstad
Abstract:
We study the recurrence relation for rational functions
whose poles are in a prescribed sequence of numbers that are real or infinite
and that are orthogonal with respect to an Hermitian positive linear functional.
We especially discuss the interplay between finite and infinite poles.
The recurrence relation will also be described in terms of a
tridiagonal matrix which is a generalization of the Jacobi matrix of the polynomial situation which corresponds to placing all the poles at infinity.
This matrix not only describes the recurrence relation, but it can be used
to give a determinant expression for the orthogonal rational functions
and it also allows for the formulation of a generalized eigenvalue problem
whose eigenvalues are the zeros of an orthogonal rational function.
These nodes can be used in rational Gauss-type quadrature formules and the
corresponding weights can be obtained from the first components of the corresponding eigenvectors.
Status:
Published: Presented at the Rome conference june 2001 on Orthogonal polynomials, special functions and applications (OPSFA01)
Full text in J. Computational and Applied Mathematics, vol 153 (2003) 89-97
BiBTeX entry:
@article{ArtBGHN01b,
author = "A. Bultheel and P. Gonz{\'a}lez-Vera and E. Hendriksen and O. Nj{\aa}stad",
title = "Orthogonal rational functions and tridiagonal matrices",
journal = "Journal of Computational and Applied Mathematics",
year = "2003",
volume = "153",
number = "1-2",
pages = "89-97",
url = "http://nalag.cs.kuleuven.be/papers/ade/tridiag/index.html",
DOI = "10.1016/S0377-0427(02)00602-7",
LIMO = "1128221",
ZBL = "1014.42017",
MR = "1985681",
}
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