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Almost everywhere convergence of a subsequence of the logarithmic means of Vilenkin-Fourier series

The main aim of this paper is to prove that the maximal operator of a subsequence of the (one-dimensional) logarithmic means of Vilenkin-Fourier series is of weak type (1, 1). Moreover, we prove that the maximal operator of the logarithmic means of quadratical partial sums of double Vilenkin-Fourier series is of weak type (1, 1), provided that the supremum in the maximal operator is taken over special indices. The set of Vilenkin polynomials is dense in L1, so by the well-known density argument the logarithmic means t2n(f) converge a.e. to f for all integrable function f. .