arXiv subjects

Evguenii A. Rakhmanov

Publications and source records attributed to Evguenii A. Rakhmanov.

4 recordsLinked to original sources

arXiv2021

Tschebyshev-Padé approximations for multivalued functions

We discuss the relation between the linear Tschebyshev-Padé approximations to analytic function $f$ and the diagonal type I Hermite-Padé polynomials for the tuple of functions $[1,f_1,f_2]$ where the pair of functions $f_1,f_2$ forms certain Nikishin system. An approach is proposed of how to extend the seminal Stahl's Theory for Padé approximations for multivalued analytic functions to the Tschebyshev-Padé approximations. The approach is based on the relation between Tschebyshev-Padé approximations and Hermite-Padé polynomials and also on a connection of Hermite-Padé polynomials and multipoint Padé approximants.

math.CV↗

arXiv2017

Zero distribution for Angelesco Hermite--Padé polynomials

We consider the problem of zero distribution of the first kind Hermite--Padé polynomials associated with a vector function $\vec f = (f_1, \dots, f_s)$ whose components $f_k$ are functions with a finite number of branch points in plane. We assume that branch sets of component functions are well enough separated (which constitute the Angelesco case). Under this condition we prove a theorem on limit zero distribution for such polynomials. The limit measures are defined in terms of a known vector equilibrium problem. Proof of the theorem is based on the methods developed by H.~Stahl, A.~A.~Gonchar and the author. These methods obtained some further generalization in the paper in application to systems of polynomials defined by systems of complex orthogonality relations. Together with the characterization of the limit zero distributions of Hermite--Padé polynomials by a vector equilibrium problem we consider an alternative characterization using a Riemann surface $\mathcal R(\vec f)$ associated with $\vec f$. In this terms we present a more general (without Angelesco condition) conjecture on the zero distribution of Hermite--Padé polynomials. Bibliography: 72 items.

math.CV↗

arXiv2016

Existence and Regularity for an Energy Maximization Problem in Two Dimensions

We consider the variational problem of maximizing the weighted equilibrium Green's energy of a distribution of charges free to move in a subset of the upper half-plane, under a particular external field. We show that this problem admits a solution and that, under some conditions, this solution is an S-curve (in the sense of Gonchar-Rakhmanov). The above problem appears in the theory of the semiclassical limit of the integrable focusing nonlinear Schrödinger equation. In particular, its solution provides a justification of a crucial step in the asymptotic theory of nonlinear steepest descent for the inverse scattering problem of the associated linear non-self-adjoint Zakharov-Shabat operator and the equivalent Riemann-Hilbert factorization problem.

math.CV↗

arXiv2010

On the convergence on nonlinear Padé--Chebyshev approximations to the multivalued analytic functions, variation of equilibrium energy and $S$-property of stationary compacts

Some new results on the convergence of nonlinear diagonal Padé--Chebyshev approximations to multivalued analytic function given on the segment $[-1,1]$, are proved. We show that these approximations converge to the given function in the "maxmimal" domain of its meromorphity and that the boundary of this maxmimal domain is an $S$-curve. The results of the work have been announced in abs/1009.4813 Bibliography: 57 items.

math.CV↗