Az Eszterházy Károly Tanárképző Főiskola Tudományos Közleményei. 1994. Sectio Mathematicae. (Acta Academiae Paedagogicae Agriensis : Nova series ; Tom. 22)

GÁT, G., On a norm convergence theorem with respect to the Vilenkin system in the Hardy spaces

On a norm convergence theorem with respect to the Vilenkin. . . 105 where A k: = Í mfc/2, 2\m

k ' \(m

f

c- l)/2, 2/mjt' for m

k > 4. If m

k < 4, then f

k : = 0. It is easy to see that n > M^+i implies Snfk = fk and n < M

k implies SVJ^ = 0. _ If M

k < n < M

k

+

1, then let y G 4(0, /), l / 1, A

f

c and a; G 4(0,1) U I

k(

0, Ajt). Consequently, y - x G h \ h+i

, jfc-i njt -1 ^nfo - = E ^ ^ ^ - *) + M

k £ ^ - *)' {i})

n(y - x) = r

k

k (y - x)) thus -1 p-0 \S

nfk\ > h (0,0 / Jt—i lé? IAI+ \i=o , , ,r

n

k

h(y-x)-l , = :(!) + (2). 1(1)1 < ^ is trivia

L (2) = ((Í - A

f

c)c

f

c) - 1 For l < [f] r

f

c((/ - l)ejt) - 1 r

f

c((Z - A

k)e

k) - 1 r

k((l - A

k)e

k) - 1 < S1H 7T That is, (2)> 1 n

k(l~

1) sm 7T— m

k m

k sin 7T —m

k m,

t — C. < C. These estimations give m

k /4] Iis-Alli > £ / * ^ Ë 0,0 (=

2 Sin 7T m

f

c — C sm 7T • mfc > c log n

k - c.