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 . . . 123 First of all we will show that Ik2'/-/|| = 0(A(s)), as 5 > oo. Since [ K%(x,t)d\{t) Jn „ =1 Ju we have <r*f(x)-f(x) = f f(t)K*(x,t)d\(t)-f(x) = [ (f(t)-f(x))K*(x,t)d\(t) JQ JU for any / G and any x G fi. For any t G I

s{x) we have \K*(x,t)\ <2

S"

1. A disjoint décomposition of fi is /s

—1 « = /.(*) IJ U I

k(x)\i

k+

l{x) \k=0 for any x G fi. Let (a:) be denoted by The following inequality holds for any x G fi and any / G Lip(a,X): !<£/(*) - /(*)l < J l/W - f(x)\\K*(x,t)\d\(t) < j \f(t)-f(x)\\K*(x,t)\d\{t)+ Is{x) + Ë / i/w-/(®)ii^i(«,oidA(<)< 2

51 j \m - f(x)\d\(t)+ /.(*) 5-1 /c=0 /x Lk s —

1 s —

1 » s —

1 x; E

2175 / i/(o - /mi n i

1+< fc = 0 7=0 rx '=0 ;=U j--