Az Eszterházy Károly Tanárképző Főiskola Tudományos Közleményei. 1998. Sectio Mathematicae. (Acta Academiae Paedagogicae Agriensis : Nova series ; Tom. 25)
GÁT, G., On a theorem of type Hardy-Littlewood with respect to the Vilenkin-like systems
On a theorem of type Hardy-Littlewood with respect to the Vilenkin-like systems GYÖRGY GÁT Abstract. In this paper we give a convergence test for generalized (by the author) Vilenkin--Fourier series. This convergence theorem is of type Hardy-Littlewood for the ordinary Vilenkin system is proved in 1954 by Yano. Introduction and the result First we introduce some necessary definitions and notations of the theory of the Vilenkin systems. The Vilenkin systems were introduced by N. J A . VILENKIN in 1947 (see e.g. [7]). Let m: = (m k,k E N) (N := {0,1,...}) be a sequence of integers each of them not less than 2. Let Z m k denote the m.k-th discrete cyclic group. Z m k can be represented by the set {0,. ... — 1}, where the group operation is the mod addition and every subset is open. The measure on Z m k is defined such that the measure of every singleton is l/m k (k E N). Let oo G rn • — X Z fn , . £=0 This gives that every x G G m can be represented by a sequence x — (a;,-, i G N), where X{ E Z m t (i G N). The group operation on G m (denoted by +) is the coordinate-wise addition (the inverse operation is denoted by —), the measure (denoted by fi) and the topology are the product measure and topology. Consequently, G m is a compact Abelian group. If sup n6 N m n < oo, then we call G m a bounded Vilenkin group. If the generating sequence m is not bounded, then G m is said to be an unbounded Vilenkin group. The boundedness of the group G m is supposed over all of this paper and denote by sup ne N m n < oo. c denotes an absolute constant (may depend only on sup n m n) which may not be the same at different occurences. A base for the neighborhoods of G m can be given as follows Io(x) : = G m, I n(x):= {y = (yi,i G N) e G m : y l = x { for i < n } Research supported by the Hungarian National Research Science Foundation, Operating Grant Number OTKA F020334.
