The Jacobi matrices approach to Nevanlinna-Pick problems

• Authors: Maxim Derevyagin
• Venue: Journal of Approximation Theory
• Year: 2011

A modification of the well-known step-by-step process for solving Nevanlinna-Pick problems in the class of R"0-functions gives rise to a linear pencil H-@lJ, 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^-^1^2HJ^-^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 Pade approximants to a unique solution @f of the Nevanlinna-Pick problem converge to @f locally uniformly in C@?R. The proposed scheme extends the classical Jacobi matrix approach to moment problems and Pade approximation for R"0-functions.