The Jacobi matrices approach to Nevanlinna–Pick problems

A modification of the well-known step-by-step process for solving Nevanlinna–Pick problems in the class of R0-functions gives rise to a linear pencil H−λJH−λJ, where HH and JJ are Hermitian tridiagonal matrices. First, we show that JJ 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−12HJ−12 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 φφ of the Nevanlinna–Pick problem converge to φφ locally uniformly in C∖RC∖R. The proposed scheme extends the classical Jacobi matrix approach to moment problems and Padé approximation for R0-functions.