Uniform convergence of double trigonometric series

It is shown that under certain conditions on ${\displaystyle \left\{c_{jk}\right\}}$, the rectangular partial sums ${\displaystyle s_{mn}(x,y)}$ converge uniformly on ${\displaystyle T^2}$. These conditions include conditions of bounded variation of order (1,0), (0,1), and (1,1) with the weights |j|, |k|, |jk|, respectively. The convergence rate is also established. Corresponding to the mentioned conditions, an analogous condition for single trigonometric series is ${\displaystyle {\sum }_{\left|k\right|=n}^{\infty }|\Delta c_k|=o\left(\frac{1}{n}\right)}$ (as n → ∞). For O-regularly varying quasimonotone sequences, we prove that it is equivalent to the condition: ${\displaystyle n{c}_n=o\left(1\right)}$ as n → ∞. As a consequence, our result generalizes those of Chaundy-Jolliffe [CJ], Jolliffe [J], Nurcombe [N], and Xie-Zhou [XZ].