Berlin Technical University
A function is (Hurwitz-)stable if all its zeros lie in the left half of the complex plane.
The classical approach to the Hurwitz stability (dating back to Hermite and Biehler) exploits
a deep relation between stable functions and mappings of the upper half of the complex
plane into itself (i.e. R-functions). Hurwitz introduced a connection between minors of the
Hurwitz matrix and the Hankel matrix built from coefficients of the related R-function
(moments), which resulted in the famous Hurwitz criterion for polynomials. This technique
also extends to the so-called strongly stable entire functions. More recent studies highlighted
another property related to the Hurwitz stability: the total nonnegativity of the corresponding
Hurwitz matrix, that is nonnegativity of all its minors. During my talk I would like to present
a complete description of power series (singly infinite or finite) that generate totally
nonnegative Hurwitz matrices. This result was obtained in Holtz and Tyaglov (2012)
and in Dyachenko (2014).