The Schur Algorithm in Terms of System Realization

TL;DR

This paper explores the Schur algorithm through the lens of System Theory, illustrating how system realizations and colligations provide new insights into rational inner functions and their algebraic properties.

Contribution

It introduces a system-theoretic approach to the Schur algorithm, connecting rational inner functions with unitary matrix realizations and Redheffer coupling.

Findings

System realizations classify rational inner functions by unitary matrices.

Redheffer coupling selects specific representatives in equivalence classes.

The approach offers algebraic insights into the Schur algorithm.

Abstract

The main goal of this paper is to demonstrate the usefulness of certain ideas from System Theory in the study of problems from complex analysis. With this paper, we also aim to encourage analysts, who might not be familiar with System Theory, colligations or operator models to take a closer look at these topics. For this reason, we present a short introduction to the necessary background. The method of system realizations of analytic functions often provides new insights into and interpretations of results relating to the objects under consideration. In this paper we will use a well-studied topic from classical analysis as an example. More precisely, we will look at the classical Schur algorithm from the perspective of System Theory. We will confine our considerations to rational inner functions. This will allow us to avoid questions involving limits and will enable us to concentrate on the algebraic aspects of the problem at hand. Given a non-negative integer (n), we describe all system realizations of a given rational inner function of degree (n) in terms of an appropriately constructed equivalence relation in the set of all unitary ((n + 1) \times (n + 1) )-matrices. The concept of Redheffer coupling of colligations gives us the possibility to choose a particular representative from each equivalence class. The Schur algorithm for a rational inner function is, consequently, described in terms of the state space representation.