The Jacobi matrices approach to Nevanlinna-Pick problems

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A modification of the well-known step-by-step process for solving Nevanlinna-Pick problems in the class of ₀-functions gives rise to a linear pencil H-λJ, 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 Padé approximants to a unique solution φ of the Nevanlinna-Pick problem converge to φ locally uniformly in. The proposed scheme extends the classical Jacobi matrix approach to moment problems and Padé approximation for ₀-functions.