Hurwitz-type matrices of doubly infinite series

TL;DR

This paper characterizes when two doubly infinite series produce a totally nonnegative Hurwitz-type matrix, linking it to specific S-functions, and provides a detailed proof highlighting properties of such matrices.

Contribution

It establishes a novel characterization of totally nonnegative Hurwitz-type matrices generated by doubly infinite series, addressing the unique challenges of the infinite case.

Findings

Doubly infinite series generate totally nonnegative Hurwitz-type matrices if and only if their ratio is an S-function.

The paper provides a detailed proof emphasizing the features of these matrices.

A forthcoming publication will present a shorter proof and applications to stability.

Abstract

This paper show that two doubly infinite series generate a totally nonnegative Hurwitz-type matrix if and only if their ratio represents an S-functions of a certain kind. The doubly infinite case needs a specific approach, since the ratios have no correspondent Stieltjes continued fraction. Another forthcoming publication (see Dyachenko, arXiv:1608.04440) offers a shorter improved version of this result as well as its application to the Hurwitz stability. Nevertheless, the proof presented here illustrates features of totally nonnegative Hurwitz-type matrices better.