摘要
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 σnf → 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 σMnf → 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 limn→∞ σnf = 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＂.
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 σnf → 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 σMnf → 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 limn→∞ σnf = 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＂.
基金
the Hungarian National Foundation for Scientific Research(OTKA),Grant No.M36511/2001 and T 048780
