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We prove that if c(j, k) -. 0 as max(IjI, lkl) -p oo and E E IA1c(j,k) < oo, j=-oo k=-oo then the series c7? _ c(j, k)ei(ij+ky) converges both pointwise for every (x,y) E (T\{0})2 and in the LP(T2)-metric for 0 < p < 1, where T is the one-dimensional torus. Both convergence statements remain valid for the three conjugate series under these same coefficient conditions.

Convergence and integrability of double trigonometric series with coefficients of bounded variation