Convergence Almost Everywhere of Partial Sums and Féjer Means of Vilenkin–Fourier Series

We characterize subsequences \(\{S_{n_k}\}\) $\{S_{n_k}\}$ of partial sums with respect to (bounded or unbounded) Vilenkin systems of \(f\in L^1(G_m)\) $f\in L^1(G_m)$ for which almost everywhere convergence holds. Moreover, we construct an explicit \(f\in L^p(G_m), \ 1\le p<\infty \) $f\in L^p(G_m),1\le p<\infty$ whose partial sums (satisfying the same conditions which guarantee almost everywhere convergence) diverges on any set of measure zero. We also prove a similar divergence result for Vilenkin–Féjer means.