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
110 István Blahota If M 0 := 1, Mfc+1 m kM k(k G N), then every n G N can be uniquely oo expressed as n = YJ njMj, where ríj G Z m j(j G N) and only a finite i=o number of n 3 's differ fíom zéro. Define on G m the generalized Rademacher functions in the following way: 27TlX k 2 x-, rk\ x) '• — E XP * :=X e G m, KÉN, m k It is known that the functions oo V>n(s):=n r£ f c(s) ( nG N) k=0 on G m are elements of the character group of G m , and ail the elements of the character group are of this form. If x,y G G m,n,m G N then it is easy to see that + y) = 1p n{x)llj n(y), and fpn+m{x) = ll> n(x)ljj m(x). The system (-0 n|n G N) is called a Vilenkin system and G m a Vilenkin group. The Dirichlet kernels are 71-1 £?(*):= X>*(x) (n G N) fc=0 with respect to the Vilenkin system for which it is known (see [4]) that: Theorem A. <.<*)={?"• Ht Let A n be the er-algebra generated by cosets I n(z ), where (n G N)(z G Gm). Let ctj, a n(k,j , n G N) be functions satisfying the following conditions: (i) a k : G m Cis^4j - measurable G N), (ii) := ÖQ :=cx-(0):=l (kJ G N) oo . oo (ni) := n <*f n )(n G N,j(n): = £ n kM k). j= 0 fc=j
