Almost everywhere convergence of a subsequence of logarithmic means of Fourier series on the group of 2-adic integers

In this paper we prove the almost everywhere convergence of a special subsequence of the logarithmic means of integrable functions, I-mn f := 1/l(mn) Sigma(mn-1)(k=1) S-k f/m(n) - f -> f for every f is an element of L-1(I), where l(n) := and Sigma(n-1)(k=1) 1/k I is the group of 2-adic integers. We suppose that Sigma(infinity)(n=1) log2(m(n) - 2([log mn]) + 1)/log m(n) < infinity. It proves that t(2n) f(x) -> f(x) a.e. as n -> infinity for every f is an element of L-1(I), too.