On almost everywhere convergence and divergence of Marcinkiewicz-like means of integrable functions with respect to the two-dimensional Walsh system

摘要

Let |n| be the lower integer part of the binary logarithm of the positive integer n and α:N{double-struck}~2→N{double-struck}~2. In this paper we generalize the notion of the two dimensional Marcinkiewicz means of Fourier series of two-variable integrable functions as t_n~αf:=1σ_(k=0)~(n-1)S_α(|n|,k)f and give a kind of necessary and sufficient condition for functions in order to have the almost everywhere relation t_n~αf→f for all fεL~1([0,1)~2) with respect to the Walsh-Paley system. The original version of the Marcinkiewicz means are defined by α(|n|,k)=(k,k) and discussed by a lot of authors. See for instance [13,8,6,3,11].