Almost everywhere convergence of sequences of
two-dimensional Vilenkin–Fejér means of integrable functions

Abstract. The aim of this paper is to prove the a.e. convergence of sequences of the Fejér means of the Vilenkin–Fourier series of two variable integrable functions on two dimensional bounded Vilenkin groups. That is, let \(a = (a_1, a_2): \mathbb N \to \mathbb N^2\) such that \(a_j(n+1)\ge\delta \sup_{k\le n}a_j(n)\) \((j=1,2, n\in\mathbb N)\) for some \(\delta>0\) and \(a_1(+\infty)=a_2(+\infty)=+\infty\). Then for each integrable function \(f\in L^1(G_m^2)\) we have the a.e. relation \(\lim_{n\to\infty}\sigma_{a_1(n), a_2(n)}f = f\). It will be a straightforward and easy consequence of this result the cone restricted a.e. convergence of the two-dimensional
Vilenkin–Fejér means of integrable functions which was proved earlier by Weisz [13] and Blahota and the author [2] independently. The trigonometric and Walsh's analogue of the main result see Gát [6], [5].