Convex invertible cones and Nevanlinna-Pick interpolation: The suboptimal case

TL;DR

This paper addresses a specific suboptimal case of Nevanlinna-Pick interpolation for positive real odd functions evaluated at real matrix points, using convex cones and linear matrix map representations.

Contribution

It provides a partial solution to a previously unresolved problem by connecting it to classical cases and analyzing positive linear matrix maps.

Findings

Solution for the suboptimal case of the interpolation problem.

Connection established with classical Nevanlinna-Pick interpolation.

Analysis of when positive linear matrix maps are completely positive.

Abstract

Nevanlinna-Pick interpolation developed from a topic in classical complex analysis to a useful tool for solving various problems in control theory and electrical engineering. Over the years many extensions of the original problem were considered, including extensions to different function spaces, nonstationary problems, several variable settings and interpolation with matrix and operator points. Here we discuss a variation on Nevanlinna-Pick interpolation for positive real odd functions evaluated in real matrix points. This problem was studied by Cohen and Lewkowicz using convex invertible cones and the Lyapunov order, but was never fully resolved. In this paper we present a solution to this problem in a special case that we refer to as `suboptimal' based on connections with the classical case. The solution requires a representation of linear matrix maps going back to R.D. Hill and an analysis of when positive linear matrix maps are completely positive, on which we reported in earlier work and which we will briefly review here.