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X+1)\ n+1 1 < Ix-11(1+2+···+n) n+1 = 11-xln <: 2 11- xln ' also fiir x -+ 1 langs des Bogens 'In l+x+···+xn 1n ~= oan n +1 - < 11-xln:i:1 nlanl 8m I = 1'In ~ an n=l (l+x+ ... +xn n+. 1 = 0(11-Xlll~XI) = -1 )\ I 0(1)-0, folglieh bei m -+ 00 § 10. Die Hardy-Littlewoodsche Umkehrung des Abelschen Stetigkeitssatzes. Ieh kehre zu dem aIten, pragnantesten W ortlaut des § 8 mit Annaherung langs der positiven Aehse zuriiek. ) ersetzt werden kann. Offenbar wiirde es geniigen, dies fiir reelle an zu beweis en.

N-k+l)x n- k = . n=k tif dk 1 ) ( dx k i-x = kT = :d' dk -kl d-k . ;J; 00 ~ xn n=O fJf' (l_x)k+1 ist daher (1- X)-I<-1 ((x) - sfJf'(l-x)-k-l fJf' (1- xtk - 1 -+ 0, x-k (Cx) - s -+ 0, (Cx) -+S. Definition. Es sei ao, all ... , an, '" werde gesetzt: h~) = eine Folge komplexer Zahlen, und es ao + ... + an, ... +h~) n+l h~O)+ h"n -- Man sagt: die Reihe h~+ ···+h~ n+l ao + a1 + ... + an + ... sei summabel kter Ordnung im Holderschen 1) Sinne, odet': der kte H (j 1d e r sche Limes der Folge Sn = ao + ...

56 - menden, in 1 miindenden Kur'IJenbogens, bei dem die Ordinate eine eindeutige, stetige, monotone Funktion der Abszisse ist (wie steil er auch miinde). 00 Behanptnng : Beweis: X ~ n=O an = O. sei ein fester Punkt des Kurvenbogens, y ein variabler Punkt auf dem Bogen zwischen x und 1. Bei Integration Hings des Bogens strebt f y x f(z)dz = 00 x n+1 ) yn+l ( ~ an n+l - n+' 1 ' n-O falls y (auf dem Bogen) nach 1 riickt, gegen den Grenzwert f l x f(z)dz = 00 ~ an n=O 1_xn+! +! 00 -"--- ~ fiir Iyl n=on+l < 1 gleichmaBig konvergiert.

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