On the Nevanlinna-Pick interpolation problem: analysis of the McMillan degree of the solutions

In this paper we investigate some aspect of the Nevanlinna-Pick and Schur interpolation problem formulated for Schur-functions considered on the right-half plane of C. We consider the well established parametrization of the solution Q = TΘ(S) := (SΘ12 + Θ22)(SΘ11 + Θ21) (see e.g. [2],[6]), where the J-inner function Θ is completely determined by the interpolation data and S is an arbitrary Schur function. We then compare the relations between the realizations of Q and S induced by Θ. We show in particular that S generates a solution with a low McMillan degree if and only if S satisfies some interpolation conditions formulated on the left-half plane of C. This analysis can be considered to be partially complementary to the results of A. Lindquist, C. Byrnes et al. on Carathéodory functions, [3], [5], [4].