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**Extra resources for Absolute Summability Of Fourier Series And Orthogonal Series**

**Example text**

4. 2. 8. 8. Let k(t), t > 0, be a positive non-decreasing function. is non-increasing and t2~'(t)/k2(t) is non-decreasing, then we have 1 Proof. cos kt dt I < A k'(k) By an integration J = k'(C/t)/tk2(C/t) increasing, by parts, C I ~ ~'(C/t) k 0 t2k--~C/t) is non-decreasing _ C [ w/k k'(C/t) - k ~0 tk2(C/t) { ~ } sin kt dt sin kt t dt sin kt dt sin kt dt. and ~'(C/t)/t2k2(C/t) we obtain C [w/k k'(C/t) I J I ~ ~ ~O t2q2(C/t) C(> ~) . 2. 8. 13) dt. J ) X ( k ) ~ ( k ) e o s ( k + l ) t } d t n-i .

7 w i t h a s u i t a b l e choice of a s e q u e n c e of s i g n s , putting + an Cn = - we c a n conclude the existence (n = 1,2,. "" ), of t h e r e q u i r e d function g(x). d. 1. Introduction. n th partial sum. Everywhere Series Let [ a n be a given infinite If {pn } is a sequence Pn = P0 + Pl ÷ "'" + Pn ÷ ~ as n ÷ ~ , series with s n as its of positive constants, and P-k = P-k = 0 for k ~ i, then the Riesz mean in of [ a n is defined by n in - pnl k=0[ PkSk (Pn ~ 0 ). 2) n=0 converges, then the series [ a n is said to be summable IR,Pn,I I or sum- mable IN,Pni.

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. 3. Sufficient Conditions. 3, we shall give some sufficient conditions for the absolute Riesz summability of orthogonal series and Fourier series.