Hurwitz matrices of doubly infinite series

TL;DR

This paper extends the criterion linking polynomial quasi-stability to the total nonnegativity of Hurwitz matrices, providing a complete description for doubly infinite series and exploring related factorization properties.

Contribution

It offers a comprehensive characterization of functions generating doubly infinite series with totally nonnegative Hurwitz matrices, expanding known results from singly infinite series.

Findings

Complete description of functions with totally nonnegative doubly infinite Hurwitz matrices

Necessary conditions for total nonnegativity of generalized Hurwitz matrices

Extension of stability criteria to doubly infinite series

Abstract

This paper aims at extending the criterion that the quasi-stability of a polynomial is equivalent to the total nonnegativity of its Hurwitz matrix. We give a complete description of functions generating doubly infinite series with totally nonnegative Hurwitz and Hurwitz-type matrices (in a Hurwitz-type matrix odd and even rows come from two distinct power series). The corresponding result for singly infinite series is known: it is based on a certain factorization of Hurwitz-type matrices, which is absent in the doubly infinite case. A necessary condition for total nonnegativity of generalized Hurwitz matrices follows as an application.