The Jacobi matrices approach to Nevanlinna-Pick problems

TL;DR

This paper introduces a Jacobi matrix-based method for solving Nevanlinna-Pick problems, establishing conditions for unique solutions and demonstrating convergence of Padé approximants using operator techniques.

Contribution

It extends the classical Jacobi matrix approach to Nevanlinna-Pick problems, providing new criteria for solution uniqueness and convergence analysis.

Findings

J is a positive operator.

Unique solution iff J^{-1/2}HJ^{-1/2} is self-adjoint.

Padé approximants converge locally uniformly.

Abstract

A modification of the well-known step-by-step process for solving Nevanlinna-Pick problems in the class of $\bR_0$-functions gives rise to a linear pencil $H-\lambda J$, where $H$ and $J$ are Hermitian tridiagonal matrices. First, we show that $J$ is a positive operator. Then it is proved that the corresponding Nevanlinna-Pick problem has a unique solution iff the densely defined symmetric operator $J^{-1/2}HJ^{-1/2}$ is self-adjoint and some criteria for this operator to be self-adjoint are presented. Finally, by means of the operator technique, we obtain that multipoint diagonal Pad\'e approximants to a unique solution $\varphi$ of the Nevanlinna-Pick problem converge to $\varphi$ locally uniformly in $\dC\setminus\dR$. The proposed scheme extends the classical Jacobi matrix approach to moment problems and Pad\'e approximation for $\bR_0$-functions.