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Abstract: In the strong or two-point Stieltjes moment problem, one has to find a positive measure on for which infinitely many moments are prescribed at the origin and at infinity. Here we consider a multipoint version in which the origin and the point at infinity are replaced by sequences of points that may or may not coincide. In the indeterminate case, two natural solutions and exist that can be constructed by a limiting process of approximating quadrature formulas. The supports of these natural solutions are disjoint (with possible exception of the origin). The support points are accumulation points of sequences of zeros of even and odd indexed orthogonal rational functions. These functions are recursively computed and appear as denominators in approximants of continued fractions. They replace the orthogonal Laurent polynomials that appear in the two-point case. In this paper we consider the properties of these natural solutions and analyze the precise behavior of which zero sequences converge to which support points. [Copyright &y& Elsevier]

Abstract: We consider rational moment problems on the real line with their associated orthogonal rational functions. There exists a Nevanlinna-type parameterization relating to the problem, with associated Nevanlinna matrices of functions having singularities in the closure of the set of poles of the rational functions belonging to the problem. We prove results related to the growth at the singularities of the functions in a Nevanlinna matrix, and in particular provide bounds on the growth analogous to the corresponding result in the classical polynomial case, when the number of singularities is finite. [ABSTRACT FROM AUTHOR]

Abstract: The main objective is to generalize previous results obtained for orthogonal Laurent polynomials and their application in the context of Stieltjes moment problems to the multipoint case. The measure of orthogonality is supposed to have support on while the orthogonal rational functions will have poles that are assumed to be “in the neighborhood of 0 and ”. In this way orthogonal Laurent polynomials will be a special case obtained when all the poles are at 0 and . We shall introduce the restrictions on the measure and the locations of the poles gradually and derive recurrence relations, Christoffel–Darboux relations, and the solution of the rational Stieltjes moment problem under appropriate conditions. [Copyright &y& Elsevier]

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. [Copyright &y& Elsevier]