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)-1(SΘ11+Θ21) (see e.g. [J.A. Ball, I. Gohberg, L. Rodman, Realization and interpolation of rational matrix functions, Operator Theory: Adv. Appl. 33 (1988) 1–72; H. Dym, J-Contractive matrix functions, reproducing kernel Hilbert spaces and interpolation, CBMS Regional Conference Series in Mathematics, No. 71, Amer. Math. Soc., Providence, RI, 1989]), 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, [C. Byrnes, A. Lindquist, On the partial stochastic realization problem, IEEE Trans. Automatic Control 42 (1997) 1049–1070; C. Byrnes, A. Lindquist, S.V. Gusev, A.S. Mateev, A complete parametrization of all positive rational extension of a covariance sequence, IEEE Trans. Automatic Control 40 (1995) 1841–1857; C. Byrnes, A. Lindquist, S.V. Gusev, A convex optimization approach to the rational covariance extension problem, TRIA/MAT, 1997].