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

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

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
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\jour Mat. Sb.
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\issue 4
\pages 131--148
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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  mathnet  crossref
  • Математический сборник Sbornik: Mathematics (from 1967)
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