Geodesic
Exact Constants in Telyakovskii's Two-Sided Estimate
Matematičeskie zametki, Tome 107 (2020) no. 6, pp. 906-921

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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 $$ \sigma(\mathbf b,x)=\frac1{m(x)}\sum_{k=1}^{m(x)-1}k^2(b_k-b_{k+1}),\qquad m(x)=\biggl[\frac\pi x\biggr]. $$ He showed that the difference $g(\mathbf b,x)-(b_{m(x)}/2)\operatorname{cot}(x/2)$ 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 $\mathbf b$ are obtained in this paper.
Keywords: sine series with monotone coefficients, convex sequence, slowly varying sequence. 
@article{MZM_2020_107_6_a10,
     author = {A. P. Solodov},
     title = {Exact {Constants} in {Telyakovskii's} {Two-Sided} {Estimate}},
     journal = {Matemati\v{c}eskie zametki},
     pages = {906--921},
     publisher = {mathdoc},
     volume = {107},
     number = {6},
     year = {2020},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/MZM_2020_107_6_a10/}
}
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A. P. Solodov. Exact Constants in Telyakovskii's Two-Sided Estimate. Matematičeskie zametki, Tome 107 (2020) no. 6, pp. 906-921. http://geodesic.mathdoc.fr/item/MZM_2020_107_6_a10/