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

BLAHOTA, I., Relation between Dirichlet kernels with respect to Vilenkin-like systems

Relation between Dirichlet kernels with respect to Vilenkin-like systems ISTVÁN BLAHOTA* Abstract. In this paper we discuss the relation between Dirrichlet-kernels with respect to Vilenkin an Vilenkin-like systems. This relation gives a useful tool in field of approximation theory on compact totally disconnected Abelian groups. Introduction Let m:=(mo,mi,...) denote a sequence of positive integers not less than 2. Denote by Z

m

j {0,1,..., rrij — 1} the additive group of integers modulo rrij (j G N). Define the group G

m as the cartesian product of the discrète cyclic groups Z

m.

, oo Gm • — X . 3=0 The elements of G

m ca

n be represented by sequences x : = (x

0 , Xi,

..., Xj,

...) (Xj G Z

m.). It easy to give a base the neighborhoods of G

m : /

0(x) G

m, In(x):={y G Gm\yo := xo,

..., y

n-\ : = } for x G G

m,n G N, k = 0,1,..., m

n — 1. Define /

n:=/

n( 0) for n G P (P := N{0}). Then I

n is a subgroup of G

m (n G N). The direct product fi of the measures Vk({j}) • = —— {jez

m

k,ke N) m

k is the Haar measure on G

m with /i(G

m) = 1. * Research supported by the Hungárián National Foundation for Scientific Research (OTKA), grant no. F007347 and the National Scientific Foundation of the Hungárián Credit Bank (Alapítvány a Magyar Felsőoktatásért és Tudományért, MHB)