Cesaro means of integrable functions with respect to unbounded Vilenkin systems

One of the most celebrated problems in dyadic harmonic analysis is the pointwise convergence of the Fejer (or (C, 1)) means of functions on unbounded Vilenkin groups. In 1999 the author proved that if f is an element of L-p (G(m)), where p > 1, then sigma(n)f --> f almost everywhere. This was the very first positive result with respect to the a.e. convergence of the Fejer means of functions on unbounded Vilenkin groups. One of the main difficulties is that the sequence of the L-1 norm of the Fejer kernels is not bounded. This is a sharp contrast between the unbounded and the bounded Vilenkin systems. The aim of this paper is to discuss the L-1 case. We prove for f is an element of L-1 (G(m)) that the relation sigma(Mn)f --> f holds a.e. (M-n is the nth generalized power). (C) 2003 Elsevier Inc. All rights reserved.