An operator approach to multipoint Pade approximations

TL;DR

This paper introduces an operator-based framework for constructing multipoint Pade approximations, providing new recurrence relations and convergence results for Nevanlinna functions.

Contribution

It presents a novel operator approach and a modified step-by-step process for multipoint Pade approximations, linking recurrence relations with spectral theory.

Findings

Established a convergence theorem for multipoint Pade approximants to Nevanlinna functions

Derived three-term recurrence relations involving spectral parameters

Reformulated relations using Jacobi matrices

Abstract

First, an abstract scheme of constructing biorthogonal rational systems related to some interpolation problems is proposed. We also present a modification of the famous step-by-step process of solving the Nevanlinna-Pick problems for Nevanlinna functions. The process in question gives rise to three-term recurrence relations with coefficients depending on the spectral parameter. These relations can be rewritten in the matrix form by means of two Jacobi matrices. As a result, a convergence theorem for multipoint Pad\'e approximants to Nevanlinna functions is proved.