Az Eszterházy Károly Tanárképző Főiskola Tudományos Közleményei. 1994. Sectio Mathematicae. (Acta Academiae Paedagogicae Agriensis : Nova series ; Tom. 22)

BLAHOTA, I., Relation between Dirichlet kernels with respect to Vilenkin-like systems

Relation between Dirichlet kernels 111 Let Xn := Tpn

an (n G N). A function system {Xn\

n £ N} of this type is called a iß a ( Vilenkin- like) system on Vilenkin group G

m. The iß and iß a systems are orthonormal and complété in i

1(G

m). The Dirichlet kernels with respect to iß a system are n —

1 D*(x, y)-=Y^ Xk(x)Xk(y) (n G N) k—0 The subsequence D

^ has a closed form (see [2]). We will use the following theorem, too (see [1]): Theorem B. DjM

t(

x>y) = ajM

t(x)â

jM

t(y)Df

M

t(x - y) (n G N, x, y G G

m). xb Tb 3~

l Lemma. Let x G G

m, j, í G N. Then DM (x) = DJ^ ix) £ 4>kM

t(x). k=o This lemma is needed in the proof of the theorem. Theorem Let x, y G G

m, n G N. Then DX(x,y) = DHx-y) holds if and only if n G {jM

t\0 < j < m

t\t,j G N}. PROOF of the lemma. Using the statements and theorems mentioned above we have the following équations: j M t — 1 j1 /(/i+l)M

t-l = E w*) = E E i*<(s) ?=0 /1=0 \ i=/iM

( j1 /(M-1)M,-1 S S + + + /i=0 \ l=hM

t j1 /(M-l)M

t-l /i=0 V i=/iM,