Az Eszterházy Károly Tanárképző Főiskola Tudományos Közleményei. 1994. Sectio Mathematicae. (Acta Academiae Paedagogicae Agriensis : Nova series ; Tom. 22)

GÁT, G., On a norm convergence theorem with respect to the Vilenkin system in the Hardy spaces

On a norm convergence theorem with respect to the Vilenkin system in the Hardy spaces GYÖRGY GÁT Abstract. In 1993. the author proved the lim X]fc=l \\Skf\\l/k = ||/||i convergence for functions / in H(G

m) (the so-called "atomic" Hardy space) with respect to ail G

m Vilenkin group. In this paper we prove that this theorem fails to hold in the case of the so-called unbounded Vilenkin group and the "maximal" Hardy space. Introduction and Resuit s First we introduce some necessary définitions and notations of the theory of the Vilenkin systems. The Vilenkin systems were introduced by N. Ja. Vilenkin in 1947 (see e.g. [8]). Let m := (m

k,k G 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 discrète cyclic group. Z

m

k can be represented by the set {0,1,. .., mjt — 1}, where the group opération is the mod m

k addition and every subset is open. The measure on Z

m

k is defined such that the measure of every singleton is € N). Let OO Gm — X . k=

0 This gives that every x G G

m can be represented by a sequence x = (x;, i G N), where X{ G Z

m

i,(i G N). The group opération on G

m (denoted by +) is the coordinate-wise addition (the inverse opération is denoted by — ), the measure (denoted by /x) and the topology are the product measure and topology. Consequently, G

m is a compact Abelian group. If sup

ne

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. A base for the neighborhoods of G

m can be given as follows I

0(x):=G

m, I

n(x):={y = (y

{,i G N) G G

m : y

l = xjoii < n} Research supported by the Hungárián National Foundation for Scientific Research (OTKA), grant no. F007347.