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

By Y. Okuyama

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5), we have easily the following corollary. 2. 13) For the trigonometric system, we shall make a remark. 2) in Chapter 3. 2, we have the following corollary. 3. 1. (i) If ~ >0 and ~( 6,f) = 0(Ls(i/6)-a/2L~0)_ (I/6)~(E)~1/6)-1) Lp+s~ for some s >0, then the Fourier series n=[0An(X) is summable IR,exp Ls(n)~,l I almost everywhere. (ii) If ~ > 0 and n(6,f) = 0(LI(I/6)I/2L (C ) (1/6 )-1) P for some s >0, then the Fourier series [ An(X) is smnmable n=0 42 IR,na,ll almost (iil) everywhere. 4. 5, these =[0An(X) integral moduli Rademacher Trigonometric Series.

2. of Varshney Let {pn } be non-negative t > 0, be a positive, non-decreasing [87], Kanno [34] proved and non-increasing. function satisfying the Let the con- dition {In/P n} is non-increaslng. 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.

4) converges for some ¢ > 0, then the series [ an#n(X) is summable IR,Ls(n)e, ll almost everywhere. (v) If ~ > 0 and the series [lanI2n Ls(n) -~ (¢). 5) Lp+s(n) converges for some ¢ > 0, then the series [ anOn(X) is summable IR,exp n/Ls(n)~,iI almost everywhere. Proof. As these results are analogously proved, we shall prove here only the case (i). i~(n)p~ { [ ~ [ n n=k X Pk-1 2 p2 n=l n p2 p2 n n-i 2 1/2 lakl } k=l n-i L(e) (n) 2 p+s Pn n 1/2 } On the other hand, if we put Pn = exp Ls(n)~ , we see that Pn ~ n-iLs (n) ~L~0) (n) -lexp Ls(n)~ .

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