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

Norm convergence of Fejér means of certain functions . . . 121 where {x^ : j G N} represents the additive digits of x. Let a

n : fi —

»C be a fonction such that each a

n is ^4

n-measurable for any n G N. A sequence {(f)

n : n G N} is said to be a UDMD system if (f)

n\=r

na

n (n G N) for some ^4

n-measurable fonctions a

n and if \<p

n{x)\ = 1 (n G N, x G H) (see [3]). The simplest UDMD system is the Rademacher system. The product system generated by the UDMD system is said to be UDMD product system. If {i/j

m : m G N} is a UDMD product system then it is orthonormal on L

2(Q). The Dirichlet kernels of the product system n G N} are defined as follows Dq(X,Î) :=0 and n-l OMeiî). j=0 for n G P- The partial sums S* f can be expressed using the Dirichlet kernels '• (s;/)(»)= / f(t)Dï(

X,t)d\(t). Ju The subsequence Dfn has a closed form. For every n G N (1) = na+ *,-(*)?*)) = {£• VtluX j=o The Fejér kernels of the product system $ are defined by Kq (x,t) := 0 and CM en) n z—' m0 for n G N. The Cesaro means of a Fourier sériés Sjf can be expressed using the Fejér kernels: K*/)(*)= f fW*(x,t)d\(t). Jn Introduce the following notation: {(f>k ®Jk)(x,t) := 4>

k(xjfa(t) (i,íeí],H N). For every n G N and x G Í2 (see [4]): n—

1 n—

1 (2) K* = 2

~

nDl + 2

3~

n II í

1 + ^ ®