By Y. Okuyama

ISBN-10: 3540133550

ISBN-13: 9783540133551

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**Example text**

K) ~(k) } X (k'J ....... } [ < 0 by virtue of the hypothesis 13 is finite. 9). Collecting the above estimations, n=l Therefore~ Itn - tn_ll Thus, by 131 and 132 , we see that we have < we complete the proof of our theorem. Some Corollaries. 5. 8. 7. Finally we consider some applications If i > ~ => 0 , 8 >= 0, ~ + ~ < I, and (log C/t)~@(t) (BV(0,w), then the series An(t ) n=O {log(n+2)} I-B is summable I N,i/(n+2){log(n+2)}~l , at t = x . 8. is due to Varshney I f a >0 and (log C/t)¢(t) c BV(0,w), then the series An(t) n=0 is summable IN,{log(n+2)}~/(n+2)l, [87].

If the conditions k=n Pklk 2 Pk 1 = 0(~ -), n : 0 , 1 , 2 , . . 5) I~ l(C/t)Id¢(t) I < ~ 0 for some constant C > 0 hold, then the series (n+l)p n Pn XnAn+l(t) n=0 is summable IN,Pnl , at t =x. 2. Summability the following Now we generalize these above theorems in form. Theorem that Factors. of these theorems. 3. Let {pn] be non-negative {~n } is a positive non-decreasing bounded function sequence such that [ p -~kXkk k = n k and non-increasing. and l(t), {~nXn/(n+l)} = 0 ( Suppose t > 0, is a positive is non-increasing, ), n = l , 2 ....

2. Equivalence Theorem Theorem. 3. to the convergence Proof. 1) by M6ricz [56]. is similarly We shall prove the converse such that Vm0(n ) < n ~ Vm0(n)+ I. 4) Let m0(n) Then we have 2} 1/2 m0(n)-i { ~ m=O > Vm+ I k=v +i = n~= 1 P n P n - 1 m0(n)-i { I m=O is equivalent o 2 2} 1/2 Pk_zlakl m Pn theorem. 4) implies by the same method implication. +l J = c 1 Pn-i i ) Pn ( P(vj-1)P-~ ~)P(v'-') j~ICj_I j= ( J by virtue of the fact that P(Vj_l)/P(vj)-P(Vj_l)/P(vj+ I) ~ 2J-i/2 j- 2J-i/2 j+l = i/2-i/4 = 1/4.

### Absolute Summability Of Fourier Series And Orthogonal Series by Y. Okuyama

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