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

On the Fejér kernel functions with respect to the Walsli-Paley system 109 is quasi-local. The conception of quasi-locality is introduced by F. Schipp [8]. Let / E L

l{I)i supp / C Ik(%°) for some k E N,x° E I and suppose that the integral of Tf on the set I \ Ik{x°) is bounded by c |f\

l. Then we call T quasi-local. That is, we prove Theorem 5. /

A4(x0

)T/ < c|/|

r Proof. If n < 2

k

, then f(n) = f

I fu

n = J

h(x0

) fu

n = w

n(2°)/

ii(j0

) / = 0, thus S

nf = 0,cr

n/ = 0. That is, we have Tf = sup

n>

2Jt |c

n/|. By Lemma 4 it follows / sup / f(x)K

n(x + y)dx dy J I\I

h(x°) n>2

k J I

k(x°) < / |/(z)| / sup |A'

n(a; + ?/)d?/| t/a: jI

k(x°) Jl\Ik(x°) n>2

k = / l/MI / sup |An(y)öíy| ^ < c l/li • • J r

k(x°) Jl\I

k n>2

h Define the Hardy space 77 as follows. Let /* :=sup

nG

N 1

"/1 be the maximal function of the integrable function / E L

l(I). Then, #(/):={/

el

1 (/):/- eL\l)}, moreover 77 is a Banach space endowed with the norm \f\jj \f*h- By standard argument (see e.g. [8]) and by the help of Theorem 5 one can prove that the operator T is of type (77, L

1) which means that \Tf\

1 < c \f\

H for all / E 77. This result with respect to the Walsh system is due to Schipp [7] and Fujii [2]. With respect to bounded Vilenkin system it is proved by Simon [6]. The noncommutative case is discussed by the author ([4]). Also by standard argument (see e.g. [8]) and by the help of Theorem 5 we have that for all / E 7>

1(7) the almost everywhere convergence cr

nf —> / (n —oo, / E 7y

l(I)) holds. This result with respect to the Walsh system is due to Fine [1]. With respect to bounded Vilenkin systems it is proved by Pál and Simon [5]. The so-called 2-adic integers and the noncommutative case are discussed by the author ([3], [4]). References [1] FINE, N. J., Cesáro summability of Walsh-Fourier series, Proc. Nat. Acad. Sei. U.S.A. 41 (1955), 558-591.