PUBLICATIONS DE L'INSTITUT MATHÉMATIQUE (BEOGRAD) (N.S.) Vol. 58(72), pp. 4350 (1995) 

On the behaviour near the origin of sine series with convex coefficientsS.A. Telyakovski\u\iSteklov Mathematical Institute of the Russian Academy of Sciences, Vavilov str. 42, Moscow 117966, GSP1, RussiaAbstract: Let a numerical sequence $\{a_k\}$ tend to zero and be convex. We obtain estimates of $$ g(x) := \sum_{k=1}^{\infty} a_k \sin kx $$ for $x\,\to\,0$ expressed in terms of the coefficients $a_k$. These estimates are of order or asymptotic character. For example, the following order equality is true: $$ g(x) \sim ma_m + \frac{1}{m} \sum_{k = 1}^{m  1} k a_k, $$ where $$ x \in \left ({\frac {\pi}{m+1}, \frac {\pi}{m}} \right ]. $$ Classification (MSC2000): 42A32 Full text of the article:
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