Pointwise Convergence of Double Trigonometric Series

The pointwise convergence problem of the rectangular partial sums of a certain type of double trigonometric series is considered, This type of series obeys certain conditions on the finite-order differences of its coefficients. We prove that if the Césaro sums of the double series converge unrestrictedly, then so do its partial sums, It is pointed out that the converse of the last statement may not hold for the same kind of double trigonometric series. As a corollary, it is shown that the double Fourier series of the mentioned type converges unrestrictedly almost everywhere. Generalizations of the above results to the restricted case are also established. These results generalize the theorems of Chen.