A Nevanlinna function is a function which is analytic in the open upper half plane and has a non-negative imaginary part there. In this paper we study a fractional linear transformation for a Nevanlinna function n with a suitable asymptotic expansion at ∞, that is an analogue of the Schur transformation for contractive analytic functions in the unit disc. Applying the transformation p times we find a Nevanlinna function np which is a fractional linear transformation of the given function n. The main results concern the effect of this transformation to the realizations of n and np, by which we mean their representations through resolvents of self-adjoint operators in Hilbert space. Our tools are block operator matrix representations, u--resolvent matrices, and reproducing kernel Hilbert spaces.
D. Alpay, A. Dijksma and H. Langer. The Schur transformation for Nevanlinna functions: operator representations, resolvent matrices, and orthogonal polynomials. Operator Theory: Advances and Applications, vol. 190 (2009), 27-63.