Mikhail Tyaglov

Structured Matrices, Continued Fractions, and Root Localization of Polynomials

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.

Hurwitz rational functions

A generalization of Hurwitz stable polynomials to real rational functions is considered. We establish an analog 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.

Sign patterns of the Schwarz matrices and generalized Hurwitz polynomials

The direct and inverse spectral problems are solved for a wide subclass of the class of Schwarz matrices. A connection between Schwarz matrices and the so-called generalized Hurwitz polynomials is found. The known results due to H. Wall and O. Holtz are briefly reviewed and obtained as particular cases.The direct and inverse spectral problems are solved for a wide subclass of the class of Schwarz matrices. A connection between the Schwarz matrices and the so-called generalized Hurwitz polynomials is found. The known results due to H. Wall and O. Holtz are briefly reviewed and obtained as particular cases.

Szegő's theorem for matrix orthogonal polynomials

We extend some classical theorems in the theory of orthogonal polynomials on the unit circle to the matrix case. In particular, we prove a matrix analogue of Szegos theorem. As a by-product, we also obtain an elementary proof of the distance formula by Helson and Lowdenslager.

On the number of real critical points of logarithmic derivatives and the Hawaii conjecture

For a given real entire function φ in the class U2n*, n ≥ 0, with finitely many nonreal zeroes, we establish a connection between the number of real zeroes of the functions Q[φ] = (φ′/φ)′ and Q1[φ] = (φ″/φ′)′. This connection leads to a proof of the Hawaii Conjecture (T. Craven, G. Csordas, and W. Smith [5]), which states that if φ is a real polynomial, then the number of real zeroes of Q[φ] does not exceed the number of nonreal zeroes of φ.

Self-interlacing polynomials II: Matrices with self-interlacing spectrum

An

Circulants and critical points of polynomials

n\times n

Maximal univalent disks of real rational functions and Hermite-Biehler polynomials

matrix is said to have a self-interlacing spectrum if its eigenvalues

Complex geometry apertures for resonant diaphragms in rectangular waveguides

\lambda_k

Linear finite difference operators preserving the Laguerre–Pólya class

,