Uniform convergence of double trigonometric series

It is shown that under certain conditions on {c(jk)}, the rectangular partial sums s(mn) (x, y) converge uniformly on 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 (\k\=n)Sigma(infinity)\Delta c(k)\=o(1/n) (as n-->infinity). For O-regularly varying quasimonotone sequences, we prove that it is equivalent to the condition: nc(n)=o(1) as n-->infinity. As a consequence, our result generalizes those of Chaundy-Jolliffe [CJ], Jolliffe [J], Nurcombe [N], and Xie-Zhou [XZ].