From f144be56b0c7ec03f74c46928b1354a959a59246 Mon Sep 17 00:00:00 2001
From: =?UTF-8?q?Andreas=20M=C3=BCller?= <andreas.mueller@ost.ch>
Date: Sun, 22 May 2022 13:36:59 +0200
Subject: add hermite application presentation

---
 vorlesungen/slides/hermite/hermiteentwicklung.tex | 69 +++++++++++++++++++++++
 1 file changed, 69 insertions(+)
 create mode 100644 vorlesungen/slides/hermite/hermiteentwicklung.tex

(limited to 'vorlesungen/slides/hermite/hermiteentwicklung.tex')

diff --git a/vorlesungen/slides/hermite/hermiteentwicklung.tex b/vorlesungen/slides/hermite/hermiteentwicklung.tex
new file mode 100644
index 0000000..e1ced30
--- /dev/null
+++ b/vorlesungen/slides/hermite/hermiteentwicklung.tex
@@ -0,0 +1,69 @@
+%
+% hermiteentwicklung.tex -- slide template
+%
+% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule
+%
+\bgroup
+\begin{frame}[t]
+\setlength{\abovedisplayskip}{5pt}
+\setlength{\belowdisplayskip}{5pt}
+\frametitle{Beliebige Polynome}
+\vspace{-20pt}
+\begin{columns}[t,onlytextwidth]
+\begin{column}{0.48\textwidth}
+\begin{block}{Polynom}
+\[
+P(x)
+=
+p_0 + p_1x + p_2x^2 + \dots + p_nx^n
+\]
+als Linearkombination von Hermite-Polynome schreiben:
+\begin{align*}
+P(x)
+&=
+a_0H_0(x)% + a_1H_1(x)
++ \dots + a_nH_n(x)
+\\
+&=
+a_0\cdot 1
+\\
+&\quad + a_1\cdot 2x
+\\
+&\quad + a_2\cdot(4x^2-2) 
+\\
+&\quad + a_3\cdot(8x^3-12x)
+\\
+&\quad + a_4\cdot(16x^4-48x^2+12)
+\\
+&\quad\;\;\vdots
+\\
+&\quad + a_n(2^nx^n + \dots)
+\end{align*}
+\end{block}
+\end{column}
+\begin{column}{0.48\textwidth}
+\begin{block}{Koeffizientenvergleich}
+führt auf ein Gleichungssystem
+\begin{center}
+\begin{tabular}{|>{$}r<{$}>{$}r<{$}>{$}r<{$}>{$}r<{$}>{$}r<{$}>{$}c<{$}|>{$}c<{$}|}
+\hline
+a_0&a_1&a_2&a_3&a_4&\dots&\\
+\hline
+ 1&  0&  0&  0&  0&\dots&p_0\\
+ 0&  2&  0&  0&  0&\dots&p_1\\
+-2&  0&  4&  0&  0&\dots&p_2\\
+ 0&-12&  0&  8&  0&\dots&p_3\\
+12&  0&-48&  0& 16&\dots&p_4\\
+\vdots&\vdots&\vdots&\vdots&\vdots&\ddots&\vdots\\
+\hline
+\end{tabular}
+\end{center}
+Dreiecksmatrix, Diagonalelement
+$\ne 0$
+$\Rightarrow$ 
+$\exists$ eindeutige Lösung
+\end{block}
+\end{column}
+\end{columns}
+\end{frame}
+\egroup
-- 
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