Read e-book online A Concise Introduction to the Theory of Integration PDF

By Daniel W. Stroock

ISBN-10: 0817637591

ISBN-13: 9780817637590

ISBN-10: 3764337591

ISBN-13: 9783764337599

Designed for the full-time analyst, physicists, engineer, or economist, this e-book makes an attempt to supply its readers with lots of the degree conception they are going to ever want. Given the alternative, the writer has always opted to improve the concrete instead of the summary features of issues taken care of.

the main new characteristic of this 3rd version is the inclusion of a brand new bankruptcy during which the writer introduces the Fourier rework. In that Hermite services play a significant position in his therapy of Parseval's identification and the inversion formulation, Stroock's strategy bears larger resemblance to that followed by means of Norbert Wiener than that utilized in most recent introductory texts. A moment function is that options to all difficulties are supplied.

As a self-contained textual content, this booklet is great for either self-study and the school room.

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Man berechne f¨ur x ∈ R die Summen der Reihen ∞ ∞ sin nx 3 n=1 n ∑ und cos nx . 4 n=1 n ∑ Aufgabe 21 C. F¨ur |x| < 1 berechne man die Summen der Reihen ∞ ∞ ∑ n2 xn , ∑ n3xn n=1 n=1 ∞ und xn . n=1 n ∑ Aufgabe 21 D*. Sei fn : [a, b] −→ R, n ∈ N, eine Folge stetiger Funktionen auf dem abgeschlossenen Intervall [a, b] ⊂ R mit fn (x) fn+1 (x) f¨ur alle x ∈ [a, b] und n ∈ N. 52 Aufgaben Es gelte lim fn (x) = 0 f¨ur alle x ∈ [a, b]. Man zeige: Die Folge ( fn ) konvergiert n→∞ auf [a, b] gleichm¨aßig gegen 0.

Sei f : ]0, 1] −→ R eine stetige Funktion. Man zeige, dass f genau dann gleichm¨aßig stetig ist, falls lim f (x) existiert. x 0 Aufgabe 11 E. Sei M eine Teilmenge von R. Die Funktion d : R −→ R sei definiert durch d(x) := inf{|x − y| : y ∈ M} f¨ur alle x ∈ R. Man zeige, dass d stetig ist. Aufgabe 11 F*. Eine stetige Funktion ϕ: [a, b] → R heißt st¨uckweise linear, wenn es eine Unterteilung a = t0 < t1 < . . < tr−1 < tr = b des Intervalls [a, b] und Konstanten α k , βk gibt, so dass f¨ur k = 0, .

Es gelte weiter: § 17 Numerische L¨osung von Gleichungen 39 a) g (x) = 0 f¨ur alle x ∈ ]a, b[, f (x) = c ∈ R, a g (x) b) lim x c) Entweder lim f (x) = lim g(x) = 0 oder x a x a Man zeige lim x a lim |g(x)| = ∞. x a f (x) = c. g(x) Aufgabe 16 J*. Gegeben sei die Funktion Fa (x) := (2 − a1/x )x , (x ∈ R∗+ ), wobei 0 < a < 1 ein Parameter sei. Man untersuche, ob die Grenzwerte lim Fa (x) und x 0 lim Fa (x) x→∞ existieren und berechne sie gegebenenfalls. § 17 Numerische L¨osung von Gleichungen Aufgabe 17 A*.

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A Concise Introduction to the Theory of Integration by Daniel W. Stroock

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