arXiv subjects
Publications and source records attributed to Mikhail Tyaglov.
We give a detailed account of various connections between several classes of objects: Hankel, Hurwitz, Toeplitz, Vandermonde and other structured matrices, Stietjes and Jacobi-type continued fractions, Cauchy indices, moment problems, total positivity, and root localization of univariate polynomials. Along with a survey of many classical facts, we provide a number of new results.
For a given real entire function $ϕ$ with finitely many nonreal zeros, we establish a connection between the number of real zeros of the functions $Q=(ϕ'/ϕ)'$ and $Q_1=(ϕ''/ϕ')'$. This connection leads to a proof of the Hawaii conjecture [T.Craven, G.Csordas, and W.Smith, The zeros of derivatives of entire functions and the Pólya-Wiman conjecture, Ann. of Math. (2) 125 (1987), 405--431] stating that the number of real zeros of $Q$ does not exceed the number of nonreal zeros of $ϕ$.
A generalization of Hurwitz stable polynomials to real rational functions is considered. We establishe an analogue of the Hurwitz stability criterion for rational functions and introduce a new type of determinants that can be treated as a generalization of the Hurwitz determinants.
We describe a wide class of polynomials, which is a natural generalization of Hurwitz stable polynomials. We also give a detailed account of so-called self-interlacing polynomials, which are dual to Hurwitz stable polynomials but have only real and simple zeroes. All proofs are given using properties of rational functions mapping the upper half-plane of the complex plane to the lower half-plane. Matrices with self-interlacing spectra and other applications of generalized Hurwitz polynomials are discussed.