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.1991 Mathematics Subject Classification. Primary: 26C15, 30C40, 30E05, 47A68; Secondary: 47B50, 93D10, 94C05.

The positive real lemma and construction of all realizations of generalized positive rational functions

“…Recall that in the scalar case Odd functions map iR to itself while Even GP functions map iR to R + . Both sets were addressed in [5] in the framework of rational functions. One can study properties of all Even and Odd functions within a prescribed set GP(r, ν, p).…”

“…in [17] and [20], see also Observation 7.1 below. The significance of (1.3) to scalar rational GP functions was recently treated in [4] and [5].…”

“…Recall that a rational function Ψ is in GP if and only if the kernel Ψ(s) + Ψ(w) * s + w * has a finite number of negative squares in its domain of definition in C + . The number of negative squares of the sum of two elements in GP is preserved if, for instance, in (1.3) Ψ 1 (s) = G(s)P 1 (s)G(−s * ) * and Ψ 2 (s) = G(s)P 2 (s)G(−s * ) * with P 1 , P 2 ∈ P and the same function G ∈ C p×p (X), see [5,Section 3] for the scalar case.…”

Interpolation by polynomials with symmetries

“…We shall denote by Odd the set of odd functions. Note that Odd ⊂ GP, for details, see [11,Proposition 4.2].…”

“…For details see [11,Section 5]. Recall that F ∈ GPE if and only if there exist G(s) so that For future reference we recall that a convex cone which in addition is closed under inversion is called a Convex Invertible Cone, cic in short 2 , see e.g.…”