arXiv subjects

Nikolay R. Ikonomov

Publications and source records attributed to Nikolay R. Ikonomov.

5 recordsLinked to original sources

arXiv2026

On Convergence of Rational Hermite-Padé Approximants

The main purpose of this paper is to compare the convergence properties of Padé approximants and rational Hermite-Padé approximants for some model class of multivalued analytic functions based of the inverse Zhoukovsky transform. We prove that in the class of analytic functions under consideration the rational Hermite-Padé approximants converge faster than the corresponding Padé approximants. In contrast to the classical vector potential-theoretic approach, which was introduced by A. A. Gonchar and E. A. Rakhmanov in 1981 and developed later by A. I. Aptekarev, V. N. Sorokin and others, the proofs here are based on some scalar mixed Green-logarithmic potential problems.

math.CV↗

arXiv2023

Rational Hermite-Padé Approximants vs Padé Approximants. Numerical Results

The main purpose of the paper is to present some powerful data on the advantage of the rational approximation procedure based on Hermite-Padé polynomials over the Padé approximation procedure. The first part of the paper is devoted to some numerical examples in this direction. The second part will be devoted to some theoretical results. In particular, we demonstrate our ideas about the advantage of rational Hermite-Padé approximants over Padé approximants analyzing the analytical structure of the frequency function $ν$ of the free Van der Pol equation.

math.CV↗

arXiv2019

Scalar equilibrium problem and the limit distribution of the zeros of Hermite-Padé polynomials of type II

The existence of the limit distribution of the zeros of Hermite-Padé polynomials of type II for a pair of functions forming a Nikishin system is proved using the scalar equilibrium problem posed on the two-sheeted Riemann surface. The relation of the results obtained here to some results of H. Stahl (1988) is discussed. Results of numerical experiments are presented. The results of the present paper and those obtained in the earlier paper of the second author [28], [32], [33] are shown to be in good accordance with both H. Stahl's results and with results of numerical experiments.

math.CV↗

arXiv2016

Zero Distribution of Hermite-Padé Polynomials and Convergence Properties of Hermite Approximants for Multivalued Analytic Functions

In the paper, we propose two new conjectures about the convergence of Hermite Approximants of multivalued analytic functions of Laguerre class ${\mathscr L}$. The conjectures are based in part on the numerical experiments, made recently by the authors in [26] and [27].

math.CV↗

arXiv2015

On the limit zero distribution of type I Hermite-Padé polynomials

In this paper are discussed the results of new numerical experiments on zero distribution of type I Hermite-Padé polynomials of order $n=200$ for three different collections of three functions $[1,f_1,f_2]$. These results are obtained by the authors numerically and do not match any of the theoretical results that were proven so far. We consider three simple cases of multivalued analytic functions $f_1$ and $f_2$, with separated pairs of branch points belonging to the real line. In the first case both functions have two logarithmic branch points, in the second case they both have branch points of second order, and finally, in the third case they both have branch points of third order. All three cases may be considered as representative of the asymptotic theory of Hermite-Padé polynomials. In the first two cases the numerical zero distribution of type I Hermite-Padé polynomials are similar to each other, despite the different kind of branching. But neither the logarithmic case, nor the square root case can be explained from the asymptotic point of view of the theory of type I Hermite-Padé polynomials. The numerical results of the current paper might be considered as a challenge for the community of all experts on Hermite-Padé polynomials theory.

math.CV↗