The Jacobi matrices approach to Nevanlinna-Pick problems

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 $J^{-1/2}H{J}^{-1/2}$ 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 Padé approximants to a unique solution $\phi$ of the Nevanlinna-Pick problem converge to $\phi$ locally uniformly in $\dC\setminus\dR$. The proposed scheme extends the classical Jacobi matrix approach to moment problems and Padé approximation for $\bR_0$-functions.

24 pages; Section 5 is modifed; some typos are corrected