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We \begin{align*} \int_a^b e^{i\lambda x} \d x &= \int_a^b \cos(\lambda x)\d x + i \int_a^b \sin(\lambda x)\d x\\ &= \frac{1}{\lambda} \sin(\lambda x)\bigg|_a^b - \frac{1}{\lambda} i \cos(\lambda x)\bigg|_a^b\\ - &\stackrel{=}{\frac{1}{i} = -i} \frac{1}{\lambda i} (\cos(\lambda x) + i \sin(\lambda x)) \bigg |_a^b\\ + &\stackrel{\frac{1}{i} = -i}{=}\quad \frac{1}{\lambda i} (\cos(\lambda x) + i \sin(\lambda x)) \bigg |_a^b\\ &= \frac{1}{\lambda i } e^{i\lambda x} \bigg|_a^b\\ \end{align*} $$\implies \int_0^{2\pi} e^{ikx} \d x = 0\qquad \forall k \in \Z \setminus \{0\}$$ @@ -72,7 +72,7 @@ Frage: Hat jede $2\pi$-periodische Funktion die Form $\sum_{k = -n}^{n}c_ke^{ixk $$\qnorm{f-s_n} = \qnorm{f} - 2\pi \sum_{k = -n}^{n}|c_k|^2$$ \end{satz} \begin{proof} - Notation $e_n(x) \coloneqq e^{ikx}$ + Notation $e_n(x) \coloneqq e^{inx}$ \begin{align*} (e_k, e_l) &= \int_0^{2\pi} e_{ikx} e^{-ikx} \d x = \int_0^{2\pi} e^{i(k-l)x}\d x = \begin{cases} 2\pi, & k = l\\ @@ -94,7 +94,7 @@ Frage: Hat jede $2\pi$-periodische Funktion die Form $\sum_{k = -n}^{n}c_ke^{ixk \end{align*} \end{proof} \begin{satz}[Besselsche Umgebung]\label{bessel} - Sei $f \in R[0, 2\pi]$ eine $2\pi$-periodische Funktion mit Fourier-Koeffizienten $c_k,\; k\in \Z$. Dann $$\exists \lim\limits_{n\to\infty}\sum_{k = -n}^{n}|c_k|^2$$ und $$\sum_{k = -\infty}^{\infty} |c_k|^2 = \lim\limits_{n\to\infty}\sum_{k = -n}^{n}|c_k|^2 \leq \qnorm{f}$$ + Sei $f \in R[0, 2\pi]$ eine $2\pi$-periodische Funktion mit Fourier-Koeffizienten $c_k,\; k\in \Z$. Dann $$\exists \lim\limits_{n\to\infty}\sum_{k = -n}^{n}|c_k|^2$$ und $$\sum_{k = -\infty}^{\infty} |c_k|^2 = \lim\limits_{n\to\infty}\sum_{k = -n}^{n}|c_k|^2 \leq \frac{\qnorm{f}}{2\pi}$$ \end{satz} \begin{proof} Aus Satz \ref{fourierungleichung} @@ -102,7 +102,7 @@ Frage: Hat jede $2\pi$-periodische Funktion die Form $\sum_{k = -n}^{n}c_ke^{ixk Die Konvergenz folgt unter Beachtung der Monotonie und Beschränktheit der Folge $$\sum_{k = -n}^{n}|c_k|^2$$ \end{proof} \begin{bem} - Sei $f\in R[0, 2\pi]$ $2\pi$-periodisch und $\qnorm{f-s_n} \underset{\to}{L^2} 0,\; n\to \infty$, d.h. die Fourier-Reihe konvergiert gegen $f$ in $L^2$. Das ist nach Satz \ref{fourierungleichung} äquivalent zu + Sei $f\in R[0, 2\pi]$ $2\pi$-periodisch und $\qnorm{f-s_n} \overset{L^2}{\to} 0,\; n\to \infty$, d.h. die Fourier-Reihe konvergiert gegen $f$ in $L^2$. Das ist nach Satz \ref{fourierungleichung} äquivalent zu $$\qnorm{f} = 2\pi \lim\limits_{n\to\infty}\sum_{k = -n}^{n}|c_k|^2 \Leftrightarrow 2 \pi \sum_{k = -\infty}^{\infty}|c_k|^2$$ \end{bem} \underline{Frage:} Unter welchen Bedingungen für $f$ gilt die Parsevalsche Gleichung @@ -260,7 +260,7 @@ $$F_n(x) = \int_\pi^x \frac{1}{2 \sin\left(\frac{y}{2}\right)} \cdot \sin\left(\ $$\implies s_n(f) \xrightarrow{L^2} f, \; n \to \infty$$ \end{proof} \begin{bem} - Konvergenz in $L^2$ ist \glqq sehr schwach \grqq\. Für \glqq glattere\grqq\ Funktionen konvergiert die Fourier-Reihe gleichmäßig. + Konvergenz in $L^2$ ist \glqq sehr schwach\grqq\ Für \glqq glattere\grqq\ Funktionen konvergiert die Fourier-Reihe gleichmäßig. \end{bem} \begin{satz} Sei $f: \R \to \C$ $2\pi$-periodisch, stetig und stückweise stetig differenzierbar, d.h. $\exists $ Unterteilung von $[0, 2\pi]$ diff --git a/analysisII.pdf b/analysisII.pdf index b39fd762798bd01d8335dcd1efe8a1c44a93e9b8..14600511208d5df16c63ed7290405a887332003c 100644 GIT binary patch delta 318447 zcmV)YK&-#{#TD}58;~RcHO!QOaUc;0 z(Lh33{Pz}db8xwPdAaZ9-O61G4Xuw9w4mVdcW+&^+EWTTL(-gpYk=6IPAo7fWJBQY za21=rdcJ(;V`;Pz$5>Jt=FOb~oq^V9nQ5k_AxwjmOlXW*=wcii3}}FM<3}-MB7elp zqMGWl>#M~&$FvrMpJpROL&BJ_4Urid`z=xqWxXVkA~m1={FJ-s4ReJ&mvOcN6O$JK z6aq9km%%;)Dw9_M4u2lMiEeF^Anj5Z+MEndqsT%W8IUa($gl5FlvK-cT6Aawco7Nm z`0n`bJwg}x{2P#RI}@ZNf?SW|$9#5ir5%x)JMVzxj|7C{V1QUHxec1+4`e}a>eX_+ z+wLP zc^pbmLkT!3*WHC2T|M6_t#6fcRavCHubK3*B>e7UNYUW=my8)_D91sF_ZyIjv6E!M 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