By EDMUND LANDAU
Im folgenden will ich zunächst über die Ziele der einzelnen acht Kapitel und die Vorgeschichte jener Fragestellungen berichten. Absichtlich ist im späteren textual content durchweg vom Einheitskreis die Rede, in dieser Einleitung vom Kreise I x I
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Extra info for Darstellung und Begründung einiger neuerer Ergebnisse der Funktionentheorie
Example text
3 ... 4 ... 2v '" v = 1 ·v v = 1v rv 1 -logn. '" § 3. Fej ers Satz, daß s" bei festem fex) nicht beschränkt zu sein braucht. Hilfssatz. Voraussetzung: Es sei m>O, bo>O, bl>O, ... , bm>O, n > 0, co> Cl > ... > C" >0. Behauptung: Alle Partialsummen im Punkte x = 1 Sk = go + gl + ... + gk der (nach dem E ne s t r ö m schen Satz für Ix I <1 konvergenten) Potenzreihe k R( ) bo+ ... + b,n x'" X = Co + ... + cn x" = go + gl X + ... + gk X + ... sind positiv. lCx) = (b o + ... + bmxm) 1 1- x CO -I CCo - Cl) x + ...
In der Tat ist für jedes feste v, wenn die Endpunkte der etwa auf 1~ gelegenen, zu unserem endlichen Vorrat gehörigen K':'> interpoliert werden und die Abszissen Ul mit die Ordinaten entstehen, (Ul' Vl) = A(Ul,Ul+I) = A (x"' X N1 ) ° Ql für < Ä. , bei denen (Ul ••• Ul+ 1) eines der ausgewählten K':/ ist, [tJ. - t [ > E), Il P ~ P ~+, -_ Q0 '(,h _ h" 1 Ul+I - UJ. +! - Vi +t l+ttJ. ) - < = h - 1 __ ( 1 1 ~ 0 Ql Ql+l 2 (1 +ttl)1 + 2 (1 + t 2) (1 + tD h-l _ _ ( (t). _q ) 1 ~ 0 Ql Ql+l 1 - 2 (1 + t 2) (1 + t~) 1 h-l 3S - h-l h-l 1=0 1=0 ) I < ~ Ql Ql+1 - -S ~ (1'lH -~) (t).
N. ~ ~n V ~