Orthogonal rational functions and quadrature on an interval
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.
scientific article; zbMATH DE number 1915259