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 Walsh—Paley system GYÖRGY GÁT* Abstract. In this paper we prove some lemmas with respect to the Fejér kernels of the Walsh-Paley system. This lemmas give a new proof for the known a.e. convergence <rnf-+f (n->oo, feL

1). Let P denote the set of positive integers, N: = PU {0} and /: = [0,1) the unit interval. Denote the Lebesgue measure of any set E C / by \E\. Denote the L

P(I

) norm of any function / by ||/||

p (1 < p < oo). Denote the dyadic expansion of n £ N and x £ / by n = X^jlo 311 x — Sj^o (i

n the cas

e of x = k,m £ N choose the expansion which terminates in zeros (these numbers are the dyadic rationals)). n

z-, X{ are the i-th coordinates of n, x, respectively. Define the dyadic addition + as oo x + y — + Vj mod2)2~

J~

1

. j=o The sets In(x) •= {y e I

: yo = X

0, . . . yVn-1 = x

n-i} for x £ I, I

n: = I

n{

0) for n £ P and f

0(x) : = I are the dyadic intervalls of I. Set e

n := (0,. . ., 0,1, 0,. . .) where the n-th coordinate of e

n is 1 the rest are zeros for all n £ N. The dyadic rationals are the finite 0,1 combinations of the elements of the set {e

n : n £ N} (which dense in I). Let (u)

n,n £ N) represent the Walsh-Paley system ([2], [8]) that is, oo «„(®) = ni1)

7

1^' riEN, xel. k=0 Denote by D

n := Yl^Zo ^fc» Walsh-Dirichlet kernels. * Research supported by the Hungarian National Research Science Foundation, Operating Grant Number OTKA T 020334.