On the uniform convergence of double sine series

Péter Kórus; Ferenc Móricz

Studia Mathematica, Tome 192 (2009), p. 79-97 / Harvested from The Polish Digital Mathematics Library

Résumé

Let a single sine series (*) $\sum_{k = 1}^{\infty} a_{k} s i n k x$ be given with nonnegative coefficients $a_{k}$ . If $a_{k}$ is a “mean value bounded variation sequence” (briefly, MVBVS), then a necessary and sufficient condition for the uniform convergence of series (*) is that $k a_{k} \to 0$ as k → ∞. The class MVBVS includes all sequences monotonically decreasing to zero. These results are due to S. P. Zhou, P. Zhou and D. S. Yu. In this paper we extend them from single to double sine series (**) $\sum_{k = 1}^{\infty} \sum_{l = 1}^{\infty} c_{k l} s i n k x s i n l y$ , even with complex coefficients $c_{k l}$ . We also give a uniform boundedness test for the rectangular partial sums of series (**), and slightly improve the results on single sine series.

Publié le : 2009-01-01

Zbl 1167.42002

EUDML-ID : urn:eudml:doc:286433

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     title = {On the uniform convergence of double sine series},
     journal = {Studia Mathematica},
     volume = {192},
     year = {2009},
     pages = {79-97},
     zbl = {1167.42002},
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Péter Kórus; Ferenc Móricz. On the uniform convergence of double sine series. Studia Mathematica, Tome 192 (2009) pp. 79-97. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-sm193-1-4/