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Seminar on Complex Analysis (Gonchar Seminar)
October 13, 2014 18:00, Moscow, Steklov Mathematical Institute, Room 411 (8 Gubkina)

Localization of zeros for Cauchy transforms

Yu. S. Belov

St. Petersburg State University, Department of Mathematics and Mechanics

Abstract: If discrete measure $\sum_n\mu_n\delta_{t_n}$ is sufficiently small (for example $\mu_n< e^{-n}$), then the zeros of Cauchy transform are localized near the $\operatorname{supp}\mu$. Moreover, it may happen that this holds for any $\nu$ such that $|\nu|<\mu$. We have found a description of such measures and attraction sets (i.e. subsets of $\operatorname{supp}\mu$ which attract zeros). We have proved that all attraction sets are ordered by inclusion. Such measures appear naturaly in the theory of canonical systems of differential equations. They correspond to the systems whose Hamiltonian consists of indivisible intervals accumulating only to the left. Moreover, this correspondence is one-to-one under some additional assumptions. This topic is connected to the problem of density of polynomials and other classical problems in analysis.