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 pa...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＂.展开更多
The aim of this paper is to prove the a.e.convergence of sequences of the Cesaro and Riesz means of the Walsh–Fourier series of d variable integrable functions.That is,let a=（a1,...,ad）：N→Nd（d∈P）such that aj（...The aim of this paper is to prove the a.e.convergence of sequences of the Cesaro and Riesz means of the Walsh–Fourier series of d variable integrable functions.That is,let a=（a1,...,ad）：N→Nd（d∈P）such that aj（n＋1）≥δsupk≤n aj（k）（j=1,...,d,n∈N）for someδ〉0 and a1（＋∞）=···=ad（＋∞）=＋∞.Then,for each integrable function f∈L1（Id）,we have the a.e.relation for the Cesaro means limn→∞σαa（n）f=f and for the Riesz means limn→∞σα,γa（n）f=f for any 0〈αj≤1≤γj（j=1,...,d）.A straightforward consequence of our result is the so-called cone restricted a.e.convergence of the multidimensional Cesaro and Riesz means of integrable functions,which was proved earlier by Weisz.展开更多
For the two-dimensional Walsh system, Gat and Weisz proved the a.e. convergence of Fejer means σnf of integrable functions, where the set of indices is inside a positive cone around the identical function, that is, ...For the two-dimensional Walsh system, Gat and Weisz proved the a.e. convergence of Fejer means σnf of integrable functions, where the set of indices is inside a positive cone around the identical function, that is, β^-1≤n1/n2 ≤β is provided with some fixed parameter ~ 〉 1. In this paper we generalize the result of Gat and Weisz. We not only generalize this theorem, but give a necessary and sufficient condition for cone-like sets in order to preserve this convergence property.展开更多
基金the Hungarian National Foundation for Scientific Research(OTKA),Grant No.M36511/2001 and T 048780
文摘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Supported by project TMOP-4.2.2.A-11/1/KONV-2012-0051
文摘The aim of this paper is to prove the a.e.convergence of sequences of the Cesaro and Riesz means of the Walsh–Fourier series of d variable integrable functions.That is,let a=（a1,...,ad）：N→Nd（d∈P）such that aj（n＋1）≥δsupk≤n aj（k）（j=1,...,d,n∈N）for someδ〉0 and a1（＋∞）=···=ad（＋∞）=＋∞.Then,for each integrable function f∈L1（Id）,we have the a.e.relation for the Cesaro means limn→∞σαa（n）f=f and for the Riesz means limn→∞σα,γa（n）f=f for any 0〈αj≤1≤γj（j=1,...,d）.A straightforward consequence of our result is the so-called cone restricted a.e.convergence of the multidimensional Cesaro and Riesz means of integrable functions,which was proved earlier by Weisz.
基金Supported by the Scientific Board of College of Nyiregyhaza
文摘For the two-dimensional Walsh system, Gat and Weisz proved the a.e. convergence of Fejer means σnf of integrable functions, where the set of indices is inside a positive cone around the identical function, that is, β^-1≤n1/n2 ≤β is provided with some fixed parameter ~ 〉 1. In this paper we generalize the result of Gat and Weisz. We not only generalize this theorem, but give a necessary and sufficient condition for cone-like sets in order to preserve this convergence property.
