Geodesic
Approximation properties of Fejér- and de la Valleé-Poussin-type means for partial sums of a special series in the system $\{\sin x\sin kx\}_{k=1}^\infty$
Sbornik. Mathematics, Tome 206 (2015) no. 4, pp. 600-617 Cet article a éte moissonné depuis la source Math-Net.Ru

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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 $\{\sin x\sin kx\}_{k=1}^\infty$, de la Valleé-Poussin means, approximation properties.
Mots-clés : Fejér means 
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     author = {I. I. Sharapudinov},
     title = {Approximation properties of {Fej\'er-} and {de~la~Valle\'e-Poussin-type} means for partial sums of a~special series in the system $\{\sin x\sin kx\}_{k=1}^\infty$},
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I. I. Sharapudinov. Approximation properties of Fejér- and de la Valleé-Poussin-type means for partial sums of a special series in the system $\{\sin x\sin kx\}_{k=1}^\infty$. Sbornik. Mathematics, Tome 206 (2015) no. 4, pp. 600-617. http://geodesic.mathdoc.fr/item/SM_2015_206_4_a5/

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