In this paper we extend and solve in the class of functions RSI mentioned in the title, a number of problems originally set for the class RS of rational functions contractive in the open right-half plane, and unitary on the imaginary line with respect to some preassigned self-adjoint matrix. The problems we consider include the Schur algorithm, the partial realization problem and the Nevanlinna-Pick interpolation problem. The arguments rely on the one-to-one correspondence between elements in a given subclass of RSI and elements in RS. Another important tool in the arguments is a new result pertaining to the classical tangential Schur algorithm.
D. Alpay, A. Melnikov, and V. Vinnikov. On the class SI of J-contractive functions intertwining solutions of linear differential equations. Integral Equations and Operator Theory, vol. 74 (2012), no. 3, pp. 313-344.