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

NAGY, K., Norm convergenc e of Fejér means of certain functions with respect to respect to UDMD product systems

120 Károly Nagy The Fourier sériés of / with respect to the system is the series oo m=0 The partial sums of order n of the Fourier series of / are defined by n-l Snf''= Xi f,Í>m}Í>m 771 = 0 for n G P The Cesaro means of order n of the Fourier series of an / are defined by £ Si (nG P). ra=1 Denote the dyadic, or 2-series, group by (G,+). Thus G consists of sequences x : = j G N) with x^ = 0, or 1 and addition + is coordinatewise, modulo 2. Let ü = [0,1) or G. By the additive digits G N) of an x G fi we shall mean the coordinates of x = (x^

0),^

1),...) if x G G and the binary coefficients of x = x ^ x ^ [0,1 ), where the finite binary expansion of x is used when x is a dyadic rational. Let A be the Lebesgue measure when fi = [0,1) and Haar measure when Í2 = G. Denote the corresponding Lebesgue spaces by L

p(Cl) for 0 < p < oo. By a dyadic interval of rank n in íi = [0,1) we mean an interval of the form ^Sr^) wher

e 0 < p < 2

n and n G N. Given a G [0,1) and n G N, there is one and only one interval of rank n which contains a. Let it be denoted by I

n{.

a)- By a dyadic interval of rank n centered at a G 0 = G we mean a set of the form I

n(a) = {x G G: x

W = a

{

k\k < n}. Denote the cr-algebra generated by the intervais I

n(a) (a G O) by A

n. The intervals I

n(a) (

a G fi) are called the atoms of A

n. Each element of A

n is a finite disjoint union of atoms. A function / defined on Í2 is said to be ^4

n-measurable if / is constant on the atoms of A

n. Let L(A

n ) denote the set of ^4

n-measurable functions on fi. Each / G L(A

n) is integrable. Let the Rademacher system on Q be denoted by {r

n : n G N}, that is r

n(x) = (-ir

(n

) (ne N)