RUS  ENG JOURNALS   PEOPLE   ORGANISATIONS   CONFERENCES   SEMINARS   VIDEO LIBRARY   PACKAGE AMSBIB
General information
Latest issue
Archive
Impact factor

Search papers
Search references

RSS
Latest issue
Current issues
Archive issues
What is RSS



Trudy Inst. Mat. i Mekh. UrO RAN:
Year:
Volume:
Issue:
Page:
Find






Personal entry:
Login:
Password:
Save password
Enter
Forgotten password?
Register


Trudy Inst. Mat. i Mekh. UrO RAN, 2017, Volume 23, Number 3, Pages 43–57 (Mi timm1436)  

Modified Bernstein function and a uniform approximation of some rational fractions by polynomials

A. G. Babenkoab, Yu. V. Kryakinc

a Ural Federal University named after the First President of Russia B. N. Yeltsin, Ekaterinburg
b Institute of Mathematics and Mechanics, Ural Branch of the Russian Academy of Sciences, Ekaterinburg
c Institute of Mathematics, Wrocław University

Abstract: P. L. Chebyshev posed and solved (1857, 1859) the problem of finding an improper rational fraction least deviating from zero in the uniform metric on a closed interval among rational fractions whose denominator is a fixed polynomial of a given degree $m$ that is positive on the interval and numerator is a polynomial of a given degree $n\ge{m}$ with unit leading coefficient. A. A. Markov solved (1884) a similar problem in the case when the denominator is the square root of a given positive polynomial. In the 20th century, this research direction was developed by S. N. Bernstein, N. I. Akhiezer, and other mathematicians. For example, in 1964 G. Szegő extended Chebyshev's result to the case of trigonometric fractions using the methods of complex analysis. In this paper, using the methods of real analysis and developing Bernstein's approach, we find the best uniform approximation on a period by trigonometric polynomials of certain order for an infinite series of proper trigonometric fractions of a special form. It turned out that, in the periodic case, it is natural to formulate some results in terms of the generalized Poisson kernel $\Pi_{\rho,\xi}(t)=(\cos\xi)P_\rho(t)+(\sin\xi)Q_\rho(t)$, which is a linear combination of the Poisson kernel $P_\rho(t)=(1-\rho^2)/[2(1+\rho^2-2\rho\cos{t})]$ and the conjugate Poisson kernel $Q_\rho(t)=\rho\sin{t}/(1+\rho^2-2\rho\cos{t})$, where $\rho\in(-1,1)$ and $\xi\in\mathbb{R}$. We find the best uniform approximation on a period by the subspace $\mathcal{T}_{n}$ of trigonometric polynomials of order at most $n$ for the linear combination $\Pi_{\rho,\xi}(t)+(-1)^{n}\Pi_{\rho,\xi}(t+\pi)$ of the generalized Poisson kernel and its shift. For $\xi=0$, this yields Bernstein's known results on the best uniform approximation on $[-1,1]$ of the fractions $1/(x^2-a^2)$ and $x/(x^2-a^2)$ by algebraic polynomials. For $\xi={\pi}/{2}$, we obtain the weight analogs (with weight $\sqrt{1-x^2}$) of these results. In addition, we find the value of the best uniform approximation on a period by the subspace $\mathcal{T}_{n}$ of a special linear combination of the mentioned Poisson kernel $P_\rho$ and the Poisson kernel $K_\rho$ for the biharmonic equation in the unit disk.

Keywords: Bernstein functions, Poisson kernels, uniform approximation.

Funding Agency Grant Number
Russian Foundation for Basic Research 15-01-02705
Ministry of Education and Science of the Russian Federation НШ-9356.2016.1
Ural Federal University named after the First President of Russia B. N. Yeltsin 2.A03.21.0006 от 27.08.2013


DOI: https://doi.org/10.21538/0134-4889-2017-23-3-43-57

Full text: PDF file (268 kB)
References: PDF file   HTML file

English version:
Proceedings of the Steklov Institute of Mathematics (Supplementary issues), 2018, 303, suppl. 1, 45–59

Bibliographic databases:

UDC: 517.51
MSC: 41A10, 42A10
Received: 17.10.2016

Citation: A. G. Babenko, Yu. V. Kryakin, “Modified Bernstein function and a uniform approximation of some rational fractions by polynomials”, Trudy Inst. Mat. i Mekh. UrO RAN, 23, no. 3, 2017, 43–57; Proc. Steklov Inst. Math. (Suppl.), 303, suppl. 1 (2018), 45–59

Citation in format AMSBIB
\Bibitem{BabKry17}
\by A.~G.~Babenko, Yu.~V.~Kryakin
\paper Modified Bernstein function and a uniform approximation of some rational fractions by polynomials
\serial Trudy Inst. Mat. i Mekh. UrO RAN
\yr 2017
\vol 23
\issue 3
\pages 43--57
\mathnet{http://mi.mathnet.ru/timm1436}
\crossref{https://doi.org/10.21538/0134-4889-2017-23-3-43-57}
\elib{http://elibrary.ru/item.asp?id=28409367}
\transl
\jour Proc. Steklov Inst. Math. (Suppl.)
\yr 2018
\vol 303
\issue , suppl. 1
\pages 45--59
\crossref{https://doi.org/10.1134/S0081543818090055}
\isi{http://gateway.isiknowledge.com/gateway/Gateway.cgi?GWVersion=2&SrcApp=PARTNER_APP&SrcAuth=LinksAMR&DestLinkType=FullRecord&DestApp=ALL_WOS&KeyUT=000453521100004}


Linking options:
  • http://mi.mathnet.ru/eng/timm1436
  • http://mi.mathnet.ru/eng/timm/v23/i3/p43

    SHARE: VKontakte.ru FaceBook Twitter Mail.ru Livejournal Memori.ru


    Citing articles on Google Scholar: Russian citations, English citations
    Related articles on Google Scholar: Russian articles, English articles
  • Trudy Instituta Matematiki i Mekhaniki UrO RAN
    Number of views:
    This page:191
    Full text:24
    References:24
    First page:10

     
    Contact us:
     Terms of Use  Registration  Logotypes © Steklov Mathematical Institute RAS, 2019