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, 2013, Volume 94, Issue 2, Pages 295–309 (Mi mz10292)  

This article is cited in 5 scientific papers (total in 5 papers)

Limit Ultraspherical Series and Their Approximation Properties

I. I. Sharapudinov

Daghestan Scientific Centre of the Russian Academy of Sciences, Makhachkala

Abstract: We study new series of the form $\sum_{k=0}^\infty f_k^{-1} \widehat P_k^{-1}(x)$ in which the general term $f_k^{-1}\widehat P_k^{-1}(x)$, $k=0,1,…$, is obtained by passing to the limit as $\alpha\to-1$ from the general term $\widehat f_k^\alpha\widehat P_k^{\alpha,\alpha}(x)$ of the Fourier series $\sum_{k=0}^\infty f_k^\alpha\widehat P_k^{\alpha,\alpha}(x)$ in Jacobi ultraspherical polynomials $\widehat P_k^{\alpha,\alpha}(x)$ generating, for $\alpha>-1$, an orthonormal system with weight $(1-x^2)^\alpha$ on $[-1,1]$. We study the properties of the partial sums $S_n^{-1}(f,x)=\sum_{k=0}^nf_k^{-1}\widehat P_k^{-1}(x)$ of the limit ultraspherical series $\sum_{k=0}^\infty f_k^{-1}\widehat P_k^{-1}(x)$. In particular, it is shown that the operator $S_n^{-1}(f)=S_n^{-1}(f,x)$ is the projection onto the subspace of algebraic polynomials $p_n=p_n(x)$ of degree at most $n$, i.e., $S_n(p_n)=p_n$; in addition, $S_n^{-1}(f,x)$ coincides with $f(x)$ at the endpoints $\pm1$, i.e., $S_n^{-1}(f,\pm1)=f(\pm1)$. It is proved that the Lebesgue function $\Lambda_n(x)$ of the partial sums $S_n^{-1}(f,x)$ is of the order of growth equal to $O(\ln n)$, and, more precisely, it is proved that $\Lambda_n(x)\le c(1+\ln(1+n\sqrt{1-x^2}\mspace{2mu}))$, $-1\le x\le 1$.

Keywords: limit ultraspherical series, Fourier series, Lebesgue function of partial sums, Jacobi polynomial, Christoffel–Darboux formula, approximation of continuous functions.

DOI: https://doi.org/10.4213/mzm10292

Full text: PDF file (469 kB)
References: PDF file   HTML file

English version:
Mathematical Notes, 2013, 94:2, 281–293

Bibliographic databases:

UDC: 517.587
Received: 11.01.2012

Citation: I. I. Sharapudinov, “Limit Ultraspherical Series and Their Approximation Properties”, Mat. Zametki, 94:2 (2013), 295–309; Math. Notes, 94:2 (2013), 281–293

Citation in format AMSBIB
\Bibitem{Sha13}
\by I.~I.~Sharapudinov
\paper Limit Ultraspherical Series and Their Approximation Properties
\jour Mat. Zametki
\yr 2013
\vol 94
\issue 2
\pages 295--309
\mathnet{http://mi.mathnet.ru/mz10292}
\crossref{https://doi.org/10.4213/mzm10292}
\mathscinet{http://www.ams.org/mathscinet-getitem?mr=3206089}
\zmath{https://zbmath.org/?q=an:06228550}
\elib{http://elibrary.ru/item.asp?id=20731777}
\transl
\jour Math. Notes
\yr 2013
\vol 94
\issue 2
\pages 281--293
\crossref{https://doi.org/10.1134/S0001434613070274}
\isi{http://gateway.isiknowledge.com/gateway/Gateway.cgi?GWVersion=2&SrcApp=PARTNER_APP&SrcAuth=LinksAMR&DestLinkType=FullRecord&DestApp=ALL_WOS&KeyUT=000323665000027}
\elib{http://elibrary.ru/item.asp?id=20456002}
\scopus{http://www.scopus.com/record/display.url?origin=inward&eid=2-s2.0-84883384455}


Linking options:
  • http://mi.mathnet.ru/eng/mz10292
  • https://doi.org/10.4213/mzm10292
  • http://mi.mathnet.ru/eng/mz/v94/i2/p295

    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. I. I. Sharapudinov, “Some special series in ultraspherical polynomials and their approximation properties”, Izv. Math., 78:5 (2014), 1036–1059  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib  elib
    2. I. I. Sharapudinov, “Nekotorye spetsialnye dvumernye ryady po sisteme $\{\sin x\sin kx\}$ i ikh approksimativnye svoistva”, Izv. Sarat. un-ta. Nov. ser. Ser. Matematika. Mekhanika. Informatika, 14:4(1) (2014), 407–412  mathnet  crossref  elib
    3. I. I. Sharapudinov, G. G. Akniev, “Diskretnye preobrazovaniya so svoistvom prilipaniya na osnove sistemy $\{\sin x\sin kx\}$ i sistemy polinomov Chebysheva vtorogo roda”, Izv. Sarat. un-ta. Nov. ser. Ser. Matematika. Mekhanika. Informatika, 14:4(1) (2014), 413–422  mathnet  crossref  elib
    4. I. I. Sharapudinov, “Approximation properties of Fejér- and de la Valleé-Poussin-type means for partial sums of a special series in the system $\{\sin x\sin kx\}_{k=1}^\infty$”, Sb. Math., 206:4 (2015), 600–617  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib
    5. M. S. Sultanakhmedov, “Spetsialnye veivlety na osnove polinomov Chebysheva vtorogo roda”, Izv. Sarat. un-ta. Nov. ser. Ser. Matematika. Mekhanika. Informatika, 16:1 (2016), 34–41  mathnet  crossref  mathscinet  elib
  • ћатематические заметки Mathematical Notes
    Number of views:
    This page:317
    Full text:95
    References:33
    First page:20

     
    Contact us:
     Terms of Use  Registration  Logotypes © Steklov Mathematical Institute RAS, 2020