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

122 Károly Nagy (3) í \K%(x,t)\d\(t)<S. Jn Let X be a Banach space with norm || ||. The space X is called a homogeneous Banach space if P Ç X Ç where P is the set of Walsh polynomials, r

xf(y):=f(y + x) and if the following three properties hold (see [1]): (0 ll/lk < 11/11 (/ € X), («') r

xfex, IMII = y/11 (xtíijex), and, for a given / G X there is a sequence of Walsh polynomials (P

n, n G N) such that (Üt) lim ||Pn - /II = o. n—+oo Define the modulus of continuity in X of an / G X by U

X(f,6):= sup II/ — Ty/II (ő> 0). |y|<í For each a > 0, Lipschitz classes in X can be defined by Lip(a,X):={/ G X: u

x{f,ô) = 0(6

a) as 6 oo}. Result s THEOREM. Suppose / G Lip(c*,X) and a > 0. Then Wnf-f || = ' 0(n~

a) 0 < a < \ as n —> oo. PROOF. Let n G P and choose s G N such that 2

s < n < 2

S+

1. Let s-1 k=

0