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 r ate is also established. Corresponding to the mentioned conditions, an analogous condition for single trigonometric series is (\k\=n)Sigma(i nfinity)\Delta c(k)\=o(1/n) (as n-->infinity). For O-regularly varying quasimonotone sequences, we prove that it is equivalent to the condit ion: nc(n)=o(1) as n-->infinity. As a consequence, our result generali zes those of Chaundy-Jolliffe [CJ], Jolliffe [J], Nurcombe [N], and Xi e-Zhou [XZ].