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+\newcommand{\norm}[1]{\left|\left|#1\right|\right|} +\begin{document} +\section{Fourier-Entwicklung} +\begin{definition}[Periodische Funktionen] + $f: \R \to \K$ heißt $L$-periodisch $(L >0)$ falls $f(x+L) = f(x)$, $\forall x \in \R (\implies f(x+kL) = f(x),\; \forall k\in \Z)$. + Sei $f$ periodisch und $p>0$. Für $\tilde{f}(x) \coloneqq f\underbrace{\left(\frac{L}{p}x\right)}_{\mathclap{\text{Variablentransformation}}}$ gilt dann $\tilde{f}: \R \to \K$ ist $p$-periodisch +\end{definition} +\begin{bsp} $p = 2\pi$ + $$\tilde{f}(x) \coloneqq f\left(\frac{L}{2\pi}x\right) \implies f(x) = \tilde{f} \left(\frac{2\pi}{L}x\right)$$ + $$\tilde{f}(x + 2\pi) = f\left(\frac{L}{2\pi}(x + 2\pi)\right) = f\left(\frac{L}{2\pi}x + L\right) \overset{f\; L\text{-per}}{=} f\left(\frac{L}{2\pi}x\right) = \tilde{f}$$ +\end{bsp} +Hier betrachten wir deshalb nur $2\pi$-periodische Funktionen $f: \R \to \K$. Weiterhin betrachen wir Funktionen $f:[0, 2\pi]\to \K$, $fin R[0,2\pi]$, $2\pi$-periodisch. +\begin{bsp}[\underline{Trigonometrische Polynome}] + Für $a_k, b_k \in \C$ betrachte + \begin{align*} + f_n(x)&\coloneqq \frac{a_0}{2} + \sum_{k = 1}^{n}(a_k\cos(kx) + b_k\sin(kx))\\ + &\stackrel{(*)}{=} \frac{1}{2}\left(a_0 + \sum_{k = 1}^{n}(a_k - i b_k)e^{i kx} + (a_k + i b_k) e^{-i kx}\right)\\ + &= \sum_{k = -n}^{n}c_k e^{i kx} \text{ mit } c_k = \begin{cases} + \frac{1}{2} (a_k-ib_k), &k\geq 0\\ + \frac{a_0}{2}, &k = 0\\ + \frac{1}{2}(a_{-k} + ib_{-k}), &k < 0 + \end{cases} + \end{align*} + $$(*): \cos(x) = \frac{1}{2}(e^{ix} + e^{-ix}),\; \sin(x) = \frac{1}{2i}(e^{ix} - e^{-ix})$$ + Für $a_k, b_k, k> 0$ ergibt sich $a_k = c_k + c_{-k},\; b_k = i(c_k - c_{-k})$. +\end{bsp} +\begin{bem} + Ist $f$ ein trigonometrisches Polynom, so kann man die Koeffizienten $a_k , b_k , c_k $ durch Integration ausrechen, d.h. $a_k, b_k, c_k$ sind eindeutig. + \begin{align*} + a_k &= \frac{1}{\pi} \int_0^{2\pi} f(x)\cos(kx) \d x, \quad k \geq 0\\ + b_k &= \frac{1}{\pi} \int_0^{2\pi} f(x)\sin(kx) \d x, \quad k > 0\\ + c_k &= \frac{1}{2\pi} \int_0^{2\pi} f(x)e^{-ikx} \d x, \quad k \in \Z\\ + &= \frac{1}{2\pi} (f, e^{ikx}), \quad \text{$L_2$-Skalarprodukt} + \end{align*} +\end{bem} +\begin{proof} + Sei zuerst $0 \neq \lambda \in \R$. + \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\\ + &= \frac{1}{\lambda i } e^{i\lambda x} \bigg|_a^b\\ + \end{align*} + $$\implies \int_0^{2\pi} e^{ikx} \d x = 0\qquad \forall k \in \Z \setminus \{0\}$$ + Dann ist $$\int_0^{2\pi} e^{ik_1x} e^{-ik_2x} \d x = \int_0^{2\pi}e^{i(k_1-k_2)x} \d x = \begin{cases} + 0, &\text{falls }k_1\neq k_2\\ + 2 \pi, &\text{falls } k_1 = k_2\; (\implies k_1 -k_2 = 0) + \end{cases}$$% + $\implies$ Behauptung für $c_k$. Für $a_k, b_k$ gilt + $$a_k = \frac{1}{\pi} \int_0^{2\pi} f(x)\frac{1}{2}\left(e^{ikx} + e^{-ikx}\right) \d x = c_{-k} + c_k = b_k \cdot i$$ +\end{proof} +\begin{bem} + Obige Formel gilt auch für + $$f(x) = \frac{a_0}{2} + \sum_{k = 1}^{\infty}(a_k\cos(kx) + b_k \sin(kx)) = \sum_{k = -\infty}^{\infty}c_k e^{ixk}$$ falls die Reihen gleichmäßig auf $[0, 2\pi]$ konvergieren. +\end{bem} +Frage: Hat jede $2\pi$-periodische Funktion die Form $\sum_{k = -n}^{n}c_ke^{ixk}$ oder kann man sie durch trigonometrische Polynome approximieren? $\implies$ Motivation für Fourier-Reihen. +\begin{definition}[\underline{Fourier-Reihe}] + Sei $f\in R[a,b]$ $2\pi$-periodisch. Die \underline{Fourier-Koeffizienten} von $f$ sind gegeben durch + $$c_k \coloneqq c_k(f) \coloneqq \frac{1}{2\pi} \int_0^{2\pi} f(x)e^{-ixk}\d x = \frac{1}{2\pi}(f,e^{ixk})$$ + Die (formale) \underline{Fourier-Reihe} von $f$ ist $$\sum_{k = -\infty}^{\infty}c_k e^{ikx}$$ mit der n-ten Partialsumme $$s_n(x) = s_n(f,x) \coloneqq \sum_{k = -n}^{n}c_ke^{ikx}.$$ Die Fourier-Reihe läßt sich in der Form schreiben + $$\frac{a_0}{2} + \sum_{k = 1}^{\infty}(a_k\cos(kx) + b_k \sin(kx))$$ wobei \begin{align*} + a_k &= \frac{1}{\pi} \int_0^{2\pi} f(x) \cos(kx) \d x\\ + b_k &= \frac{1}{\pi} \int_0^{1\pi} f(x) \sin(kx)\d x + \end{align*} +\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 $n\in \N$ + $$\qnorm{f-s_n} = \qnorm{f} - 2\pi \sum_{k = -n}^{n}|c_k|^2$$ +\end{satz} +\begin{proof} + Notation $e_n(x) \coloneqq e^{ikx}$ + \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\\ + 0 , & k \neq l + \end{cases}\\ + c_k &= \frac{1}{2\pi} \int_0^{2\pi}f(x)e^{-ikx}\d x = \frac{1}{2\pi}(f, e_k)\quad \implies (f, e_k) = 2\pi c_k\\ + (f, s_n) &= \sum_{k = -n}^{n}(f, c_ke_k) = \sum_{k = -n}^{n}\overline{c_k}(f, e_k) = \sum_{k = -n}^{n}\overline{c_k}2\pi c_k = 2\pi \sum_{k = -n}^{n}|c_k|^2\\ + (s_n, s_n) &= \sum_{k = -n}^{n}\sum_{l = -n}^{n}\underbrace{c_k\overline{c_k}}_{|c_k|^2} \cdot \underbrace{(e_k, e_l)}_{\mathrlap{\scalebox{0.5}{$=\begin{cases} + 0, &k\neq l\\ + 2\pi, &k = l + \end{cases}$}}} + \end{align*} + Dann + \begin{align*} + \qnorm{f-s_n} &= (f-s_n, f-s_n)\\ + &= (f,f) - (f, s_n) - (s_n, f) + (s_n, s_n)\\ + &= \qnorm{f} - 2\pi \sum_{k = -n}^{n}|c_k|^2 - 2\pi \sum_{k = -n}^{n}|c_k|^2 + 2\pi \sum_{k = -n}^{n}|c_k|^2\\ + &= \qnorm{f} - 2\pi \sum_{k = -n}^{n}|c_k|^2 + \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 $$\sum_{k = -\infty}^{\infty} |c_k|^2 = \lim\limits_{n\to\infty}\sum_{k = -n}^{n}|c_k|^2 \leq \qnorm{f}$$ +\end{satz} +\begin{proof} + Aus Satz \ref{fourierungleichung} + $$2\pi \sum_{k = -n}^{n}|c_k|^2 = \qnorm{f} - \underbrace{\qnorm{f-s_n}}_{\geq 0} \leq \qnorm{f}$$ + 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 + $$\qnorm{f} = 2\pi \lim\limits_{n\to\infty}\sum_{k = -n}^{n}|c_k|^2 \Leftrightarrow 2 \pi \sum_{k = -\infty}^{\infty}|c_k|^2$$ +\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 $c_k$.\\ +\underline{Ziel:} Zeige die Parsevalsche Gleichung für die Fourier-Koeffizienten +$c_k$, $\{e_k = e^{ikx}, k\in \Z\}$, $f\in R[a,b]$ +$\implies$ Konvergenz der Fourier-Reihe in $L^2$. +\begin{lemma}\label{HilfslemmaA} + $\forall t\in \R\setminus \{2 \pi k| k \in \Z\}$ gilt $$\sum_{k = 1}^{n}\cos(kt) = \frac{\sin\left((n + \frac{1}{2})t\right)}{2 \sin\left(\frac{1}{2}t\right)} - \frac{1}{2}$$ +\end{lemma} +\begin{proof} + $\cos(kt) = \frac{1}{2}\left(e^{ikt} + e^{-ikt}\right)$ + \begin{align*} + \implies \frac{1}{2} + \sum_{k = 1}^{n}\cos(kt) &= \frac{1}{2} \sum_{k = -n}^{n}e^{ikt}\\ + &= \frac{1}{2}e^{-int}\underbrace{\sum_{k = 0}^{2n}e^{ikt}}_{\mathrlap{\text{geometrische Summenformel}}}\\ + &= \frac{1}{2}e^{-int} \frac{1 - e^{(2n+1)it}}{1 - e^{it}}\\ + &= \frac{1}{2} \frac{e^{-int} - e^{(n+1)it}}{1 - e^{it}}\\ + &\stackrel{\text{Erweitern}}{=}\quad \; \frac{1}{2}\frac{e^{i(n+\frac{1}{2})t} - e^{-i(n+\frac{1}{2})t}}{e^{i\frac{1}{2}t} - e^{-i\frac{1}{2}t}}\\ + &= \frac{1}{2}\frac{\sin\left(\left(n + \frac{1}{2}\right)t\right)}{\sin\left(\frac{1}{2}t\right)} + \end{align*} +\end{proof} +\begin{lemma}\label{HilfslemmaB} + Sei $f:[a,b] \to \R$ eine stetig differenzierbare Funktion. Es gilt für $x\in [a,b]$ und $s\in \R$ + $$F_s(x)\coloneqq \int_a^x f(y)\sin(sy) \d y$$ konvergiert gleichmäßig gegen 0 für $|s| \to \infty$ und $x\in [a,b]$. +\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 $x\in [a,b]$. +\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 $\delta > 0$. +\end{lemma} +\begin{proof} + Aus Hilfslemma \ref{HilfslemmaA} folgt für $0 < x < 2\pi$ und $n\in \N$ + \begin{align*} + \sum_{k = 1}^{n}\frac{\sin(kx)}{k} &= \sum_{k = 1}^{n}\int_\pi^x \cos(ky)\d y\\ + &= \int_\pi^x\left(\sum_{k = 1}^{n}\cos(ky)\right)\d y\\ + &\stackrel{\text{Lemma }\ref{HilfslemmaA}}{=} \qquad \int_\pi^x\left(\frac{\sin\left((n + \frac{1}{2})y\right)}{2 \sin\left(\frac{1}{2}y\right)} - \frac{1}{2}\right) \d y\\ + &= \underbrace{\int_\pi^x \frac{\sin\left((n + \frac{1}{2})y\right)}{2 \sin\left(\frac{1}{2}y\right)} \d y}_{=: F_n(x)} - \frac{1}{2}(x-\pi) + \end{align*} +Z.Z.: $F_n(x)$ konvergiert gleichmäßig gegen 0 für $n \to \infty$. Die Funktion $f(y) = \frac{1}{2\sin\left(\frac{y}{2}\right)}$ ist auf dem Intervall $[\delta, 2\pi-\delta]$ stetig differenzierbar, weil $\frac{y}{2}\neq 0$ auf $[\delta, 2\pi - \delta]$, sodass aus Hilfslemma \ref{HilfslemmaB} folgt +$$F_n(x) = \int_\pi^x \frac{1}{2 \sin\left(\frac{y}{2}\right)} \cdot \sin\left(\left(n + \frac{1}{2}\right) y\right) \d y \overset{\text{glm.}}{\to} 0$$ für $n\to \infty$ und $x\in [\delta, 2\pi - \delta]$. +\end{proof} +\begin{lemma}\label{HilfslemmaD} + Die Reihe $$\sum_{k = 1}^{\infty} \frac{\cos(kx)}{k^2} = \frac{(x-\pi)^2}{4} - \frac{\pi^2}{12}$$ konvergiert gleichmäßig $\forall x, 0 \leq x \leq 2\pi.$ Insbesondere gilt für $x= 0$ $$\sum_{k = 1}^{\infty}\frac{1}{k^2}= \frac{\pi^2}{6}$$ +\end{lemma} +\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\\ + &\stackrel{\ref{HilfslemmaC}}{=} -\int_y^x \sum_{k = 1}^{\infty}\frac{\sin(kt)}{k} \d t\\ + &\stackrel{\text{glm. Konv.}}{\stackrel{\text{Satz reference 1.3.2}}{=}} \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}\\ + \end{align*} + $$\xRightarrow{y \text{ fest}} \frac{(x-\pi)^2}{4} =\sum_{k = 1}^{\infty} \frac{\cos(kx)}{k^2} + C\quad \forall x\in (0, 2\pi), C \text{ konst}$$ + 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} &= C \cdot 2\pi\\ + C &= \frac{\pi^2}{12} + \end{align*} + $$\implies \sum_{k = 1}^{\infty}\frac{\cos(kx)}{k^2} = \frac{(x - \pi)^2}{4} - \frac{\pi^2}{12},\quad x \in (0,2\pi)$$ + Für $x = 0$ oder $x = 2\pi$ folgt die Behauptung durch Grenzübergang, da \underline{beide} Seiten stetig sind auf $[0, 2\pi]$ +\end{proof} +\begin{lemma}\label{HilfslemmaE} + Sei $f$ Treppenfunktion, $f\in R[0, 2\pi]$, $2\pi$ periodisch. Dann $s_n \underset{n\to \infty}{\to} f$ in $L^2[0, 2\pi]$, d.h. Fourier-Reihe von $f$ konvergiert im quadratischen Mittel gegen $f$. +\end{lemma} +\begin{proof} + Zunächst Spezialfall + $$f_a(x) \coloneqq \begin{cases} + 1, &0 0$. Dann gibt es zu $\varepsilon$ $2\pi$-periodische Treppenfunktionen $\varphi_\varepsilon, \psi_\varepsilon: \R \to \R$ mit Eigenschaften + $$-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$$ + Konstruktion von $\varphi_\varepsilon, \psi_\varepsilon$ siehe Rannacher. Dann, $$|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}$$ + Dann gilt $\forall n \geq n_\varepsilon$ + \begin{align*} + \norm{f-s_n(f)} &= \norm{f-s_n(f-\varphi_\varepsilon) - s_n(\varphi_\varepsilon) - \varphi_\varepsilon +\varphi_\varepsilon}\\ + &\leq \norm{(f-\varphi_\varepsilon) - s_n(f - \varphi_\varepsilon)} + \norm{\varphi_\varepsilon - s_n(\varphi_\varepsilon)}\\ + &\leq \frac{\varepsilon}{2} + \frac{\varepsilon}{2}\\ + &= \varepsilon + \end{align*} + $$\implies s_n(f) \xrightarrow{L^2} f, \; n \to \infty$$ +\end{proof} +\begin{bem} + Konvergenz in $L^2$ ist \glqq sehr schwach \grqq\. Für \glqq glattere\grqq\ Funktionen konvergiert die Fourier-Reihe gleichmäßig. +\end{bem} +\begin{satz} + Sei $f: \R \to \C$ $2\pi$-periodisch, stetig und stückweise stetig differenzierbar, d.h. $\exists $ Unterteilung von $[0, 2\pi]$ + $$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 $\sum_{k \in \Z} |\gamma_k|^2 = \frac{1}{2\pi}\qnorm{\phi} < \infty$. + 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\\ + &= \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| \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 $s_n(f) \xrightarrow{\text{glm.}} g,\; n\to \infty,\; \Rightarrow s_n(f) \xrightarrow{L^2} g,\; n\to \infty$. Andererseits $s_n(f) \xrightarrow{L^2} f,\; n\to \infty$ + \begin{align*} + &\implies \norm{f-g}_{L^2} = 0\\ + &\implies f\equiv g,\text{ weil $f$ und $g$ stetig sind}\\ + &\implies s_n(f)\xrightarrow{\text{glm.}} f,\; n\to \infty + \end{align*} + \end{proof} +\end{document}