Convex cones of generalized positive rational functions and the Nevanlinna–Pick interpolation

Abstract

Scalar rational functions with a non-negative real part on the right half plane, called positive, are classical in the study of electrical networks, dissipative systems, Nevanlinna–Pick interpolation and other areas. We here study generalized positive functions, i.e. with a non-negative real part on the imaginary axis. These functions form a convex invertible cone, cic in short, and we explore two partitionings of this set: (i) into (infinitely many non-invertible) convex cones of functions with prescribed poles and zeroes in the right half plane and (ii) each generalized positive function can be written as a sum of even and odd parts. The sets of even generalized positive and odd functions form subcics.It is well known that the Nevanlinna–Pick interpolation problem is not always solvable by positive functions. Unfortunately, there is no computationally simple procedure to carry out this interpolation in the framework of generalized positive functions. Through examples it is illustrated how the two above partitionings of generalized positive functions can be exploited to introduce simple ways to carry out the Nevanlinna–Pick interpolation.Finally we show that only some of these properties are carried over to rational generalized bounded functions, mapping the imaginary axis to the unit disk.