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Abstract

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 $∑_{|k|= n}^∞ |Δc_k| = o(1/n)$ (as n → ∞). For O-regularly varying quasimonotone sequences, we prove that it is equivalent to the condition: $nc_{n} = o(1)$ as n → ∞. As a consequence, our result generalizes those of Chaundy-Jolliffe [CJ], Jolliffe [J], Nurcombe [N], and Xie-Zhou [XZ].