Exact Constants in Telyakovskii’s Two-Sided Estimate of the Sum of a Sine Series with Convex Sequence of Coefficientsстатья
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Аннотация:It is known that the sum of the sine series $g(\mathbf b,x)=\sum_{k=1}^\infty b_k\sin kx$ whose coefficients constitute a convex sequence $\mathbf b$ is positive on the interval $(0,\pi)$. To estimate its values in a neighborhood of zero, Telyakovskii 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 the difference $g(\mathbf b,x)-(b_{m(x)}/2)\ctg(x/2)$ in a neighborhood of zero admits a two-sided estimate in terms of the function $\sigma(\mathbf b,x)$ with absolute constants. The exact values of these constants for the class of convex sequences b are obtained in this paper.