Orthogonal rational functions and tridiagonal matrices

摘要

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 formulas and the corresponding weights can be obtained from the first components of the corresponding eigenvectors.