RUS  ENG JOURNALS   PEOPLE   ORGANISATIONS   CONFERENCES   SEMINARS   VIDEO LIBRARY   PERSONAL OFFICE
General information
Latest issue
Archive

Search papers
Search references

RSS
Latest issue
Current issues
Archive issues
What is RSS



Chebyshevskii Sb.:
Year:
Volume:
Issue:
Page:
Find






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


Chebyshevskii Sb., 2016, Volume 17, Issue 2, Pages 170–183 (Mi cheb487)  

This article is cited in 1 scientific paper (total in 1 paper)

Approximation of analytic periodic functions by linear means of Fourier series

O. G. Rovenskaa, O. A. Novikovb

a Donbass State Engineering Academy, Kramatorsk
b Donbass State Pedagogical University

Abstract: The work concerns the questions of approximation of periodic differentiable functions of high smoothness by repeated arithmetic means of Fourier sums. One of the classifications of periodic functions nowadays is the classification suggested by A. Stepanets which is based on the concept of $(\psi,\beta)$-differentiation. The given classification allows to distinguish all classes of summable periodic functions from the functions where the Fourier series can deviate to infinitely differentiable functions including analytical and entire ones. When choosing the parameters properly, classes of $(\psi,\beta)$-differentiable functions exactly coincide with the well-known classes of Vail differentiable functions, Sobolev classes $W^l_p$ and classes of convolutions with integral kernels.
De la Vallee Poussin sums and their special cases (Fourier sums and Fejer sums) were extensively studied for many decades by many prominent experts in the theory of functions. At present, the large amount of factual material is accumulated in numerous publications. One of the most important directions in this field is the investigation of approximation properties of these sums for various classes of functions. The aim of the present paper is to systematize the known results related to the approximation properties of de la Vallee Poussin sums on classes of Poisson integrals and to present new facts obtained for their generalizations.
In certain cases asymptotic equalities are found for upper bounds of deviations in the uniform metric of the repeated de la Vallee Poussin sums on the classes $C^\psi_{\beta,\infty}$ and $C^\psi_\beta H_\omega$ which are generated by multiplicators $\psi(k)$ and shifts on argument $\beta$ provided that sequences $\psi(k)$ which define the specified classes tend to zero with the rate of geometrical progression. In doing so classes $C^\psi_{\beta,\infty}$ and $C^\psi_\beta H_\omega$ consist of analytic functions which can be regularly extended in the corresponding strip.
We introduce generalized de la Vallee Poussin sums and study their approximation properties for the classes of analytic periodic functions. We obtain asymptotic equalities for upper bounds of deviations of the repeated de la Vallee Poussin sums on classes of Poisson integrals. Under certain conditions, these equalities guarantee the solvability of the Kolmogorov–Nikol’skiy problem for the repeated de la Vallee Poussin sums and classes of Poisson integrals. We indicate conditions under which the repeated sums guarantee a better order of approximation than ordinary de la Vallee Poussin sums.
Bibliography: 16 titles.

Keywords: Fourier series, de la Valee Poussin method, asymptotic formula.

Full text: PDF file (625 kB)
References: PDF file   HTML file

Document Type: Article
UDC: 517.5
Received: 10.03.2016
Accepted:10.06.2016

Citation: O. G. Rovenska, O. A. Novikov, “Approximation of analytic periodic functions by linear means of Fourier series”, Chebyshevskii Sb., 17:2 (2016), 170–183

Citation in format AMSBIB
\Bibitem{RovNov16}
\by O.~G.~Rovenska, O.~A.~Novikov
\paper Approximation of analytic periodic functions by linear means of Fourier series
\jour Chebyshevskii Sb.
\yr 2016
\vol 17
\issue 2
\pages 170--183
\mathnet{http://mi.mathnet.ru/cheb487}
\elib{http://elibrary.ru/item.asp?id=26254432}


Linking options:
  • http://mi.mathnet.ru/eng/cheb487
  • http://mi.mathnet.ru/eng/cheb/v17/i2/p170

    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

    This publication is cited in the following articles:
    1. L. K. Dodunova, A. A. Ageikin, “Approximation of analytic functions by universal Vallee-Poussin sums on the Chebyshev polynomials”, Izvestiya Irkutskogo gosudarstvennogo universiteta. Seriya Matematika, 24 (2018), 12–23  mathnet  crossref
  • Number of views:
    This page:90
    Full text:34
    References:13

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