Classes of realizable Herglotz-Nevanlinna functions

Abstract

In this chapter we are going to introduce three distinct subclasses N0(R), N1(R), and N01(R) of the class of functions N(R) realizable as impedance functions of L-systems, that was studied in Chapter 6. We give complete proofs of direct and inverse realization theorems in each subclass. We show that each subclass is characterized by a different property of the state-space operator in the corresponding realizing L-system. Based on this partition of the class N(R), we introduce the corresponding structure of subclasses Ω0(R, J), Ω1(R, J), and Ω01(R, J) on the class Ω (R, J) consisting of functions realizable as transfer functions of L-systems. Multiplication and coupling theorems are then proved for each subclass of Ω (R, J).