On the index theory of pseudo-Carathéodory functions. Application to the linear system stability theorem
type: Item
instance of: scholarly article
title: On the index theory of pseudo-Carathéodory functions. Application to the linear system stability theorem
zbMATH Open document ID: 0643.30028
DOI: 10.1007/BF01195795
published in: Integral Equations and Operator Theory
publication date: 1987-01-01T00:00:00Z
review text: The class C of pseudo-Carathéodory functions consists of the functions f(z) expressible as the ratio of two \(H^{\infty}\)-functions and whose radial limit has a nonnegative real part almost everywhere on the unit circle: Re f(e\({}^{i\theta})\geq 0\) for \(0\leq \theta \leq 2\pi\). Such a function need not be analytic on the unit disk; a measure of the ``amount of singularity'' of f(z) in the unit disk is an integer valued parameter (called the index of f). In particular the class of Carathéodory functions (C-functions which are analytic on the unit disk) consists exactly of the C-functions having index 0. Other interesting classes are density functions (C-functions f with positive real values on the unit circle). In an earlier paper, the author together with P. Delsarte and Y. Kamp obtained an essentially unique factorization \(h=fg\) for a C-function h, where f is a Carathéodory function and g is a density function and where h and g have the same index. This is analogous, but not equivalent via fractional linear change of variable, to the factorization of a pseudo-Schur function w (i.e. the ratio of \(H^{\infty}\)-functions with values of modulus less than 1 on the unit circle) as \(w=uv^{-1}\) where u is a Schur function and v is inner, and is one of the main tools for the present paper. The main goal of the present paper is to describe how the index determination problem for the class of pseudo-lossless functions turns out to be equivalent to the polynomial zeros location problem. Various decomposition techniques are described for the index computation of rational pseudo-lossless functions. Various known tests to count zeros of a polynomial in a standard region (e.g. unit disk or right half plane), such as Routh-Hurwitz test, Schur-Cohn test and Bistritz partial extraction, are shown to fit into the framework of index theory for C-functions. Also by extracting an elementary C-factor of a special form, it is shown that the algorithms can be continued even in deg…
Mathematics Subject Classification ID: 30D50
Mathematics Subject Classification ID: 30D55
Mathematics Subject Classification ID: 30C15
Mathematics Subject Classification ID: 30D05
Mathematics Subject Classification ID: 65H05
zbMATH DE Number: 4048042
zbMATH Keywords: pseudo-Carathéodory functions
zbMATH Keywords: Schur function
zbMATH Keywords: Routh-Hurwitz test
zbMATH Keywords: Schur-Cohn test
author: Yves V. Genin
reviewed by: Joseph A. Ball
MaRDI profile type: Publication
cites work: ANALYTIC PROPERTIES OF SCHMIDT PAIRS FOR A HANKEL OPERATOR AND THE GENERALIZED SCHUR-TAKAGI PROBLEM
cites work: Algebraic Structure of Generalized Positive Real Matrices
cites work: Q3309525
cites work: Q5560003
cites work: Q3746076
cites work: Pseudo-lossless functions with application to the problem of locating the zeros of a polynomial
cites work: Q3677074
cites work: Analysis of the Lyapunov equation using generalized positive real matrices
cites work: Minimal realizations of pseudo-positive and pseudo-bounded rational matrices
cites work: Q3254327
cites work: Q5795016
cites work: Q4065382
cites work: Über einige Fortsetzungsprobleme, die eng mit der Theorie hermitescher Operatoren im RaumeIIx zusammenhängen. I. Einige Funktionenklassen und ihre Darstellungen
cites work: Q3954052
cites work: Q5546070
cites work: Q3960778
cites work: Q3274185
description: scientific article; zbMATH DE number 4048042