Convex Cones of Generalized Positive Rational Functions and Nevanlinna-Pick Interpolation

TL;DR

This paper explores the structure of generalized positive rational functions, their partitionings, and develops a new approach for Nevanlinna-Pick interpolation within this broader class, extending classical results.

Contribution

It introduces a convex cone framework for generalized positive functions, analyzes their partitionings, and proposes a simple Nevanlinna-Pick interpolation method for these functions.

Findings

Generalized positive functions form a convex invertible cone (cic).

Any positive function can be decomposed into even and odd parts within the generalized positive set.

A new simple procedure for Nevanlinna-Pick interpolation over generalized positive functions is proposed.

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 a pair of even and odd subcics and (ii) to (infinitely many non-invertible) convex cones of functions with prescribed poles and zeroes in the right half plane. It is then shown that a positive function can always be written as a sum of even and odd part, only over the larger set of generalized positive. It is well known that over positive functions Nevanlinna-Pick interpolation is not always feasible. Over generalized positive, there is no easy way to carry out this interpolation. The second partitioning is subsequently exploited to introduce a simple procedure for Nevanlinna-Pick interpolation. Finally we show that only some of these properties are carried over to generalized bounded functions, mapping the imaginary axis to the unit disk.