On the convergence of Fejér means of some subsequences of partial sums of Walsh–Fourier
series

Abstract. The aim of this paper is to improve a result of the author (see [7]). That is, to prove some a.e. convergence results of Fejér means of subsequences of partial sums of Walsh–Fourier series of integrable functions. We prove for sequences \(a\) satisfying the condition \(a(n+1)\ge \left(1 + \frac{1}{(n+1)^{\delta}}\right)a(n)\) for some \(0 < \delta < 1/2\) for every \(n\in \mathbb{N}\), that the \((C,1)\) means of the partial sums \(S_{a(n)}f\) converge to \(f\) a.e. Thus, this a.e. convergence result also holds for lacunary sequences \(a\) and this was earlier verified in [7].