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

By Y. Okuyama

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Let f(x) { L2(0,2w) and oo f(x) ~--~-i a0 + oo ~ (anCOS nx + b sin nx) -= [0An (x). 2) 1/2 [f(x+t)- f(x-t)]2dx)dt} , o ,o I ( ={~ o,o [f(x+2t) + f ( x - 2 t ) - 2 f ( x ) ] 2 d x ] d t } 1/2 Leindler [43] established the following equivalence theorem for the trigonometric system. 5. Let 0 < 8 < 2. tone function such that [ i k=n kSl (k) Let l(x) (x ~ I) be a positive mono- < A i = n6-1l (n) Then the conditions I1 1 [f(x+t) - f ( x - t ) ] 2 d x ] 6 / 2 0 t21 (i/t) I i 0 t21(i/t) J0 ( dt < ~ , [f(x+2t) + f(x-2t) - 2f(x)]2dx] B/2 dt < ;0 I i 6 n=l and [ 1 E(2)( }B f) < .

Pn = i, then we have IR,Pn, II=IC,I I and v n = 2 n . 1. 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.

N=l ~-777~ { n are mutually equivalent. 2, we have the following corollary. 3. ~i/2-~L(e)(i/6)-I/k ~° p (iv) ~(~,r) . . (0) Ls-lt±/~J s+q" (v) ~(~,f) : 0(~I/2Ls+I(I/6)I/kL~0)(I/~) i/k-iL(e) s+q+l'tl/,)-i/k "~ ) 32 respectively. The case k = i and p : 2 in the results (i) ~ (iii) are due to Ul'yanov [85], and the case s = q = k = I and r = 0 in (iv) is due to Okuyama [61]. 5. Rademacher Trigonometric Series. system. 3. 6. Let k ~ i and let {pn } be a positive sequence such that for any fixed integer J0 > 0, Pn_j(Pn/Pn-Pn_j/Pn_j) = 0(i) for n ~ J0 ~ j ~ i.

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