  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 $\infty$, 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 $n_p$ 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 $n_p$,
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.
