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\d x + i \int_{a}^{b} \text{Im} f(x) \d x + .\] + +\end{definition} + +\begin{bem} + \begin{enumerate} + \item Analog: Definitionen von uneigentlichen Riemann-integralen für + komplexwertige Funktionen + \item Die Rechenregeln f+r das reelle Riemann-integral übertragen sich auf komplexwertige + Integrale, insbesondere gilt: + \begin{align*} + \int_{a}^{b} \overline{f(x)} \d x &= \int_{a}^{b} \left( \text{Re}f(x) - i \cdot \text{Im}f(x) \right) \d x \\ + &= \int_{a}^{b} \text{Re}f(x) \d x - i \int_{a}^{b} \text{Im}f(x) \d x \\ + &= \overline{\int_{a}^{b} f(x) \d x } + .\end{align*} + \end{enumerate} +\end{bem} + +\begin{definition} + Eine Funktion $f\colon [a,b] \to \mathbb{K}$ ($\mathbb{K} = \R$ oder $\mathbb{K} = \mathbb{C}$) + heißt stückweise stetig, falls + \begin{enumerate}[1)] + \item $f$ in $[a,b]$ bis auf endlich viele Ausnahmestellen stetig und beschränkt ist. + \item in jeder dieser Unstetigkeitsstellen $\xi \in [a,b]$ die links- bzw. + rechtsseitigen Grenzwerte + \[ + f(\xi_{\pm} := \lim_{h \searrow 0} f(\xi \pm h) + .\] existieren. Für $\xi \in (a,b)$ wird + \[ + f(\xi) := \frac{f(\xi_{-} + f(\xi_{+})}{2} + .\] gesetzt. + \end{enumerate} +\end{definition} + +\begin{bem} + Stückweise stetige Funktionen sind Riemann-integrierbar. + + Die Menge der in diesem Sinne auf $[a,b]$ stückweise stetigen (Riemann-integrierbaren) + Funktionen bilden einen Vektorraum $R[a,b]$. +\end{bem} + +\begin{definition} + Wir definieren + \[ + (f, g) := \int_{a}^{b} f(x) \overline{g(x)} \d x \qquad (\text{,,Sesquilinearform''}) + .\] Dies ist wohldefiniert da für $f, g \in R[a,b]$ das Produkt $f(x) \cdot \overline{g(x)} \in R[a,b]$ ist. +\end{definition} + +\begin{definition}[Skalarprodukt] + Sei $V$ Vektorraum über $\mathbb{K}$. Die Abbildung $<\cdot, \cdot >\colon V \times V \to \mathbb{K}$ + heißt Skalarprodukt auf $V$, falls $\forall u, v, w \in V$ und $\alpha \in \mathbb{K}$ gilt: + \begin{enumerate}[(S1)] + \item $\langle v, u\rangle = \overline{\langle u, v\rangle}$ (Symmetrie, + hermitesch falls $\mathbb{K} = \mathbb{C}$ + symmetrisch falls $\mathbb{K} = \R$) + \item $\langle\alpha v, u\rangle = \alpha \langle v, u\rangle$ \\ + $\langle v, \alpha u\rangle = \overline{\alpha}\langle v, u\rangle$ \\ + $\langle v, u + w\rangle = \langle v, u\rangle + \langle v, w\rangle$ \\ + $\langle v + u, w\rangle = \langle v, w\rangle + \langle u, w\rangle$ + \item Positivdefinitheit: $\langle v, v \rangle \ge 0$ \\ + $\langle v, v \rangle = 0 \iff v = 0$ + \end{enumerate} +\end{definition} + +\begin{bem} + Auf $R[a,b]$ besitzt $(\cdot , \cdot )$ die Eigenschaften eines Skalarprodukts, denn + es gilt $\forall \alpha, \beta \in \mathbb{C}$, $\forall f, g \in R[a,b]$, $f_1, f_2 \in R[a,b]$, + $g_1, g_2 \in R[a,b]$: + \begin{enumerate}[(1)] + \item $(\alpha f_1 + \beta f_2, g) = (\alpha f_1, g) + (\beta f_2, g) = \alpha (f_1, g) + \beta (f_2, g)$ + \item $(f, \alpha g_1 + \beta g_2) = (f, \alpha g_1) + (f, \beta g_2) = \overline{\alpha}(f, g_1) + \overline{\beta} (f, g_2)$ + \item $\displaystyle (f, g) + = \int_{a}^{b} f \cdot \overline{g} \d x + = \int_{a}^{b} \overline{\overline{f} g} \d x + = \overline{\int_{a}^{b} \overline{f} g} \d x + = \overline{\int_{a}^{b} g \overline{f} \d x } + = \overline{(g, f)}$ + \item $(f,f) = \displaystyle \int_{a}^{b} f \overline{f} \d x = \int_{a}^{b} |f(x)|^2 \d x \ge 0$ + \item Aus (4) und der Definition von $R[a,b]$ folgt: $(f,f) = 0 \implies f \equiv 0$ auf $[a,b]$. + \end{enumerate} + $(\cdot , \cdot )$ wird auf $R[a,b]$ $L^2$-Skalarprodukt genannt. +\end{bem} + +\begin{lemma} + Für ein $L^2$-Skalarprodukt $(\cdot , \cdot )$ auf $R[a,b]$ gilt die Cauchy-Schwarz Ungleichung: + \[ + |(f,g)|^2 \le (f,f)\cdot (g,g) + .\] +\end{lemma} + +\begin{proof} + \begin{enumerate}[1)] + \item Falls $g \equiv 0$ gilt trivialerweise + \[ + |(f,g)|^2 = 0 = (f,f) \cdot (g,g) + .\] + \item Falls $g \not\equiv 0$, sei $\alpha \in \mathbb{K}$ beliebig + \[ + 0 \le (f + \alpha g, f + \alpha g) = (f,f) + \alpha(g,f) + \overline{\alpha}(f,g) + + \alpha \cdot \overline{\alpha}(g,g) + .\] Setze $\alpha := - \frac{(f,g)}{(g,g)} = - \frac{\overline{(g,f)}}{(g,g)}$. Dann gilt + \begin{align*} + 0 &\le (f,f) - \frac{(f,g) \cdot (g, f)}{(g,g)} - \frac{(g, f) \cdot (f,g)}{(g,g)} + + \frac{(f,g)(g,f)(g,g)}{(g,g)(g,g)} \\ + &= (f,f) - \frac{(f,g)(g, f)}{(g,g)} \\ + &= (f,f) - \frac{\overline{(f,g)}(f,g)}{(g,g)} \\ + &= (f,f) - \frac{|(f,g)|^2}{(g,g)} \\ + \implies 0 &\le (f,f)(g,g) - |(f,g)|^2 + .\end{align*} + \end{enumerate} +\end{proof} + +\begin{definition}[$L^2$-Norm] + Das $L^2$-Skalarprodukt $(\cdot , \cdot )$ induziert die $L^2$-Norm auf $R[a,b]$ mit + \[ + \Vert f \Vert = \Vert f \Vert_{L^2} := (f,f)^{\frac{1}{2}} = \left(\int_{a}^{b} f \cdot \overline{f} \d x\right)^{\frac{1}{2}} + .\] +\end{definition} + +\begin{bem} + Normeigenschaften von $L^2$ auf $R[a,b]$ sind erfüllt: + \begin{enumerate}[(N1)] + \item Definitheit: $\Vert f \Vert = 0 \implies (f,f) = 0 \implies f = 0$ auf $[a,b]$ + \item Homogenität: $\Vert \alpha f \Vert = (\alpha f, \alpha f)^{\frac{1}{2}} = (|\alpha|^2 (f,f))^{\frac{1}{2}} = |\alpha| \cdot \Vert f \Vert$ + \item Dreiecksungleichung: + \begin{align*} + \Vert f + g \Vert &= (f + g, f + g)^{\frac{1}{2}} \\ + &= \left( \Vert f \Vert^2 + (f,g) + (g,f) + \Vert g \Vert^2 \right)^{\frac{1}{2}} \\ + &\stackrel{\text{CSU}}{\le} \left( \Vert f \Vert^2 + 2 \Vert f \Vert \Vert g \Vert + + \Vert g \Vert^2\right)^{\frac{1}{2}} \\ + &= \Vert f \Vert + \Vert g \Vert + .\end{align*} + \end{enumerate} +\end{bem} + +\begin{definition}[Konvergenz im Quadratischen Mittel ($L^2$-Konvergenz)] + Seien $f_n \in R[a,b], n \in \N, f \in R[a,b]$. $f_n$ konvergiert gegen $f$ im Quadratischen + Mittel $f_n \xrightarrow[L^2]{n \to \infty}$, wenn gilt + \[ + \Vert f_n - f \Vert_{L^2} \xrightarrow{n \to \infty} 0 + .\] Das heißt, dass die quadratische Abweichung zwischen $f_n$ und $f$ gegen Null konvergiert: + \[ + \int_{a}^{b} |f_n(x) - f(x)|^2 \d x \xrightarrow{n \to \infty} 0 + .\] +\end{definition} + +\begin{bem} + \begin{enumerate}[(1)] + \item Es gilt: + \[ + \Vert f_n - f \Vert_{L^2}^2 = \int_{a}^{b} |f_n(x) - f(x)|^2 \d x + \le \Vert f_n - f \Vert_{\infty}^2 (b-a) + .\] Damit folgt + \[ + \Vert f_n - f \Vert_{\infty} \xrightarrow{n \to \infty} 0 + \implies \Vert f_n - f \Vert_{L^2} \xrightarrow{n \to \infty} 0 + .\] + Die Umkehrung gilt i.A. nicht! Beispiel: $f_n(x) := x^{n}$, $x \in [-1, 1]$ + \begin{figure}[h!] + \centering + \begin{tikzpicture} + \begin{axis}% + [grid=both, + minor tick num=4, + grid style={line width=.1pt, draw=gray!10}, + major grid style={line width=.2pt,draw=gray!50}, + axis lines=middle, + enlargelimits={abs=0.2}, + ymax=1, + ymin=-1, + ] + \addplot[domain=-1:1,samples=50,smooth,red] {x^1}; + \addplot[domain=-1:1,samples=50,smooth,purple] {x^2}; + \addplot[domain=-1:1,samples=50,smooth,green] {x^3}; + \legend{$n=1$, $n=2$, $n=3$} + \end{axis} + \end{tikzpicture} + \begin{tikzpicture} + \begin{axis}% + [grid=none, + minor tick num=4, + grid style={line width=.1pt, draw=gray!10}, + major grid style={line width=.2pt,draw=gray!50}, + axis lines=middle, + enlargelimits={abs=0.2}, + ymax=1, + ymin=0, + ytick={0}, + xtick = {0.2, 0.5, 0.9}, + xticklabels = {$a$, $\xi$, $b$} + ] + \addplot[domain=0:1,samples=50,smooth,red] {0}; + \node[red,circle,fill,inner sep=0.5pt] at (axis cs:0.5,0.5) {}; + \node[black,circle,fill,inner sep=0.5pt] at (axis cs:0.5,0) {}; + \end{axis} + \end{tikzpicture} + \caption{Links: $f_n(x) = x^{n}$, Rechts: $f(x) \not\equiv 0$} + \label{abb:nichtvollstaendig} + \end{figure} + \[ + \Vert f_n \Vert^2_{L^2} = \int_{-1}^{1} x^{2n} \d x + = 2 \int_{0}^{1} x^{2n} \d x + = 2 \frac{x^{2n+1}}{2n+1} \Big|_{0}^{1} = \frac{2}{2n+1} \xrightarrow{n \to \infty} 0 + .\] Damit folgt $f_n \xrightarrow[L^2]{n \to \infty} f \equiv 0$. + + Aber wegen $f_n(1) = 1$ für $x = 1$, $n \in \N$, konvergiert $f_n$ nicht punktweise gegen + $f \equiv 0$ und wegen $f_n(-1) = (-1)^{n}, n \in \N, x = -1$ konvergiert $f_n$ nicht. + \item Der Raum $R[a,b]$ mit $L^2$-Norm $\Vert \cdot \Vert$ ist \textbf{nicht vollständig}, d.h. + es existieren Cauchy-Folgen in $R[a,b]$, die keinen Grenzwert in $R[a,b]$ haben. Beispiel: siehe + Abb. \ref{abb:nichtvollstaendig} (Rechts). Hier ist + $f(x) \not\equiv 0$, $x \in [a,b]$. + \[ + \int_{a}^{b} |f(x)|^2 \d x = 0 = \Vert f \Vert_{L^2} + ,\] aber $f(x) \not\in R[a,b]$, denn + \[ + f(\xi) \neq 0 = \frac{\lim_{h \searrow 0} f(\xi + h) - \lim_{h \searrow 0} f(\xi - h)}{2} + .\] + \end{enumerate} +\end{bem} + +\begin{definition}[Orthogonalität] + $f, g \in R[a,b]$ heißen orthogonal, wenn gilt $(f, g) = 0$. + + Eine Teilmenge $S \subset R[a,b]$ heißt Orthogonalsystem, wenn alle Elemente + aus $S$ paarweise orthogonal sind, d.h. + \[ + (f_i, f_j) = \begin{cases} + \Vert f_i \Vert^2 & i = j \\ + 0 & i \neq j + \end{cases} \quad \forall f_i, f_j \in S + .\] +\end{definition} + +\begin{satz} + Die trigonometrischen Funktionen, für $k, l \in \N$ + \begin{align*} + c_k(x) &:= \begin{cases} + 1 & k = 0 \\ + \cos(k x) & \text{sonst} + \end{cases} \\ + s_l(x) &:= \sin (l x) + \end{align*} bilden auf $R[a,b]$ bezüglich des $L^2$-Skalarprodukts ein + Orthogonalsystem und es gilt + \begin{align*} + &\int_{0}^{2 \pi} c_k(x) \d x = \int_{0}^{2 \pi} s_l(x) \d x = \int_{0}^{2\pi} c_k(x) s_l(x) \d x = 0 \\ + &\int_{0}^{2\pi} c_k(x) c_l(x) \d x = \pi \delta_{kl} \\ + &\int_{0}^{2\pi} s_k(x) s_l(x) \d x = \pi \delta_{kl} + \intertext{Hier sei} + &\delta_{kl} := \begin{cases} + 1 & k = l \\ + 0 & k \neq l + \end{cases} \qquad \text{Kroneckersymbol} + .\end{align*} +\end{satz} + +\begin{proof} + \begin{align*} + \int_{0}^{2\pi} c_k(x) \d x + &= \int_{0}^{2\pi} \cos(k x) \d x + = \frac{1}{k} \sin(k x) \Big|_{0}^{2\pi} = 0 \\ + \int_{0}^{2\pi} s_k(x) \d x &= \int_{0}^{2\pi} \sin(kx) \d x = - \frac{1}{k} \cos(k x) \Big|_{0}^{2\pi} + = \frac{1}{k}(1-1) = 0 + \intertext{Damit folgt} + \int_{0}^{2\pi} c_k(x) s_l(x) \d x \quad &= \quad + \int_{0}^{2\pi} \underbrace{\cos(k x)}_{u'} \cdot \underbrace{\sin(l x)}_{v} \d x \\ + &\stackrel{\text{part. Int.}}{=} + \quad + \underbrace{\underbrace{\frac{1}{k} \sin(k x)}_{u} \cdot \underbrace{\sin(l x)}_{v} \Big|_{0}^{2\pi}}_{= 0} + - \int_{0}^{2\pi} \underbrace{\frac{1}{k} \sin(k x)}_{u} \underbrace{l \cos(l x)}_{v'} \d x \\ + &= \quad - \frac{l}{k} \int_{0}^{2\pi} s_k(x) c_l(x) \d x + \intertext{Für $l = k$ gilt} + \int_{0}^{2\pi} c_k(x) s_k(x) \d x &= - \int_{0}^{2\pi} c_k(x) s_k(x) \d x \\ + \implies 2 \int_{0}^{2\pi} c_k(x) s_k(x) \d x &= 0 \\ + \implies \int_{0}^{2\pi} c_k(x) s_k(x) \d x &= 0 + \intertext{Analog folgt mit partieller Integration} + \int_{0}^{2\pi} c_k(x) c_l(x) \d x + &= \frac{l}{k} \int_{0}^{2\pi} s_k(x) s_l(x) \d x \\ + \stackrel{l = k}{\implies} \int_{0}^{2\pi} c_k^2 \d x + &= \int_{0}^{2\pi} s_k^2 \d x = \int_{0}^{2\pi} (1- c_k^2(x)) \d x + = 2\pi - \int_{0}^{2\pi} c_k^2(x) \d x \\ + \implies \int_{0}^{2\pi} c_k^2(x) \d x &= \pi = \int_{0}^{2\pi} s_k^2(x) \d x + \intertext{Wenn $k \neq l$, dann folgt} + \int_{0}^{2\pi} c_k(x) c_l(x) \d x \quad + &= \quad \frac{l}{k} \int_{0}^{2\pi} s_k(x) s_l(x) \d x \\ + &\stackrel{\text{part. Int.}}{=} \quad \frac{l^2}{k^2} \int_{0}^{2\pi} c_k(x) c_l(x) \d x \\ + \implies \int_{0}^{2\pi} c_k(x) c_l(x) \d x \quad &= \quad 0 + \intertext{Analog} + \int_{0}^{2\pi} s_k(x) s_l(x) \d x &= 0 \\ + \int_{0}^{2\pi} c_k(x) s_l(x) \d x &= 0 + .\end{align*} +\end{proof} + +\end{document} diff --git a/analysisII.pdf b/analysisII.pdf index f6002c02a8d092eff041ab7b64c33ba883a93829..33e51f309a31973bcc25ae9d62446cde9a0eb267 100644 GIT binary patch delta 128981 zcmV(`K-0gVo)4I`5wI!&0yQv~0SEypf8|w8bJH*oz57?_C?Bj!tCeI8Cn#x~!gRp# z^gxf-IkI7OZh;zaT5Sn*!gSYLYm(~GM@J<j-G z!SJ)#TfUr^MHMHS`r1^J*2s$4e>6C$0PyAZ1=}w+)z|fGSv3GaZ7hejqS_xdk@v_F zkOEDpDyXiK0F-81txIzsDCj_ejB?&|1Sy{`ZZzVJc5YS{ZR|tiCraRm9hpT<-#^5O zafSUAXJg|Y5Gkj%05+5_({ib#rKi3Jb{7=B>;Q2P7+ z1QN}(W0#SS0Ti=o19i{?H8VGp;nOOATXPgQ7Jm1y=)8?9X`!RL2y9gnAdsY-U2xc{ z1s+V?9tL~Hwc~M;<=5{~U!@-PWiHqz@PeBejarv;`Mz_Gbnp22KOvc39nfT&(B#L1 z?~f1eJ&`i8#44teOgZO&li!XW z6%WrJKYLnz_%d%){^!YWM?W7w`}N3u36x1Sd3N;lyI&5Cm>j=&`S|&N9zXlx>Cv+n 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z0&Vz8xlXR)+NrRa33z;kBP?_H+%4%ytJ_Qub2}xjv66UszBMc zNpBn0&Xetq0L>>mhMpEY)2&eJc|mCmSzU3~D3&jW9xwcd5A26=li|=Iwfet*6^6mK zWII0Ct$lkv)&QG~;VLjEeh|@yG5B)5#}#$3?+>3sp1F|0=ndh(FaYCWxbu!Mu@zBI zr?hu|?)poshNnEGET#6rXWOfG6z*kVQLagG@hPo(jEN&rNhDCGw8+53&UOaaj=Nao zwxAnx3_7C~S16?=GLN#pnFErr-j$w|)nv?9xTR28XlKXO4{hlE#w{Y;`q!Gm@aWhENdN!< diff --git a/analysisII.tex b/analysisII.tex index 41d839d..6ce94f9 100644 --- a/analysisII.tex +++ b/analysisII.tex @@ -1,4 +1,4 @@ -\documentclass[titlepage]{../../../lecture} +\documentclass[titlepage]{lecture} \usepackage{standalone} \usepackage{tikz} @@ -18,5 +18,6 @@ \newpage \input{ana1.tex} +\input{ana2.tex} \end{document} diff --git a/lecture.cls b/lecture.cls new file mode 100644 index 0000000..0edddc7 --- /dev/null +++ b/lecture.cls @@ -0,0 +1,136 @@ +\ProvidesClass{lecture} +\LoadClass[a4paper, titlepage]{article} + +\RequirePackage[utf8]{inputenc} +\RequirePackage[T1]{fontenc} +\RequirePackage{textcomp} +\RequirePackage[german]{babel} +\RequirePackage{amsmath, amssymb, amsthm} +\RequirePackage{mdframed} +\RequirePackage{fancyhdr} +\RequirePackage{geometry} +\RequirePackage{import} +\RequirePackage{pdfpages} +\RequirePackage{transparent} +\RequirePackage{xcolor} +\RequirePackage{array} +\RequirePackage[shortlabels]{enumitem} +\RequirePackage{tikz} +\RequirePackage{pgfplots} +\RequirePackage[nobottomtitles]{titlesec} +\RequirePackage{listings} +\RequirePackage{mathtools} +\RequirePackage{forloop} +\RequirePackage{totcount} + +\usetikzlibrary{quotes, angles} + +\DeclareOption*{\PassOptionsToClass{\CurrentOption}{article}} +\DeclareOption{uebung}{ + \makeatletter + \lhead{\@title} + \rhead{\@author} + \makeatother +} +\ProcessOptions\relax + +% PAGE GEOMETRY +\geometry{ + left=15mm, + right=40mm, + top=20mm, + bottom=20mm +} + +% PARAGRAPH no indent but skip +\setlength{\parskip}{3mm} +\setlength{\parindent}{0mm} + +\theoremstyle{definition} +\newmdtheoremenv{satz}{Satz}[section] +\newmdtheoremenv{lemma}[satz]{Lemma} +\newmdtheoremenv{korrolar}[satz]{Korrolar} +\newmdtheoremenv{definition}[satz]{Definition} + +\newtheorem{bsp}[satz]{Beispiel} +\newtheorem{bem}[satz]{Bemerkung} +\newtheorem{aufgabe}{Aufgabe} + +% enable aufgaben counting +\regtotcounter{aufgabe} + +\newcommand{\N}{\mathbb{N}} +\newcommand{\R}{\mathbb{R}} +\newcommand{\Z}{\mathbb{Z}} +\newcommand{\Q}{\mathbb{Q}} +\newcommand{\C}{\mathbb{C}} + +% HEADERS + +\pagestyle{fancy} + +\newcommand{\incfig}[1]{% + \def\svgwidth{\columnwidth} + \import{./figures/}{#1.pdf_tex} +} +\pdfsuppresswarningpagegroup=1 + +% horizontal rule +\newcommand\hr{ + \noindent\rule[0.5ex]{\linewidth}{0.5pt} +} + +% punkte tabelle +\newcommand{\punkte}{ + \@punkten{\totvalue{aufgabe}} +} + +\def\@punkten#1{ + \newcounter{n} + \begin{tabular}{|c|*{#1}{m{1cm}|}m{1cm}|@{}m{0cm}@{}} + \hline + Aufgabe + \forloop{n}{1}{\not{\value{n} > #1}}{ + & \centering A\then + } + & \centering $\sum$ & \\[5mm] \hline + Punkte + \forloop{n}{1}{\not{\value{n} > #1}}{ + & + } + & & \\[5mm] \hline + \end{tabular} +} + +% code listings, define style +\lstdefinestyle{mystyle}{ + commentstyle=\color{gray}, + keywordstyle=\color{blue}, + numberstyle=\tiny\color{gray}, + stringstyle=\color{black}, + basicstyle=\ttfamily\footnotesize, + breakatwhitespace=false, + breaklines=true, + captionpos=b, + keepspaces=true, + numbers=left, + numbersep=5pt, + showspaces=false, + showstringspaces=false, + showtabs=false, + tabsize=2 +} + +% activate my colour style +\lstset{style=mystyle} + +% better stackrel +\let\oldstackrel\stackrel +\renewcommand{\stackrel}[2]{% + \oldstackrel{\mathclap{#1}}{#2} +}% + +% integral d sign +\makeatletter \renewcommand\d[1]{\ensuremath{% + \;\mathrm{d}#1\@ifnextchar\d{\!}{}}} +\makeatother