On the uniform convergence of double sine series

Péter Kórus ¹   ; Ferenc Móricz ¹

¹ University of Szeged Bolyai Institute Aradi vértanúk tere 1 6720 Szeged, Hungary

Studia Mathematica, Tome 193 (2009) no. 1, pp. 79-97

Résumé

Let a single sine series ($*$) $\sum^\infty_{k=1} a_k \sin kx$ 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 $ka_k\to 0$ as $k\to \infty$. 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 $(**)$ $\sum^\infty_{k=1} \sum^\infty_{ l =1} c_{k l }$ $\sin kx \sin l y$, 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.

Détail

DOI : 10.4064/sm193-1-4

Keywords: single sine series * sum infty sin given nonnegative coefficients mean value bounded variation sequence briefly mvbvs necessary sufficient condition uniform convergence series * infty class mvbvs includes sequences monotonically decreasing zero these results due zhou zhou paper extend single double sine series ** sum infty sum infty sin sin even complex coefficients uniform boundedness test rectangular partial sums series nbsp ** slightly improve results single sine series