On the absolute convergence of multiple fourier series

Let f : R-N C be a periodic function with period 2 pi in each variable. We prove sufficient conditions for the absolute convergence of the multiple Fourier series of f in terms of moduli of continuity, of bounded variation in the sense of Vitali or Hardy and Krause, and of the mixed partial derivative ill case f is an absolutely continuous function. Our results extend the classical theorems of Bernstein and Zygmund from single to multiple Fourier series.