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

On a theorem of type Hardy-Littlewood. . . 87 and gives . k-1 / f(y) V Xj(y)Xj(x + y)d/*(x) = 0 J = M

n . Ar-1 (5) /(») = / (f(x + y)-f(y)) £ Xi(s/)Xi(® + j

Gm • »Í j=M„ In (5) we integrate over G

m which is the disjoint union of /

n, /

n

o \ I

n and Gm \ In

0- Since sequence m is bounded, then we have (6) / (/(* +

s/) -/(!/)) E + < (k — M

n) f \f{x + y) -f{y)\dfi{x) < c£

n/log M

r Jin By (1) we have (7) k1 I (/(* + y) - /(y)) £ Xi(y)Xi(® + '»oV» i=M

n < 71 — 1 p 71 — 1 Y,

c

M* / I /(* + y)- f(y)I < E S — 71 q Jl

s\Is+

1 5-n

0 C£. log A/

s Finally, we have x E G

m \ /

n

o

. This by (2) imphes 7i k, -1 A4",, —1 + = o. 5 — 710 j=0 / = 0 Denote by no —1 A:, —1 M

a — 1 + y,y): = ^ J] ^ Xfct'+D+^+i^ + I/lx^+D+jM^iW' s=0 j= 0 i=0