On one interpolating rational process of Fejer – Hermite

Ya A Rouba,K A Smatrytski,Ya V Dirvuk

doi:10.29235/1561-2430-2020-56-3-263-274

Abstract

In this paper, a new approach to the definition of the interpolating rational process of Fejer – Hermite with first-type Chebyshev – Markov nodes on a segment is studied and some of its approximating properties are described. In the introduction a brief analysis of the results on the topic of the research is carried out. Herein, the methods of the construction of interpolating processes, in particular, Fejer – Hermite processes, in the polynomial and rational approximation are analysed. A new method to determine the interpolating rational Fejer – Hermite process is proposed. One of the main results of this paper is the proof of the uniform convergence of this process for an arbitrary function, which is continuous on the segment, under some restrictions for the poles of approximating functions. This result is preceded by some auxiliary statements describing the properties of special rational functions. The classic methods of mathematical analysis, approximation theory, and theory of functions of a complex variable are used to prove the results of the work. Moreover, we present the numerical analysis of the effectiveness of the application of the constructed interpolating Fejer – Hermite process for the approximation of a continuous function with singularities. The choice of parameters, on which the nodes of interpolation depend, is made in several standard ways. The obtained results can be applied to further study the approximating properties of interpolating processes.

Highlights

A new approach to the definition of the interpolating rational process of Fejer – Hermite with first-type Chebyshev – Markov nodes on a segment is studied and some of its approximating properties are described
One of the main results of this paper is the proof of the uniform convergence of this process for an arbitrary function, which is continuous on the segment, under some restrictions for the poles of approximating functions. This result is preceded by some auxiliary statements describing the properties of special rational functions
The classic methods of mathematical analysis, approximation theory, and theory of functions of a complex variable are used to prove the results of the work

Summary

Рассмотрим линейный оператор

Наряду с числам= и ak, k 1, 2,..., 2n −1, будем рассматривать числа ck, связанные с данными числами условиями ( ) ck. Здесь выбираем так ветвь квадратного корня, чтобы | ck |< 1, =k 1, 2,..., 2n −1. При выполнении условий (20) для любых x ∈[−1,1] справедливы неравенства. Будем обозначать положительные постоянные, зависящие только от ρ . Так как x ∈[−1,1], то в случае действительного ak будем иметь ( ) (1− | ck |)2 ≤ 2| ck |. В случае комплексных значений параметров ak, аналогичный результат получится при рассмотрении двух комплексно сопряженных слагаемых. Суммируя найденные неравенства по k от 1 до 2n – 1 и учитывая условие (20), придем к соотношениям (21).

Qn l

Для числителя последней дроби получим

Рассмотрим функцию

Findings

Список использованных источников