From e795795f66fea1c0f005daa76edb063a3437d770 Mon Sep 17 00:00:00 2001 From: salagne Date: Wed, 6 May 2020 12:07:23 +0200 Subject: [PATCH] fixed some mistakes --- ana3.tex | 36 ++++++++++++++++++------------------ analysisII.pdf | Bin 353017 -> 405041 bytes 2 files changed, 18 insertions(+), 18 deletions(-) diff --git a/ana3.tex b/ana3.tex index 19c87d1..6ab0a4b 100644 --- a/ana3.tex +++ b/ana3.tex @@ -41,7 +41,7 @@ Hier betrachten wir deshalb nur $2\pi$-periodische Funktionen $f: \R \to \K$. 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*} $$\implies \int_0^{2\pi} e^{ikx} \d x = 0\qquad \forall k \in \Z \setminus \{0\}$$ @@ -68,13 +68,13 @@ Frage: Hat jede $2\pi$-periodische Funktion die Form $\sum_{k = -n}^{n}c_ke^{ixk \end{definition} \begin{satz} \label{fourierungleichung} - Sei $f\in R[a,b]$ eine $2\pi$-periodische Funktion mit den Fourier-Koeffizienten $c_k,\; k\in \Z$ und $s_n = \sum_{k = 1}^{n}c_ke^{ikx}$. Dann gilt für alle $n\in \N$ + Sei $f\in R[a,b]$ eine $2\pi$-periodische Funktion mit den Fourier-Koeffizienten $c_k,\; k\in \Z$ und $s_n = \sum_{k = -n}^{n}c_ke^{ikx}$. Dann gilt für alle $n\in \N$ $$\qnorm{f-s_n} = \qnorm{f} - 2\pi \sum_{k = -n}^{n}|c_k|^2$$ \end{satz} \begin{proof} - Notation $e_n(x) \coloneqq e^{ikx}$ + Notation $e_k(x) \coloneqq e^{ikx}$ \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} + (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\\ 0 , & k \neq l \end{cases}\\ @@ -102,8 +102,8 @@ 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 - $$\qnorm{f} = 2\pi \lim\limits_{n\to\infty}\sum_{k = -n}^{n}|c_k|^2 \Leftrightarrow 2 \pi \sum_{k = -\infty}^{\infty}|c_k|^2$$ + Sei $f\in R[0, 2\pi]$ $2\pi$-periodisch und $\qnorm{f-s_n} \underset{L^2}{\to} 0,\; n\to \infty$, d.h. die Fourier-Reihe konvergiert gegen $f$ in $L^2$. Das ist nach Satz \ref{fourierungleichung} äquivalent zu + $$\qnorm{f} = 2\pi \lim\limits_{n\to\infty}\sum_{k = -n}^{n}|c_k|^2 \Leftrightarrow \qnorm{f} = 2 \pi \sum_{k = -\infty}^{\infty}|c_k|^2$$ \end{bem} \underline{Frage:} Unter welchen Bedingungen für $f$ gilt die Parsevalsche Gleichung $$\qnorm{f} = 2\pi \sum_{k = -\infty}^{\infty}|c_k|^2$$ mit den Fourier-Koeffizienten $c_k$.\\ @@ -130,8 +130,8 @@ $\implies$ Konvergenz der Fourier-Reihe in $L^2$. \end{lemma} \begin{proof} Sei $s \neq 0$. - $$F_s(x) = \int_a^x f(y) \sin(sy) \d y \overset{\text{part. Integr.}}{=} f(y) \frac{1}{s} \cos(sy) \bigg|_a^x + \int_a^x \frac{1}{s} \cos(sy) f'(y) \d y.$$ - $f,f'$ stetig auf $[a,b] \implies \exists M > 0, \text{s.d. } |f(y)| \leq M, |f'(y)\leq M$ mit $y\in[a,b]$. Dann gilt $|F_s(x)| \leq \frac{2M}{|s|} + \frac{M}{|s|}\cdot (b-a), \; \forall x\in [a,b]$. Also konvergiert $|F_s(x)|$ gleichmäßig gegen 0 für $|s| \to \infty$ und $x\in [a,b]$. + $$F_s(x) = \int_a^x f(y) \sin(sy) \d y \overset{\text{part. Integr.}}{=} -f(y) \frac{1}{s} \cos(sy) \bigg|_a^x + \int_a^x \frac{1}{s} \cos(sy) f'(y) \d y.$$ + $f,f'$ stetig auf $[a,b] \implies \exists M > 0, \text{s.d. } |f(y)| \leq M, |f'(y)|\leq M$ mit $y\in[a,b]$. Dann gilt $|F_s(x)| \leq \frac{2M}{|s|} + \frac{M}{|s|}\cdot (b-a), \; \forall x\in [a,b]$. Also konvergiert $|F_s(x)|$ gleichmäßig gegen 0 für $|s| \to \infty$ und $x\in [a,b]$. \end{proof} \begin{lemma}\label{HilfslemmaC} Es gilt $\frac{\pi - x}{2} = \sum_{k = 1}^{\infty}\frac{\sin(kx)}{k}$ für $0 < x < 2\pi$ mit gleichmäßiger Konvergenz auf allen Intervallen $[\delta, 2\pi - \delta]$ für $\delta > 0$. @@ -153,7 +153,7 @@ $$F_n(x) = \int_\pi^x \frac{1}{2 \sin\left(\frac{y}{2}\right)} \cdot \sin\left(\ \begin{proof} Lemma \ref{HilfslemmaC} $\implies$ $\forall x, y \in (0, 2\pi)$ \begin{align*} - \frac{(x - \pi)^2}{4} - \frac{(y-4)^1}{4} &= \int_y^x \frac{t-\pi}{2}\d t\\ + \frac{(x - \pi)^2}{4} - \frac{(y - \pi)^2}{4} &= \int_y^x \frac{t-\pi}{2}\d t\\ &\stackrel{\ref{HilfslemmaC}}{=} -\int_y^x \sum_{k = 1}^{\infty}\frac{\sin(kt)}{k} \d t\\ &\stackrel{\text{glm. Konv.}}{\stackrel{\text{Satz \ref{permutesumint}}}{=}} \qquad - \sum_{k = 1}^{\infty}\int_y^x\frac{\sin(kt)}{k}\d t\\ &= \sum_{k = 1}^{\infty}\frac{\cos(kx)}{k^2} - \sum_{k = 1}^{\infty}\frac{\cos(ky)}{k^2}\\ @@ -162,7 +162,7 @@ $$F_n(x) = \int_\pi^x \frac{1}{2 \sin\left(\frac{y}{2}\right)} \cdot \sin\left(\ Die Reihe $\sum_{k = 1}^{\infty}\frac{\cos(kx)}{k^2}$ konvergiert gleichmäßig auf $[0, 2\pi]$ mit Majorante $\sum_{k = 1}^{\infty}\frac{1}{k^2}$. Bestimme die Konstante $C$: \begin{align*} \int_0^{2\pi} \frac{(x-\pi)^2}{4}\d x &= \int_0^{2\pi} \left(\sum_{k = 1}^{\infty}\frac{\cos(kx)}{k^2} + C\right) \d x\\ - \frac{\pi^3}{6} &= \sum_{k = 1}^{\infty}\underbrace{\int_0^{2\pi} \frac{\cos(kx)}{k^2} \d x}_{=0} + \int_0^{2\pi} C \d x + \frac{\pi^3}{6} &= \sum_{k = 1}^{\infty}\underbrace{\int_0^{2\pi} \frac{\cos(kx)}{k^2} \d x}_{=0} + \int_0^{2\pi} C \d x \\ \frac{\pi^3}{6} &= C \cdot 2\pi\\ C &= \frac{\pi^2}{12} \end{align*} @@ -237,19 +237,19 @@ $$F_n(x) = \int_\pi^x \frac{1}{2 \sin\left(\frac{y}{2}\right)} \cdot \sin\left(\ $f_{a_j}$ Spezielle Treppenfunktion mit Sprungstelle $a = a_j$ und $f_{a_j}(x) \in \{0,1\}\; \forall j, x\neq a_j$. Dann $\norm{f_{a_j} - s_n(f_{a_j})} \xrightarrow{L^2} 0, n\to \infty$. Betrachte $$s_n(f) = \sum_{j = 1}^{l}d_js_n(f_{a_j})$$ und $$\norm{f-s_n(f)} = \norm{\sum_{j = 1}^{l}d_j(f_{a_j} - s_n(f_{a_j}))} \leq \sum_{j = 1}^{l}|d_j|\underbrace{\norm{f_{a_j}-s_n(f_{a_j})}}_{\xrightarrow{L^2}0} \xrightarrow{L^2} 0, \; n\to \infty$$ \end{proof} \begin{satz} - Sei $f\in R[0,2\pi]$ eine $2\pi$-periodische Treppenfunktion. Dann konvergiert die Fourier-Reihe von $f$ im quadratischen Mittel gegen $f$ und es gilt die Parsevalsche Gleichung (sog. Vollständigkeitsrelation) + Sei $f\in R[0,2\pi]$ eine $2\pi$-periodische Funktion. Dann konvergiert die Fourier-Reihe von $f$ im quadratischen Mittel gegen $f$ und es gilt die Parsevalsche Gleichung (sog. Vollständigkeitsrelation) $$\frac{1}{2\pi} \underbrace{\int_0^{2\pi} |f(x)|^2\d x}_{=\qnorm{f}} = \sum_{k = -\infty}^{\infty}|c_k|^2$$ \end{satz} \begin{proof} O.B.d.A. sei $f$ reellwertig (sonst werden Real- und Imaginärteil getrennt behandelt) und $|f(x)| \leq 1 \forall x\in [0, 2\pi]$ (sonst betrachte $\overline{f}(x) \coloneqq \frac{f(x)}{M},\; M = \sup\limits_{x\in [0, 2\pi]} |f(x)|$). Sei $\varepsilon > 0$. Dann gibt es zu $\varepsilon$ $2\pi$-periodische Treppenfunktionen $\varphi_\varepsilon, \psi_\varepsilon: \R \to \R$ mit Eigenschaften - $$-1 \leq \varphi_\varepsilon \leq f \leq \psi_\varepsilon\leq 1$$ und $$\max\limits_{x\in [0, 2\pi]} |\psi_\varepsilon(x) - \varphi_\varepsilon| \leq \frac{1}{16\pi}\varepsilon^2$$ + $$-1 \leq \varphi_\varepsilon \leq f \leq \psi_\varepsilon\leq 1$$ und $$\max\limits_{x\in [0, 2\pi]} |\psi_\varepsilon(x) - \varphi_\varepsilon(x)| \leq \frac{1}{16\pi}\varepsilon^2$$ Konstruktion von $\varphi_\varepsilon, \psi_\varepsilon$ siehe Rannacher. Dann, $$|f-\varphi_\varepsilon|^2 \leq |\psi_\varepsilon - \varphi_\varepsilon|^2 \leq (|\psi_\varepsilon| + |\varphi_\varepsilon|)(\psi_\varepsilon - \varphi_\varepsilon) \underset{|\varphi_\varepsilon| < 1}{\underset{|\psi_\varepsilon| < 1}{\leq}} 2 (\psi_\varepsilon - \varphi_\varepsilon)$$ und $$\qnorm{f-\varphi_\varepsilon} = \int_0^{2\pi} |f-\varphi_\varepsilon|^2 \d x\leq 2 \int_0^{2\pi}(\psi_\varepsilon - \varphi_\varepsilon) \d x\leq 2 \frac{\varepsilon^2}{16\pi}\cdot 2\pi = \frac{\varepsilon^2}{4}.$$ Weiter gilt: $\varphi_\varepsilon$ Treppenfunktion $\xRightarrow{\ref{HilfslemmaE}}$ Fourier-Reihe von $\varphi_\varepsilon$ konvergiert gegen $\varphi_\varepsilon$ in $L^2$, d.h. $$\forall \varepsilon > 0 \exists n_\varepsilon:\; \forall n \geq n_\varepsilon:\; \norm{s_n(\varphi_\varepsilon) - \varphi_\varepsilon} \leq \frac{\varepsilon}{2}$$ Aus Satz \ref{bessel} folgt - $$\qnorm{(f-\varphi_\varepsilon) - sn(f - \varphi_\varepsilon)} \leq \qnorm{f - \varphi_\varepsilon} \leq \frac{\varepsilon^2}{4}$$ + $$\qnorm{(f-\varphi_\varepsilon) - s_n(f - \varphi_\varepsilon)} \leq \qnorm{f - \varphi_\varepsilon} \leq \frac{\varepsilon^2}{4}$$ Dann gilt $\forall n \geq n_\varepsilon$ \begin{align*} \norm{f-s_n(f)} &= \norm{f-s_n(f-\varphi_\varepsilon) - s_n(\varphi_\varepsilon) - \varphi_\varepsilon +\varphi_\varepsilon}\\ @@ -260,25 +260,25 @@ $$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 $[0, 2\pi]$ $$0 = t_0 < t_1 <\dots < t_m = 2\pi$$ mit $f\big|_{[t_{j-1}, t_j]}$ ist stetig differenzierbar für $j = 1,\dots, m$. Dann konvergiert die Fourier-Reihe von $f$ gleichmäßig gegen $f$. \end{satz} \begin{proof} - $f$ stetig $\implies f \in R[0, 2\pi] \implies$ Fourier-Reihe von $f$ konvergiert gegen $f$ in $L^2$, d.h. $\norm{s_n(f) - f} \xrightarrow{L^2} 0,\; n\to \infty$. Betrachte $\phi: \R \to \C, \phi(x) = \phi_j(x),\; x\in (t_{j-1}, t_j),\; \phi_j: [t_{j-1}, t_j] \to \C$ stetige Abbildung von $f\big|_{[t_{j-1}, t_j]}$. Definiere $\phi$ in $t_j$ entsprechend (möglich, da $\phi$ eine stückweise stetige Funktion ist). - Defininition von $R[0, 2\pi] \implies \phi \in [0,2\pi] \implies$ Für die Fourier-Koeffizienten von $\phi$ gilt: $\gamma_k\coloneqq \frac{1}{2\pi}\int_0^{2\pi} \phi(x) e^{-ikx} \d x$ und $\sum_{k \in \Z} |\gamma_k|^2 = \frac{1}{2\pi}\qnorm{\phi} < \infty$. + $f$ stetig $\implies f \in R[0, 2\pi] \implies$ Fourier-Reihe von $f$ konvergiert gegen $f$ in $L^2$, d.h. $\norm{s_n(f) - f} \xrightarrow{L^2} 0,\; n\to \infty$. Betrachte $\phi: \R \to \C, \phi(x) = \phi_j(x),\; x\in (t_{j-1}, t_j),\; \phi_j: [t_{j-1}, t_j] \to \C$ stetige Ableitung von $f\big|_{[t_{j-1}, t_j]}$. Definiere $\phi$ in $t_j$ entsprechend (möglich, da $\phi$ eine stückweise stetige Funktion ist). + Defininition von $R[0, 2\pi] \implies \phi \in R[0,2\pi] \implies$ Für die Fourier-Koeffizienten von $\phi$ gilt: $\gamma_k\coloneqq \frac{1}{2\pi}\int_0^{2\pi} \phi(x) e^{-ikx} \d x$ und $\sum_{k \in \Z} |\gamma_k|^2 = \frac{1}{2\pi}\qnorm{\phi} < \infty$. Berechne Fourier-Koeffizienten $c_k$ von $f$ \begin{align*} c_k &= \frac{1}{2\pi} \int_0^{2\pi} f(x) e^{-ikx}\d x\\ - &\stackrel{\text{part.Integr.}}{=}\qquad \frac{1}{2\pi} f(x) \frac{i}{k}\underbrace{e^{-ikx}\big|_0^{2\pi}}_{=0} - \frac{1}{2\pi} \int_0^{2\pi} \underbrace{f'(x)}_{\phi(x)} \frac{i}{k}e^{-ikx}\d x\\ + &\stackrel{\text{part.Integr.}}{=}\qquad \underbrace{\left. \frac{1}{2\pi} f(x) \frac{i}{k}e^{-ikx}\right|_0^{2\pi}}_{=0} - \frac{1}{2\pi} \int_0^{2\pi} \underbrace{f'(x)}_{\phi(x)} \frac{i}{k}e^{-ikx}\d x\\ &= \frac{-i}{2\pi k} \int_0^{2\pi} \phi(x) e^{-ikx} \d x\\ &= \frac{-i}{k}\gamma_k\\ &\implies |c_k| = \frac{1}{k}|\gamma_k| \intertext{Es gilt $|\alpha \cdot \beta| \leq \frac{1}{2}|\alpha|^2 + \frac{1}{2}|\beta|^2$, da Quadrate größer 0 sind} &\leq \frac{1}{2}\frac{1}{k^2} + \frac{|\gamma_k|^2}{2}\\ - &\implies \sum_{k = -\infty}^{\infty} |c_k| \leq \frac{1}{2}\sum_{k = -\infty}^{\infty} + \frac{1}{2} \sum_{k = -\infty}^{\infty}|\gamma_k|^2 < \infty\\ + &\implies \sum_{k = -\infty}^{\infty} |c_k| \leq \frac{1}{2}\sum_{k = -\infty}^{\infty}\frac{1}{k^2} + \frac{1}{2} \sum_{k = -\infty}^{\infty}|\gamma_k|^2 < \infty\\ &\implies \sum_{k = -\infty}^{\infty}|c_k| \text{ absolut konvergent}\\ &\implies \underbrace{\sum_{k = -\infty}^{\infty}c_ke^{ikx}}_{\mathclap{\text{Fourier-Reihe von} f}} \end{align*} konvergiert gleichmäßig gegen eine Funktion $g$, die stetig ist. Also $s_n(f) \xrightarrow{\text{glm.}} g,\; n\to \infty,\; \Rightarrow s_n(f) \xrightarrow{L^2} g,\; n\to \infty$. 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