On the class SI of J-contractive functions intertwining solutions of linear differential equations

TL;DR

This paper extends the class SI of J-contractive functions, related to 2D systems, and solves classical problems like Schur algorithm, realization, and interpolation within this class using new theoretical tools.

Contribution

It introduces extensions of classical problems to the SI class of J-contractive functions and develops new methods based on a correspondence with the SC class.

Findings

Extended Schur algorithm for SI class

Solved partial realization problem in SI class

Established interpolation results for SI functions

Abstract

In the PhD thesis of the second author under the supervision of the third author was defined the class SI of J-contractive functions, depending on a parameter and arising as transfer functions of overdetermined conservative 2D systems invariant in one direction. In this paper we extend and solve in the class SI, a number of problems originally set for the class SC of functions contractive in the open right-half plane, and unitary on the imaginary line with respect to some preassigned signature matrix J. The problems we consider include the Schur algorithm, the partial realization problem and the Nevanlinna-Pick interpolation problem. The arguments rely on a correspondence between elements in a given subclass of SI and elements in SC. Another important tool in the arguments is a new result pertaining to the classical tangential Schur algorithm.