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Let {c(n)} be a complex null sequence such that for some integer m > 1 and some p £ ( 1, 2] oo J2 \Amc(n)\p < co and JZ |A(c(n) c(-«))| lgn< oo. |rt|<oo n=\ It is shown that the series ( * ) J2 c(n)e"", teT = |n|<oo 2ttZ converges a.e. and that the well-known condition Cw of J. W. Garrett and C. V. Stanojevic [4, 3] implies that the series (*) is the Fourier series of its sum. This generalizes results of W. O. Bray and C. V. Stanojevic [1]. An important consequence of the main result is that nAc(n) = 0(1), |n| —» oo, implies that the condition Cw is equivalent to the de la Vallée Poussin summability of partial sums (S„(c)} as conjectured in [8].

Convergence and integrability of trigonometric series with coefficients of bounded variation of order (𝑚,𝑝)