Geodesic
On the behaviour near the origin of sine series with convex coefficients
Publications de l'Institut Mathématique, _N_S_58 (1995) no. 72, p. 43 Cet article a éte moissonné depuis la source eLibrary of Mathematical Institute of the Serbian Academy of Sciences and Arts

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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 łeft ({\frac {\pi}{m+1}, \frac {\pi}{m}} \right ]. $
Classification : 42A32
Keywords: sine series 
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     author = {S.A. Telyakovskii},
     title = {On the behaviour near the origin of sine series with convex coefficients},
     journal = {Publications de l'Institut Math\'ematique},
     pages = {43 },
     year = {1995},
     volume = {_N_S_58},
     number = {72},
     zbl = {0945.42004},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/PIM_1995_N_S_58_72_a5/}
}
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S.A. Telyakovskii. On the behaviour near the origin of sine series with convex coefficients. Publications de l'Institut Mathématique, _N_S_58 (1995) no. 72, p. 43 . http://geodesic.mathdoc.fr/item/PIM_1995_N_S_58_72_a5/