Matematicheskie Zametki
RUS  ENG    JOURNALS   PEOPLE   ORGANISATIONS   CONFERENCES   SEMINARS   VIDEO LIBRARY   PACKAGE AMSBIB  
General information
Latest issue
Forthcoming papers
Archive
Impact factor
Subscription
Guidelines for authors
License agreement
Submit a manuscript

Search papers
Search references

RSS
Latest issue
Current issues
Archive issues
What is RSS



Mat. Zametki:
Year:
Volume:
Issue:
Page:
Find






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


Mat. Zametki, 2021, Volume 110, Issue 3, Pages 434–449 (Mi mz12833)  

Approximation of Functions by Discrete Fourier Sums in Polynomials Orthogonal on a Nonuniform Grid with Jacobi Weight

M. S. Sultanakhmedov

Daghestan Scientific Centre of Russian Academy of Sciences, Makhachkala

Abstract: Let there be given a partition of the closed interval $[-1,1]$ by arbitrary nodes $\{\eta_j\}_{j=0}^N$, where $\lambda_N=\max_{0\le j \le N-1} (\eta_{j+1}-\eta_{j})$. For a continuous function $f(t)$ given on an arbitrary grid $\Omega_N=\{t_j \mid \eta_{j} \le t_j \le \eta_{j+1}\}_{j=0}^{N-1}$, the approximation properties of the discrete Fourier sums $\Lambda^{\alpha,\beta}_{n,N}(f,t)$ in polynomials $\widehat P^{\alpha,\beta}_{n, N} (t)$ are investigated in the case of nonnegative integer parameters $\alpha$, $\beta$; these polynomials are orthogonal to $\Omega_N$ with Jacobi weight $\kappa^{\alpha,\beta}(t)=(1-t)^{\alpha}(1+t)^{\beta}$. Given the restriction $n=O(\lambda_N^{-1/3})$ on the order of the Fourier sums, a pointwise estimate of the Lebesgue function $L^{\alpha,\beta}_{n, N}(t)$ is obtained; it depends on $n$ and the position of the point $t \in [-1,1]$:
$$ L^{\alpha,\beta}_{n,N}(t)=O[\ln{(n+1)}+ |\widehat P^{\alpha,\beta}_{n,N}(t)|+ |\widehat P^{\alpha,\beta}_{n+1,N}(t)|]. $$


Keywords: Jacobi polynomials, Fourier sum, nonuniform grid, Lebesgue function, orthogonal polynomials, approximation properties.

DOI: https://doi.org/10.4213/mzm12833

Full text: PDF file (544 kB)
First page: PDF file
References: PDF file   HTML file

English version:
Mathematical Notes, 2021, 110:3, 418–431

Bibliographic databases:

UDC: 517.518.82+517.521
Received: 07.07.2020
Revised: 04.03.2021

Citation: M. S. Sultanakhmedov, “Approximation of Functions by Discrete Fourier Sums in Polynomials Orthogonal on a Nonuniform Grid with Jacobi Weight”, Mat. Zametki, 110:3 (2021), 434–449; Math. Notes, 110:3 (2021), 418–431

Citation in format AMSBIB
\Bibitem{Sul21}
\by M.~S.~Sultanakhmedov
\paper Approximation of Functions by Discrete Fourier Sums in Polynomials Orthogonal on a Nonuniform Grid with Jacobi Weight
\jour Mat. Zametki
\yr 2021
\vol 110
\issue 3
\pages 434--449
\mathnet{http://mi.mathnet.ru/mz12833}
\crossref{https://doi.org/10.4213/mzm12833}
\transl
\jour Math. Notes
\yr 2021
\vol 110
\issue 3
\pages 418--431
\crossref{https://doi.org/10.1134/S0001434621090108}
\isi{http://gateway.isiknowledge.com/gateway/Gateway.cgi?GWVersion=2&SrcApp=PARTNER_APP&SrcAuth=LinksAMR&DestLinkType=FullRecord&DestApp=ALL_WOS&KeyUT=000711049900010}
\scopus{https://www.scopus.com/record/display.url?origin=inward&eid=2-s2.0-85118125749}


Linking options:
  • http://mi.mathnet.ru/eng/mz12833
  • https://doi.org/10.4213/mzm12833
  • http://mi.mathnet.ru/eng/mz/v110/i3/p434

    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
  • Математические заметки Mathematical Notes
    Number of views:
    This page:73
    References:6
    First page:2

     
    Contact us:
     Terms of Use  Registration to the website  Logotypes © Steklov Mathematical Institute RAS, 2021