diff options
Diffstat (limited to 'vorlesungen/slides/5')
| -rw-r--r-- | vorlesungen/slides/5/Makefile.inc | 5 | ||||
| -rw-r--r-- | vorlesungen/slides/5/approximation.tex | 56 | ||||
| -rw-r--r-- | vorlesungen/slides/5/chapter.tex | 4 | ||||
| -rw-r--r-- | vorlesungen/slides/5/plan.tex | 198 | ||||
| -rw-r--r-- | vorlesungen/slides/5/planbeispiele.tex | 103 | ||||
| -rw-r--r-- | vorlesungen/slides/5/stoneweierstrass.tex | 63 |
6 files changed, 425 insertions, 4 deletions
diff --git a/vorlesungen/slides/5/Makefile.inc b/vorlesungen/slides/5/Makefile.inc index 4ca3de4..bea2feb 100644 --- a/vorlesungen/slides/5/Makefile.inc +++ b/vorlesungen/slides/5/Makefile.inc @@ -5,6 +5,8 @@ # (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule # chapter5 = \ + ../slides/5/plan.tex \ + ../slides/5/planbeispiele.tex \ ../slides/5/verzerrung.tex \ ../slides/5/motivation.tex \ ../slides/5/charpoly.tex \ @@ -36,9 +38,12 @@ chapter5 = \ ../slides/5/satzvongelfand.tex \ \ ../slides/5/stoneweierstrass.tex \ + ../slides/5/swbeweis.tex \ ../slides/5/potenzreihenmethode.tex \ ../slides/5/logarithmusreihe.tex \ ../slides/5/exponentialfunktion.tex \ ../slides/5/hyperbolisch.tex \ + \ + ../slides/5/approximation.tex \ ../slides/5/chapter.tex diff --git a/vorlesungen/slides/5/approximation.tex b/vorlesungen/slides/5/approximation.tex new file mode 100644 index 0000000..a35bae7 --- /dev/null +++ b/vorlesungen/slides/5/approximation.tex @@ -0,0 +1,56 @@ +% +% approximation.tex +% +% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule +% + +\begin{frame}[t] +\frametitle{Approximation einer reellen Funktion} +\vspace{-18pt} +\begin{columns}[t,onlytextwidth] +\begin{column}{0.5\textwidth} +\begin{block}{Gegeben} +Eine stetige Funktion $f\colon[a,b]\to\mathbb{R}$ +\end{block} +\end{column} +\begin{column}{0.5\textwidth} +\uncover<2->{% +\begin{block}{Gesucht} +Approximationspolynome $p_n\to f$ gleichmässig auf $[a,b]$ +\end{block}} +\end{column} +\end{columns} +\uncover<3->{% +\begin{block}{Lösungsmöglichkeiten} +\vspace{-3pt} +\begin{center} +\renewcommand{\arraystretch}{1.3} +\begin{tabular}{|p{4.2cm}|l|} +\hline +Familie&Approximationspolynom für $[a,b]=[0,1]$ +\\ +\hline +\uncover<4->{% +\raggedright +Lagrange-Interpolationspolynom} +&\uncover<5->{% +$\displaystyle\begin{aligned} +l(x)&=(x-x_0)(x-x_1)\dots(x-x_n),\quad x_k = \frac{k}{n} +\\ +p_n(x)&= \sum_{k=0}^n f(x_k)\frac{l(x)}{x-x_k} +\end{aligned}$} +\\ +\hline\uncover<6->{% +\raggedright +Approximation mit Bernstein-Polynomen} +&\uncover<7->{$\displaystyle \begin{aligned} +B_{k,n}(t) &= \frac{1}{(b-a)^n}\binom{n}{k}(t-a)^k(b-t)^{n-k} +\\ +B_n(f)(t) &= \sum_{k=0}^n B_{k,n}(t) \cdot f\biggl(\frac{k}{n}\biggr) +\end{aligned}$} +\\ +\hline +\end{tabular} +\end{center} +\end{block}} +\end{frame} diff --git a/vorlesungen/slides/5/chapter.tex b/vorlesungen/slides/5/chapter.tex index 96eea29..314269d 100644 --- a/vorlesungen/slides/5/chapter.tex +++ b/vorlesungen/slides/5/chapter.tex @@ -3,6 +3,8 @@ % % (c) 2021 Prof Dr Andreas Müller, Hochschule Rapperswi % +\folie{5/plan.tex} +\folie{5/planbeispiele.tex} \folie{5/verzerrung.tex} \folie{5/motivation.tex} \folie{5/charpoly.tex} @@ -28,9 +30,11 @@ \folie{5/Aiteration.tex} \folie{5/satzvongelfand.tex} \folie{5/stoneweierstrass.tex} +\folie{5/swbeweis.tex} \folie{5/potenzreihenmethode.tex} \folie{5/logarithmusreihe.tex} \folie{5/exponentialfunktion.tex} \folie{5/hyperbolisch.tex} \folie{5/spektrum.tex} \folie{5/normal.tex} +\folie{5/approximation.tex} diff --git a/vorlesungen/slides/5/plan.tex b/vorlesungen/slides/5/plan.tex new file mode 100644 index 0000000..23b1b93 --- /dev/null +++ b/vorlesungen/slides/5/plan.tex @@ -0,0 +1,198 @@ +% +% plan.tex +% +% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule +% +\bgroup +\definecolor{darkgreen}{rgb}{0,0.5,0} +\definecolor{darkred}{rgb}{0.8,0.0,0} +\begin{frame}[t] +\frametitle{Was ist $f(A)$?} +\vspace{-5pt} +\begin{center} +\begin{tikzpicture}[>=latex,thick] + +\uncover<7->{ + \fill[color=blue!20] (-1.5,0.7) rectangle (11.5,3.8); +} + +\uncover<4->{ + \fill[color=darkgreen!20] (-1.5,-0.7) rectangle (11.5,0.7); +} + +\uncover<12->{ + \fill[color=darkred!20] (-1.5,-0.7) rectangle (11.5,-3.8); +} + +\begin{scope}[xshift=-1cm] +\node at (0,0) [left] {$A$}; +\end{scope} + +%\foreach \x in {1,...,20}{ +% \only<\x>{ \node at (-1,3) {\x}}; +%} + +% +% Blauer Ast +% + +\uncover<2->{ + \draw[->,color=blue,shorten <= 0.3cm, shorten >= 0.0cm] + (-1.2,0) -- (0,1.3); + + \begin{scope}[xshift=0cm,yshift=1.5cm] + \fill[color=white,opacity=0.7] (0,-0.6) rectangle (3.4,0.6); + \draw[color=blue] (0,-0.6) rectangle (3.4,0.6); + \node at (0,0) [right] {$\begin{aligned} + f&=p\in\mathbb{R}[X]\\ + f(A)&=p(A) + \end{aligned} + $}; + \end{scope} +} + +\uncover<7->{ + \draw[->,color=blue] (1.8,2.1) -- (3.6,3); + + \begin{scope}[xshift=3.6cm,yshift=3cm] + \fill[color=white,opacity=0.7] (0,-0.6) rectangle (3.7,0.6); + \draw[color=blue] (0,-0.6) rectangle (3.7,0.6); + \node at (0,0) [right] {\begin{minipage}{3cm}\raggedright + $f$ durch $p_n\in\mathbb{R}[X]$\\ + approximieren + \end{minipage}}; + \end{scope} +} + +\uncover<8->{ + \draw[->,color=blue] (7.3,3) -- (9.5,1.9); + + \begin{scope}[xshift=7.6cm,yshift=1.5cm] + \fill[color=white,opacity=0.7] (0,-0.35) rectangle (3.8,0.4); + \draw[color=blue] (0,-0.35) rectangle (3.8,0.4); + \node at (0,0) [right] {$\displaystyle f(A) = \lim_{n\to\infty}p_n(A)$}; + \end{scope} +} + +\uncover<9->{ + \node[color=blue] at (3.6,1.6) [right] {\begin{minipage}{4cm} + \raggedright + Konvergenz $p_n\to f$\\ + auf Spektrum $\operatorname{Sp}(A)\subset\mathbb{R}$ + \end{minipage}}; +} + +\uncover<11->{ + \node[color=blue] at (-1.5,3.8) [below right] + {$A$ symmetrisch: $A=A^*$}; +} +\uncover<10->{ + \node[color=blue] at (11.5,3.8) [below left] {$A$ diagonalisierbar}; +} + +% +% Roter Ast +% + +\uncover<12->{ + \draw[->,color=darkred,shorten <= 0.3cm, shorten >= 0.0cm] (-1.2,0) -- (0,-1.3); + + \begin{scope}[xshift=0cm,yshift=-1.5cm] + \fill[color=white,opacity=0.7] (0,-0.6) rectangle (3.4,0.6); + \draw[color=darkred] (0,-0.6) rectangle (3.4,0.6); + \node at (0,0) [right] {$\begin{aligned} + f&=p\in\mathbb{C}[Z,\overline{Z}]\\ + f(A)&=p(A,A^*) + \end{aligned}$}; + \end{scope} +} + +\uncover<13->{ + \node[color=darkred] at (1.7,-2.1) [below left] + {Für $|Z|^2 = Z\overline{Z}$}; +} + +\uncover<14->{ + \draw[->,color=darkred] (1.8,-2.1) -- (3.6,-3); + + \begin{scope}[xshift=3.6cm,yshift=-3cm] + \fill[color=white,opacity=0.7] (0,-0.6) rectangle (3.7,0.6); + \draw[color=darkred] (0,-0.6) rectangle (3.7,0.6); + \node at (0,0) [right] {\begin{minipage}{3.5cm}\raggedright + $f$ durch $q_n\in\mathbb{C}[Z,\overline{Z}]$\\ + approximieren + \end{minipage}}; + \end{scope} +} + +\uncover<15->{ + \draw[->,color=darkred] (7.3,-3) -- (9.5,-1.85); + + \begin{scope}[xshift=7.6cm,yshift=-1.5cm] + \fill[color=white,opacity=0.7] (0,-0.35) rectangle (3.8,0.4); + \draw[color=darkred] (0,-0.35) rectangle (3.8,0.4); + \node at (0,0) [right] + {$\displaystyle f(A) = \lim_{n\to\infty}q_n(A,A^*)$}; + \end{scope} +} + +\uncover<16->{ + \node[color=darkred] at (3.6,-1.8) [right] {\begin{minipage}{4cm} + \raggedright + Konvergenz $p_n\to f$\\ + auf $\operatorname{Sp}(A)\cup\operatorname{Sp}(A^*)$ + \end{minipage}}; +} + +\uncover<17->{ + \node[color=darkred] at (11.5,-3.8) [above left] {% + \begin{minipage}{3.5cm}\raggedleft + nur sinnvoll definiert wenn + $AA^*=A^*A$ + \end{minipage}}; +} + +\uncover<18->{ + \node[color=darkred] at (-1.5,-3.8) [above right] + {$A$ normal: $AA^*=A^*A$}; +} + +% +% Grüner Ast +% + +\uncover<3->{ + \draw[->,color=darkgreen,shorten <= 0.0cm, shorten >= 0.0cm] + (-1,0) -- (0,0); + + \begin{scope}[xshift=0cm,yshift=0cm] + \fill[color=white,opacity=0.7] (0,-0.6) rectangle (2.9,0.6); + \draw[color=darkgreen] (0,-0.6) rectangle (2.9,0.6); + \node at (0,0) [right] {$\displaystyle + f(z)=\sum_{k=0}^\infty a_kz^k$}; + \end{scope} +} + +\uncover<5->{ + \node[color=darkgreen] at (5.9,0) [above] {$f(z)$ analytisch!}; +} +\uncover<6->{ + \node[color=darkgreen] at (5.9,0) [below] + {$\varrho(A)<\text{Konvergenzradius}$}; +} + +\uncover<4->{ + \draw[->,color=darkgreen] (2.9,0) -- (8.5,0); + + \begin{scope}[xshift=8.5cm] + \fill[color=white,opacity=0.7] (0,-0.6) rectangle (2.9,0.6); + \draw[color=darkgreen] (0,-0.6) rectangle (2.9,0.6); + \node at (0,0) [right] {$\displaystyle + f(A)=\sum_{k=0}^\infty a_kA^k$}; + \end{scope} +} + +\end{tikzpicture} +\end{center} +\end{frame} +\egroup diff --git a/vorlesungen/slides/5/planbeispiele.tex b/vorlesungen/slides/5/planbeispiele.tex new file mode 100644 index 0000000..7b98a95 --- /dev/null +++ b/vorlesungen/slides/5/planbeispiele.tex @@ -0,0 +1,103 @@ +% +% planbeispiele.tex +% +% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule +% +\bgroup +\definecolor{darkgreen}{rgb}{0,0.6,0} +\definecolor{darkred}{rgb}{0.8,0,0} +\setlength{\abovedisplayskip}{5pt} +\setlength{\belowdisplayskip}{5pt} +\begin{frame}[t] +\frametitle{Beispiele} +\vspace{-15pt} +\begin{columns}[t] +\begin{column}{0.33\textwidth} +\setbeamercolor{block body}{bg=blue!20} +\setbeamercolor{block title}{bg=blue!20} +\uncover<2->{% +\begin{block}{$A$ diagonal, $\operatorname{Sp}(A)\subset\mathbb{R}$\strut} +Beispiele: +\begin{align*} +f(x) +&= +x^k, +\\ +f(x)&= +\sqrt{x}, +\sqrt[k]{x} +\\ +f(x)&=|x| +\end{align*} +\vspace{43pt} +\end{block}} +\end{column} +\begin{column}{0.33\textwidth} +\setbeamercolor{block body}{bg=darkgreen!20} +\setbeamercolor{block title}{bg=darkgreen!20} +\uncover<1->{% +\begin{block}{$f(z)$ analytisch\strut} +Beispiele: +\begin{align*} +e^z +&= +\sum_{k=0}^\infty \frac{z^k}{k!} +\\ +\cos z +&= +\sum_{k=0}^\infty (-1)^k\frac{z^{2k}}{2k!} +\\ +\sin z +&= +\sum_{k=0}^\infty (-1)^k\frac{z^{2k+1}}{(2k+1)!} +\end{align*} +\end{block}} +\end{column} +\begin{column}{0.33\textwidth} +\setbeamercolor{block body}{bg=darkred!20} +\setbeamercolor{block title}{bg=darkred!20} +\uncover<3->{% +\begin{block}{$A$ normal, $AA^*=A^*A$\strut} +Beispiele: +\begin{align*} +f(z)&=\sqrt{z\overline{z}}=|z| +\end{align*} +\vspace{76pt} +\end{block}} +\end{column} +\end{columns} +\vspace{-10pt} +\begin{columns}[t] +\begin{column}{0.33\textwidth} +\setbeamercolor{block body}{bg=blue!20} +\setbeamercolor{block title}{bg=blue!20} +\uncover<5->{% +\begin{block}{} +\vspace{-6pt} +$f(A)$ wohldefiniert für {\color{blue}diagonalisierbare} +Matrizen $A\in M_n(\mathbb{R})$ +\end{block}} +\end{column} +\begin{column}{0.33\textwidth} +\setbeamercolor{block body}{bg=darkgreen!20} +\setbeamercolor{block title}{bg=darkgreen!20} +\uncover<4->{% +\begin{block}{} +\vspace{-6pt} +$f(A)$ wohldefiniert für {\color{darkgreen}jedes} $A\in M_n(\mathbb{C})$ +\vspace{14pt} +\end{block}} +\end{column} +\begin{column}{0.33\textwidth} +\setbeamercolor{block body}{bg=darkred!20} +\setbeamercolor{block title}{bg=darkred!20} +\uncover<6->{% +\begin{block}{} +\vspace{-6pt} +$f(A)$ wohldefiniert für {\color{darkred}normale} +Matrizen $A\in M_n(\mathbb{C})$ +\end{block}} +\end{column} +\end{columns} +\end{frame} +\egroup diff --git a/vorlesungen/slides/5/stoneweierstrass.tex b/vorlesungen/slides/5/stoneweierstrass.tex index 3f9cab5..e2e9e30 100644 --- a/vorlesungen/slides/5/stoneweierstrass.tex +++ b/vorlesungen/slides/5/stoneweierstrass.tex @@ -3,9 +3,64 @@ % % (c) 2021 Prof Dr Andreas Müller, Hochschule Rapperswil % +\bgroup +\definecolor{darkgreen}{rgb}{0,0.6,0} \begin{frame}[t] -\frametitle{Stone-Weierstrass} - -TODO XXX - +\frametitle{Allgemeiner Approximationssatz} +\vspace{-20pt} +\begin{columns}[t] +\begin{column}{0.5\textwidth} +\begin{theorem}[Stone-Weierstrass, $\mathbb{R}$] +$A$ eine {\color{darkgreen}$\mathbb{R}$}-Algebra +von stetigen Funktionen auf einem +%abgeschlossenen und beschränkten +kompakten +Definitionsgebiet $D\subset {\color{darkgreen}\mathbb{R}}$, +\begin{itemize} +\item<2-> konstante Funktion $c\in A$, +\item<3-> für $d_1,d_2\in D$ gibt es ein $s\in A$ mit +$s(d_1)\ne s(d_2)$. +\end{itemize} +\uncover<4->{% +Dann lässt sich jede stetige Funktion durch Funktionen aus $A$ +approximieren} +\end{theorem} +\uncover<5->{ +\begin{block}{Anwendung} +\uncover<6->{$A={\color{darkgreen}\mathbb{R}}[X]$}\uncover<7->{, +$s(X)=X$}\uncover<8->{, +jede stetige Funktion kann durch +Polynome in $X$ approximiert werden} +\end{block}} +\end{column} +\begin{column}{0.5\textwidth} +\uncover<9->{% +\begin{theorem}[Stone-Weierstrass, $\mathbb{C}$] +$A$ eine {\color<10->{red}$\mathbb{C}$}-Algebra von stetigen Funktionen +auf einem +%abgeschlossenen und beschränkten +kompakten +Definitionsgebiet $D\subset {\color<10->{red}\mathbb{C}}$, +\begin{itemize} +\item konstante Funktion $c\in A$, +\item für $d_1,d_2\in D$ gibt es ein $s\in A$ mit +$s(d_1)\ne s(d_2)$. +\only<11->{ +\item {\color{red}$f\in A\Rightarrow \overline{f}\in A$} +} +\end{itemize} +Dann lässt sich jede stetige Funktion durch Funktionen aus $A$ +approximieren +\end{theorem}} +\vspace{-5pt} +\uncover<12->{% +\begin{block}{Anwendung} +$A={\color{red}\mathbb{C}}[Z,\overline{Z}]$\uncover<13->{, +$s(Z{\color{red},\overline{Z}})=Z$}\uncover<14->{, +jede stetige Funktion +lässt sich durch Polynome in $Z{\color{red},\overline{Z}}$ approximieren} +\end{block}} +\end{column} +\end{columns} \end{frame} +\egroup |