Cesàro means of integrable functions with respect to unbounded Vilenkin systems Original Research Article

One of the most celebrated problems in dyadic harmonic analysis is the pointwise convergence of the Fejér (or (C,1)) means of functions on unbounded Vilenkin groups. In 1999 the author proved that if f∈Lp(Gm), where p>1, then σnf→f almost everywhere. This was the very first “positive” result with respect to the a.e. convergence of the Fejér means of functions on unbounded Vilenkin groups. One of the main difficulties is that the sequence of the L1 norm of the Fejér 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 L1 case. We prove for f∈L1(Gm) that the relation σMnf→f holds a.e. (Mn is the nth generalized power).