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 Mat. Sb., 2015, Volume 206, Number 4, Pages 131–148 (Mi msb8391)

Approximation properties of Fejér- and de la Valleé-Poussin-type means for partial sums of a special series in the system $\{\sin x\sin kx\}_{k=1}^\infty$

I. I. Sharapudinov

Daghestan Scientific Centre of the Russian Academy of Sciences, Makhachkala

Abstract: This paper is concerned with series of the form
$$\Phi(\theta)=A_\Phi(\theta)+\sin\theta\sum_{k=1}^\infty\varphi_k\sin k\theta,$$
where $\Phi(\theta)$ is an even $2\pi$-periodic function with finite values $\Phi(0)$ and $\Phi(\pi)$,
\begin{gather*} A_\Phi(\theta)=\frac{\Phi(0)+\Phi(\pi)}{2}+\frac{\Phi(0)-\Phi(\pi)}{2}\cos\theta, \qquad \varphi(\theta)=\Phi(\theta)-A_\Phi(\theta),
\varphi_k=\frac{2}{\pi}\int_0^\pi\varphi(t)\frac{\sin kt}{\sin t} dt. \end{gather*}
Series of this type appear as a particular case of more general special series in ultraspherical Jacobi polynomials, which were first introduced and studied by the author. Partial sums of the form $\Pi_n(\Phi)=\Pi_n(\Phi,\theta) =A_\Phi(\theta)+\sin\theta\sum_{k=1}^{n-1}\varphi_k\sin k\theta$ are shown to have a number of important properties, which give them an advantage over trigonometric Fourier sums of the form $S_n(\Phi,\theta)=\frac{a_0}{2}+\sum_{k=1}^na_k\cos k\theta$. Approximation properties of Fejér- and de la Valleé-Poussin-type means for the partial sums $\Pi_n(\Phi,\theta)$ are studied.
Bibliography: 7 titles.

Keywords: special series in the system $\{\sin x\sin kx\}_{k=1}^\infty$, Fejér means, de la Valleé-Poussin means, approximation properties.

 Funding Agency Grant Number Russian Foundation for Basic Research 10-01-00191 This research was carried out with the financial support of the Russian Foundation for Basic Research (grant no. 10-01-00191).

DOI: https://doi.org/10.4213/sm8391

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English version:
Sbornik: Mathematics, 2015, 206:4, 600–617

Bibliographic databases:

UDC: 517.538
MSC: Primary 41A17; Secondary 42C10, 46E30, 46E35
Received: 02.06.2014 and 28.11.2014

Citation: I. I. Sharapudinov, “Approximation properties of Fejér- and de la Valleé-Poussin-type means for partial sums of a special series in the system $\{\sin x\sin kx\}_{k=1}^\infty$”, Mat. Sb., 206:4 (2015), 131–148; Sb. Math., 206:4 (2015), 600–617

Citation in format AMSBIB
\Bibitem{Sha15} \by I.~I.~Sharapudinov \paper Approximation properties of Fej\'er- and de~la~Valle\'e-Poussin-type means for partial sums of a~special series in the system $\{\sin x\sin kx\}_{k=1}^\infty$ \jour Mat. Sb. \yr 2015 \vol 206 \issue 4 \pages 131--148 \mathnet{http://mi.mathnet.ru/msb8391} \crossref{https://doi.org/10.4213/sm8391} \mathscinet{http://www.ams.org/mathscinet-getitem?mr=3354987} \zmath{https://zbmath.org/?q=an:06464996} \adsnasa{http://adsabs.harvard.edu/cgi-bin/bib_query?2015SbMat.206..600S} \elib{https://elibrary.ru/item.asp?id=23421639} \transl \jour Sb. Math. \yr 2015 \vol 206 \issue 4 \pages 600--617 \crossref{https://doi.org/10.1070/SM2015v206n04ABEH004471} \isi{http://gateway.isiknowledge.com/gateway/Gateway.cgi?GWVersion=2&SrcApp=PARTNER_APP&SrcAuth=LinksAMR&DestLinkType=FullRecord&DestApp=ALL_WOS&KeyUT=000356313700006} \scopus{https://www.scopus.com/record/display.url?origin=inward&eid=2-s2.0-84931293076} 

• http://mi.mathnet.ru/eng/msb8391
• https://doi.org/10.4213/sm8391
• http://mi.mathnet.ru/eng/msb/v206/i4/p131

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This publication is cited in the following articles:
1. G. G. Akniev, “Approximation properties of Fourier sums for $2\pi$-periodic piecewise linear continuous functions”, Dagestanskie elektronnye matematicheskie izvestiya, 2016, no. 5, 13–19
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