Exact Constants in Telyakovskii’s Two-Sided Estimate of the Sum of a Sine Series with Convex Sequence of Coefficients

Abstract

It is known that the sum of the sine series $g\left( {{\rm{b,}}\,x} \right) = \sum\nolimits_{k = 1}^\infty {{b_k}}$b_k sin kx whose coefficients constitute a convex sequence b is positive on the interval (0, π). To estimate its values in a neighborhood of zero, Telyakovskii used the piecewise continuous function

$$\sigma \left( {{\mathop{\rm b}\nolimits} ,\,x} \right) = {1 \over {m\left( x \right)}}\sum\limits_{k = 1}^{m\left( x \right) - 1} {{k^2}\left( {{b_k} - {b_{k + 1}}} \right),\,\,\,\,\,\,\,\,\,\,\,m\left( x \right) = \left[ {{{\rm{\pi }} \over x}} \right].} $$

He showed that the difference g(b, x) − (b_m(_x)/2)cot(x/2) in a neighborhood of zero admits a two-sided estimate in terms of the function a(b,x) with absolute constants. The exact values of these constants for the class of convex sequences b are obtained in this paper.