On the uniform convergence of double sine series
Volume 193 / 2009
Abstract
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.
