Asymptotics of Sums of Sine Series with Fractional Monotonicity Coefficients

Abstract

We study the following question: which monotonicity order implies upper and lower estimates of the sum of a sine series \(g\left( {{\boldsymbol{b}},x} \right) = \sum\nolimits_{k = 1}^\infty {{b_k}} \) sin kx near zero in terms of the function \(v\left( {{\boldsymbol{b}},x} \right) = x\sum\nolimits_{k = 1}^{\left[ {\pi /x} \right]} {k{b_k}} \). Our results complete, on a qualitative level, the studies began by R. Salem and continued by S. Izumi, S. A. Telyakovskiĭ and A. Yu. Popov.