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Fourier serieses with respect to Vilenkin multiplicative systems are studied. It is proved that a Dini-Lipschitz condition imposed on the group or group integral modulus of continuity is sufficient for the convergence of the series in the normC() orL(), respectively. Necessity in the classesHω andHω1 of that condition is also proved. A boundedness criterion of the sequence of partial sums inHω andHω1 is also obtained.

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