Sharp Two-Sided Estimate for the Sum of a Sine Series with Convex Slowly Varying Sequence of Coefficientsстатья
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Аннотация:The sum of a sine series $g(\mathbf b,x)=\sum_{k=1}^\infty b_k\sin kx$ with coefficients forming a convex sequence $\mathbf b$ is known to be positive on the interval $(0,\pi)$. To estimate its values near zero Telyakovski\u{\i} used the piecewise-continuous function $\sigma(\mathbf b,x)=\bigl(1/m(x)\bigr)\sum_{k=1}^{m(x)-1}k^2(b_k-b_{k+1})$, $m(x)=[\pi/x]$. He showed that in some neighborhood of zero the difference $g(\mathbf b,x)-(b_{m(x)}/2)\cot(x/2)$ can be estimated from both sides two-sided in terms of the function $\sigma(\mathbf b,x)$ with absolute constants. In the present paper, sharp values of these constants on the class of convex slowly varying sequences $\mathbf b$ 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.