Az Eszterházy Károly Tanárképző Főiskola Tudományos Közleményei. 1997. Sectio Mathematicae. (Acta Academiae Paedagogicae Agriensis : Nova series ; Tom. 24)

GÁT, G., On the Fejér kernel functions with respect to the Walsh-Paley system

108 György Gát As a straightforward consequence of Lemma 2 we get Lemma 3. sup|

n|=

m l-fi^c.+i)^' dx < eV2

s+

t

, where m > s, í E N are fixed. Proof. If s > t, then by Lemma 2 it follows that / sup' \K

n

i.+D

i

2.(x)\dx= I 2

a+

t1dx = 2

t~

1. Jl

t\I

t+

1 \n\=m J I

3(e

t) On the other hand, if s < t, then also by Lemma 2 we have / sup \K

n(s+i) 2'{x)\dx < c c2

s+

t < JIt\I

t

+i |n|=m Jl

t\I

t+i c2

s

. Lemma 4. fj\

I

k sup|

n|

>

A \K

n(x) \ dx < eV2

k~

A

, for all A > k G N. Proof. By Lemma 1 we have n \K

n\ < I ? 5 = 0 consequently, - k — 1 p oo / sup \K

n(x)\dx <2_ \ / / S

UP IK

n(x)\dx Jl\h \n\>A t=

Q Jl

t\I

t +! m=

A |n|=m /c —

1 oo -IlYI^ sup n\Kn(x)\dx t=0m=A JI

t\It+i \n\=m k—

1 oo \ ( ^ f t=

0 m = \s-Q J

lt\lt+1 N=m + SU

P 2^(^)1 dx\ s=t+

1 ^ Wf+i |n| = m / /c —

1 oo 1 m fc —

1 oo <cvy y <<=2^ 2

? i—

0 m=A m=A The following Theorem shows that the maximal operator Tf: = sup \o

nf\ ne p