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

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.
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