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

124 Károly Nagy s-l , s-l & p o — 1 2'12'w*(/, 2~

s) + £ / |/(í) - /(x)| f] |lf /C=0 rx '=0 'S S-l k=0 /

s —1 s —1 n ii+M*WIMM < 2s) + rx <=° fc=0 Lk From this inequality we have s-l 114/ - /II < "*(/, 2"

s) + E 22~

k) = 0(A(s)) as s - oo. k=0 for any / 6 Lip(o;,X). We have used the following result: T X ( =0 Lk l^k S-l To prove this, let /) (1-|-(J)i(x)4>i(t)) and suppose for a moment 1=0 lïk that (4) ^jf Jtfi-iOM)l

2dA(t) < V*. Using the Cauchy-Buniakovski inequahties we have / < ^) // iff^oi'dAío ^k V * Now we will prove (4) (see [4]): /

. s —

1 s —

1 r"í '=0 J=0 'Jfc