Sharp Two-Sided Estimate for the Sum of a Sine Series with Convex Slowly Varying Sequence of Coefficients

Abstract

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.