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By Wulf-Dieter Geyer

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1 y =f (x) y =f (x) 0 0 1 28 . ... .... .. .... .. ... .. . . .. . . .. . . .. . . .. . . . .. . . .. . .. .. 16 . . .. .. . .. .. .. . .. . . .. .. .. . .. .. . .. .. .. . .. . . .. . 0 0 1 37. Die Folge (Pi (t)) der rekursiv durch P = 0 ; Pi (t) = Pi (t) + 21 (t Pi (t)) 1 (2) 2 +1 de niertenpPolynome Pi konvergiert auf dem Intervall 0; 1] monoton und gleichma ig gegen die Funktion t . Beweis: Zunachst zeigen wir mit Induktion nach n Pi (t) p fur 0 t 1; i 2 IN: t Der Fall i = 1 ist trivial, den Induktionsschritt liefert die Gleichung h p p p p t Pi (t) = t Pi (t) 12 t Pi (t) = t Pi (t) 1 21 t + Pi (t) 2 +1 p aus der fur t 1 mit der Induktionsvoraussetzung Pi (t) p t Pi (t) +1 p t Pi (t) 1 i ; t die Abschatzung p t 0 p folgt.

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Analysis II für Physiker by Wulf-Dieter Geyer


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