PUBLICATIONS DE L'INSTITUT MATHÉMATIQUE (BEOGRAD) (N.S.)Vol. 58(72), pp. 43--50 (1995)
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PUBLICATIONS DE L'INSTITUT MATHÉMATIQUE (BEOGRAD) (N.S.) Vol. 58(72), pp. 43--50 (1995) |
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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, GSP-1, 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:
Electronic fulltext finalized on: 1 Nov 2001. This page was last modified: 16 Nov 2001.
© 2001 Mathematical Institute of the Serbian Academy of Science and Arts
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