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
106 György Gát It is well-known that ([2], [8]) Snf(y) = J I{x)D n(y + x)dx = f * D n{y) (y G J, n 6 P) the n-th partial sum of the Walsh-Fourier series. Moreover, ([8], p. 28.) (1) D 2n{x): = 2 n, if xein, 0, otherwise, (2) D n{x) =co n(x)^n k(D 2k+i(x) - D 2k(x)) = co n(x)^2n k(-l) X k D 2k(x), k=o k=0 n G N, x G I. Define the n-th Fejér means [8] of function / G L 1(/) as °nf(y)--= - y2 Skf(y) n z—' for y G / and n G P and define n-th Fejér kernel [8] ^ n — 1 K n(x):= - V 77 L / for £ G / and n G P. This gives ^n/(y) = J^ f(x)K n(x + y)dx = / * ^„(y) (y G /, n G P). Set 6-1 Kg i b:=y^Dj a,6GN and n (s ):=^n z-2 8 (n,5GN). j-a Also set for n G N |n| : = max{j G N : nj ± 0}. That is, <n< 2H + 1. In this paper c denotes an absolute constant which may not be the same at different occurences. Then we have by an easy calculation that Lemma 1. riK n = 2 s f° r aü « G P. •
