new slides

7 files changed, 429 insertions, 7 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
diff --git a/vorlesungen/slides/test.tex b/vorlesungen/slides/test.tex
index 99f9cda..90138a1 100644
--- a/vorlesungen/slides/test.tex
+++ b/vorlesungen/slides/test.tex
@@ -4,17 +4,18 @@
 % (c) 2019 Prof Dr Andreas Müller, Hochschule Rapperswil
 %
 %\folie{5/verzerrung.tex}
+%\folie{5/plan.tex}
+\folie{5/planbeispiele.tex}
+%\folie{5/approximation.tex}
  
 % XXX Visualisierung Cayley-Hamilton-Produkte
 % XXX \folie{5/chvisual.tex}
 
 % XXX stone weierstrass incomplete
 %\folie{5/stoneweierstrass.tex}
+%\folie{5/swbeweis.tex}
 
 % XXX polynome auf dem spektrum
 % XXX Motiviation für *-Operation
 %\folie{5/normal.tex}
 
-\folie{4/char2.tex}
-\folie{4/frobenius.tex}
-\folie{4/qundr.tex}