By Y. Okuyama

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5. 5) in m, n. Let {~(n)} be a positive series 0<~ i ~(n)/n n=l converges. 6) then the series ~ n~(n)An(t) n=l is summable IN,Pnl , at t =x. 5 is an extension the case B = 0. 6. 5. theorem. 8) and 2~/t)Idf(t) I < ~ J0 are mutually Also, proved exclusive. Okuyama in Chapter Theorem dition [59] proved the following theorem, which will be 6. 7. Let ~(t) be a non-decreasing function. 2) {~(n)~(n)/n} function. 1). 5. 5 in form. 8. ~(t) , t > 0, t~'(t)/~2(t) is equivalent Let is a {pn } be non-negative positive is non-increasing, is non-increasing, and non-increasing.

And l(t), {~nXn/(n+l)} = 0 ( Suppose t > 0, is a positive is non-increasing, ), n = l , 2 .... 1) n and I T X(C/t)Id¢(t)l 0 < ~ , for a constant C(>2w). 2) Then the series n=O is summable IN,Pnl , at t =x. 1. 1. [17]. 2. 3. then for any x, n-i k+l [ APkX k=m of 49 where m and n are integers such that n > m > 0. This lemma is easily obtained. 3. Let t be the n th N~rlund mean of the n series [~n~nAn+l(t). Then we have = = n LI k =[0pk~n-kXn-kAn+l-k(x)" tn where Ak(X) - 0 ¢(t)cos kt dt. ~ Henc e, tn-i = k ~= l ( P ~ k tn- Pn-k-l) ~klkAk+l(X) Pn-i n = ~- (t) { - -i [ (pnPn_ k _ Pn_kPn)UklkCOS(k+l)t}d t PnPn-1 k=l n = Y 0d~(t){PnPn-ll k=l[(PnPn-k- Pn-kPn)Wkkk sin(k+l)t~k_~ ,.

1(v). 1 and the case can be shown It is sufficient s e r i e s [ anrn(t)cos to show the existence nx which is non-summable IN,Pnlk for almost every (t,x) in (0,1)x(0,2w) and the series is convergent for E = 0. For this purpose we put a Ls+l(n) I/k = n n L(0)(n) 1-17k~(°) s , ,l/k_ Ls+q+l~n) , )i/k . Ls+q+2~n Then we have [ I an 12 n Ls+l( n -2/kL(0)(,2-2/k~(0) (n)2/k-I ) s n) Ls+q+ I = [ n T(0) ~n ~IjLs+q+2(n)2/k ~s+q+l' < ~ (v) of similarly. foI' 1 ~ k < 2. 5) 35 On the other hand, since Pn = (n+2)-l~(0)(n+2)-l--~Ls, we see that Pn Z Ls+l(n) and Pn Pn-j - 0(j L(0)(j)) s+l Pn-j Pn for j <= n <= 2j.