Almost everywhere convergence of sequences of Cesàro and Riesz means of integrable functions with respect to the multidimensional Walsh system

The aim of this paper is to prove the a.e. convergence of sequences of the Cesàro and Riesz means of the Walsh-Fourier series of d variable integrable functions. That is, let a = (a 1, ...,a d ): . . .d (d . .) such that a j (n + 1) . . sup k.n a j (k) (j = 1, ..., d, n . .) for some . > 0 and a 1(+.) = ... = a d (+.) = +.. Then, for each integrable function f . L 1(I d), we have the a.e. relation for the Cesàro means limn.. . .a(n) f = f and for the Riesz means limn.. . .,.a(n) f = f for any 0 < . j . 1 . . j (j = 1, ..., d). A straightforward consequence of our result is the so-called cone restricted a.e. convergence of the multidimensional Cesàro and Riesz means of integrable functions, which was proved earlier by Weisz.