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

88 György Gát Then, (8) < + < c / (/(* + y) - f(y)) E xAy)xj(x + v) J G

m \In

0 j = M

n \ (f(x + y)~ f(y))J{x + y,y)d^x) JG

m\I

n

o I (f{x + y) - f(y))J(x + y, y) dp(x) Ji

n

o í (f(x + y)-f(y))J(x + y,y)d

fi{x) JG

m M

n

o í If(x + y)-f(y)\dfi(x) JIn

0 / /(* + y)J(x + y,y)dfj.(x) 'Gm TIQ — 1 k, —1 M, -1 /log M

n

o+ £ E E I/V

S+1

)+J'M

S + /)| s=0 j=0 1=0 C \

n0 < C£

n

o +C

n

o2á

n < C£

n

o + At last by (4), (6), (7), (8), we get IS

kf(y) - S

M

nf(y)I < IS

MJ(y) - f(y)\ + + + r ^ / (f(x + y)-f(y)) Xj{y)Xj{x + y)dfi{x) Ji

n j = M

n fc-1 j (/(* + y) - f(y)) £ xAv)xj(x + y)M*) k1 Gm\In o j = M

n - +C£N/ LOS + £ BIX+<*- + 71-1 C£, C \

n o 2^1 s=n

0 < C£

n

0 + C£

n + sup E

s(l/n

0 + • • • + l/n) + . s>no \ ^