From 70ac96eb428c0415908942cb3af605d882635f92 Mon Sep 17 00:00:00 2001
From: =?UTF-8?q?Andreas=20M=C3=BCller?= <andreas.mueller@ost.ch>
Date: Sun, 28 Mar 2021 18:00:00 +0200
Subject: new slides

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 vorlesungen/slides/5/approximation.tex | 56 ++++++++++++++++++++++++++++++++++
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+%
+% 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}
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