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There are some gaps between the conditions for the sequences of coefficients {a,} in these two theorems. In ?2, we shall prove that the conditions for {an } in Theorem B can be replaced by those in Theorem A. On the other hand Reves and Szasz [2 ] have proved the analogues of Cantor-Lebesgue's theorem and Denjoy-Lusin's theorem for the double trigonometric series. In ?3, we shall generalize these theorems: the former is on the line of Mazur-Orlicz' generalization for CantorLebesgue's theorem [1, Theorem 1 and its corollary] and the latter is in the direction of Salem's generalization for Denjoy-Lusin's theorem [3, Theorem X].

Notes on Fourier analysis. XXX. On the absolute convergence of certain series of functions