Pointwise Convergence of Multiple Trigonometric Series

This is devoted to the study of pointwise convergence of the rectangular partial sums of multiple trigonometric series with coefficients satisfying certain conditions involving finite-order differences. Three types of such conditions are posed. Convergence may be understood both in the unrestricted sense and in the restricted sense. We prove that if the first arithmetic mean of an n-fold trigonometric series of the above type converges, then so does its rectangular partial sum. As a corollary, we obtain that any multiple Fourier series of the above type converges restrictedly almost everywhere. These results extend those of Chen, Chen and Hsieh, Móricz, and Móricz and Waterman.