The sum of a sine series $$g\left({b,x} \right) = \sum\nolimits_{k = 1}^\infty {}$$ bk sin kx with coefficients forming a convex sequence b is known to be positive on the interval (0,π). To estimate its values near zero Telyakovskiĭ used the piecewise-continuous function $$\sigma \left({{\bf{b}},x} \right) = \left({1/m\left(x \right)} \right)\sum\nolimits_{k = 1}^{m\left(x \right) - 1} {{k^2}\left({{b_k} - {b_{k + 1}}} \right),\,\,m\left(x \right) = \left[{\pi /x} \right]}$$ . He showed that in some neighborhood of zero the difference g(b,x) − (bm(x)/2) cot(x/2) can be estimated from both sides two-sided in terms of the function σ(b,x) with absolute constants. In the present paper, sharp values of these constants on the class of convex slowly varying sequences b are found. A sharp two-sided estimate for the sum of a sine series on this class is obtained. Examples that demonstrate good accuracy of the obtained two-sided estimates are given.

中文翻译：

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具有凸缓变系数序列的正弦级数之和的尖锐两侧估计

正弦序列 $$g\left({b,x} \right) = \sum\nolimits_{k = 1}^\infty {}$$ bk sin kx 的和与系数形成凸序列 b 已知为在区间 (0,π) 上为正。为了估计其接近零的值 Telyakovskiĭ 使用分段连续函数 $$\sigma \left({{\bf{b}},x} \right) = \left({1/m\left(x \right)} \right)\sum\nolimits_{k = 1}^{m\left(x \right) - 1} {{k^2}\left({{b_k} - {b_{k + 1}}} \right ),\,\,m\left(x \right) = \left[{\pi /x} \right]}$$ 。他表明，在零的某个邻域中，可以根据函数 σ(b,x) 从两侧两侧估计差 g(b,x) − (bm(x)/2) cot(x/2) ) 与绝对常数。在本文中，发现了这些常数在凸慢变序列 b 类上的尖锐值。获得了对此类的正弦序列之和的尖锐两侧估计。给出了证明所获得的两侧估计具有良好准确性的示例。