Az Eszterházy Károly Tanárképző Főiskola Tudományos Közleményei. 1998. Sectio Mathematicae. (Acta Academiae Paedagogicae Agriensis : Nova series ; Tom. 25)

GÁT, G., On a theorem of type Hardy-Littlewood with respect to the Vilenkin-like systems

86 György Gát {x e Gm, ne N, f e L

l(G

m)). Then, y - x $ I

s gives (1) \D

n(y,x)

I < cM

s (s e N) ([2]). It is also known ([2]) that for y - x £ I

s M

s — 1 (2) . Y, XjM

a+t{y)xjM,+t(x) = o (je N). t=

0 Moreover

, Sim = / JG

m (n e N, y e G

m

). For more details on Vilenkin-like systems see e.g. [2]-[4j. The following theorem of type Hardy-Littlewood for the ordinary Vilenkin system is proved in 1954 by YANO ([8]). We generalize this result for Vilenkin-like systems. Theorem. Suppose that the following two conditions hold for function f e L

l(G

m) and for a y e G

m(1) M

nlogM

n J

I

n I f(x + y)~ f(y)\ dp(x) - 0 (n - oo), (2) \f(k)\ < ck~

6 for some 8 > 0. Then S

nf(y

) converges to f(y)Proof. Denote by (3) M

n log M

nJ^ \f(x + y)~ f(y)\ dp(x) = '.e

n - 0. (3) imphes that (4) \S

MJ(y)-f(y)\ = M

r I f(x) - f(y)dp(x) Uv) < log M

n for n e N. Let k e N and n e N for which M

n < k < M

n+Also, let n > no e N be some integer depend on n for which r < n/n

0 that is the ratio of n and n

0 has a lower bound, where constant r £ N is discussed later. Skf(y)= / f{x)y2xj(y)xj(x)dn(x) JGm j=

0 . k~ 1 = / f(z+ y)Y^Xj(y)Xj{x+y)dfi(x) A— r\