On the index theory of pseudo-Carathéodory functions. Application to the linear system stability theorem
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 degenerate situations where the classical algorithms break down.
scientific article; zbMATH DE number 4048042
scientific article