Almost Everywhere Convergence of Fejér Means of L 1 Functions on Rarely Unbounded Vilenkin Groups.

A highly celebrated problem in dyadic harmonic analysis is the pointwise convergence of the Fejér (or ( C, 1)) means of functions on unbounded Vilenkin groups. There are several papers of the author of this paper concerning this. That is, we know the a.e. convergence σ _n f → f ( n → ∞) for functions f ∈ L ^p , where p > 1 ( Journal of Approximation Theory, 101(1), 1–36, (1999)) and also the a.e. convergence σ M _n f → f ( n → ∞) for functions f ∈ L ¹ ( Journal of Approximation Theory, 124(1), 25–43, (2003)). The aim of this paper is to prove the a.e. relation lim _n → σ _n f = f for each integrable function f on any rarely unbounded Vilenkin group. The concept of the rarely unbounded Vilenkin group is discussed in the paper. Basically, it means that the generating sequence m may be an unbounded one, but its "big elements" are not "too dense". [ABSTRACT FROM AUTHOR]