Asymptotics of the Sum of a Sine Series with a Convex Slowly Varying Sequence of Coefficients

We study the asymptotic behavior in a neighborhood of zero of the sum of a sine series g ( b , x ) = ∑ k = 1 ∞ b k sin k x whose coefficients constitute a convex slowly varying sequence b . The main term of the asymptotics of the sum of such a series was obtained by Aljančić, Bojanić, and Tomić. To estimate the deviation of g ( b , x ) from the main term of its asymptotics b m ( x ) / x , m ( x ) = [ π / x ] , Telyakovskiĭ used the piecewise-continuous function σ ( b , x ) = x ∑ k = 1 m ( x ) − 1 k 2 ( b k − b k + 1 ) . He showed that the difference g ( b , x ) − b m ( x ) / x in some neighborhood of zero admits a two-sided estimate in terms of the function σ ( b , x ) with absolute constants independent of b . Earlier, the author found the sharp values of these constants. In the present paper, the asymptotics of the function g ( b , x ) on the class of convex slowly varying sequences in the regular case is obtained.