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

Aleksei Solodov

doi:10.3390/math9182252

Abstract

We study the asymptotic behavior in a neighborhood of zero of the sum of a sine series g(b,x)=∑k=1∞bksinkx 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 bm(x)/x, m(x)=[π/x], Telyakovskiĭ used the piecewise-continuous function σ(b,x)=x∑k=1m(x)−1k2(bk−bk+1). He showed that the difference g(b,x)−bm(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.

Highlights

We refine the asymptotics of the sum of a sine series with convex slowly varying coefficients, obtained by Aljancic, Bojanic, and Tomic [1] and strengthened by Telyakovskiı [2,3]
In [4,5], results reducing the asymptotic behavior of trigonometric series with general monotone coefficients to ones with monotone coefficients were obtained
Note that it is possible to improve the asymptotics for the last case using the results of the present paper

Summary

Introduction

We refine the asymptotics of the sum of a sine series with convex slowly varying coefficients, obtained by Aljancic, Bojanic, and Tomic [1] and strengthened by Telyakovskiı [2,3].The result of Aljancic, Bojanic, and Tomicwas generalized for more extensive classes of trigonometric series. Asymptotics of the Sum of a Sine Series with a Convex Slowly Varying Sequence of Coefficients. We refine the asymptotics of the sum of a sine series with convex slowly varying coefficients, obtained by Aljancic, Bojanic, and Tomic [1] and strengthened by Telyakovskiı [2,3]. A sequence {βk}∞k=1 is called slowly varying (see [13,14]) if: lim β[δk] = 1 (1)

Results

Conclusion