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
112 István Blahota j — 1 rrit — 1 m o—1 X X "' X ^loMo( x) m--Í'lt-iM t.A x)' tl JhM t(x) = h=0 l t-i=0 l 0= o j — 1 m t—1 mo—lX^tM S fc-iJlf.-iW-E^oW 3 /1=0 íj _ 1 =0 Z 0=0 /j-1 \ í-1 mjfc-1 [J J] V>Z f cM f c(s) = \/i=0 / k—0 l k-0 / J—1 \ m t— 1 m 0-l X^ M<( X)) X ^M^W'" X &oM 0( x) = \h=0 / i«_i=0 i 0=0 / j—1 \ m„-1 m 0-l X '*' X ^MoW-^-iM,.!^)\h=0 ) /<_i=0 i 0-0 /i—1 \ m t-1 m 0-l \/i=0 / lt-1=0 l 0= o /j-1 \ Mt-1 J-1 (X^^w X M*) = ^M.^íX^«^)' \h=0 / Z=0 This complétés the proof of the Lemma. PROOF of the theorem The form n = jM t is not unique. (For example jM t +1 = (jm t)Mt •) In our présentation let n = jM t be that expression, in which j is the least. 1. Sufficiency. Suppose that n G {jMt\t,j G N; 0 < j < m t} : and x,y G G m. 1.1. Let x — y çji It- In this case by the theorem A. we have D^^x — y) = 0. By lemma D^ M t(x — y) = 0. The theorem B. shows that D*hsS x>y) = "jMtiicîâjMtiy^M.C® - y)So DJ m (X, y) = 0, too. Consequently if x — y £ It,t,j G N, then D* M t(x,y) = Df M i(x-y) = 0.
