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diff --git a/vorlesungen/slides/2/hilbertraum/plancherel.tex b/vorlesungen/slides/2/hilbertraum/plancherel.tex new file mode 100644 index 0000000..73dd46b --- /dev/null +++ b/vorlesungen/slides/2/hilbertraum/plancherel.tex @@ -0,0 +1,102 @@ +% +% plancherel.tex -- slide template +% +% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule +% +\bgroup +\begin{frame}[t] +\setlength{\abovedisplayskip}{5pt} +\setlength{\belowdisplayskip}{5pt} +\frametitle{Plancherel-Gleichung} +\vspace{-20pt} +\begin{columns}[t,onlytextwidth] +\begin{column}{0.48\textwidth} +\begin{block}{Hilbertraum mit Hilbert-Basis} +$H$ Hilbertraum mit Hilbert-Basis +$\mathcal{B}=\{b_k\;|\; k>0\}$, $x\in H$ +\end{block} +\uncover<2->{% +\begin{block}{Analyse: Fourier-Koeffizienten} +\begin{align*} +a_k = \hat{x}_k &=\langle b_k, x\rangle +\\ +\uncover<3->{\hat{x}&=\mathcal{F}x} +\end{align*} +\end{block}} +\vspace{-10pt} +\uncover<4->{% +\begin{block}{Synthese: Fourier-Reihe} +\begin{align*} +\tilde{x} +&= +\sum_k a_k b_k +\uncover<5->{= +\sum_k \langle x,b_k\rangle b_k} +\end{align*} +\end{block}} +\vspace{-6pt} +\uncover<6->{% +\begin{block}{Analyse von $\tilde{x}$} +\begin{align*} +\langle b_l,\tilde{x}\rangle +&= +\biggl\langle +b_l,\sum_{k}\langle b_k,x\rangle b_k +\biggr\rangle +\uncover<7->{= +\sum_k \langle b_k,x\rangle\langle b_l,b_k\rangle} +\uncover<8->{= +\sum_k \langle b_k,x\rangle\delta_{kl}} +\uncover<9->{= +\langle b_l,x\rangle} +\uncover<10->{= +\hat{x}_l} +\end{align*} +\end{block}} +\end{column} +\begin{column}{0.48\textwidth} +\uncover<11->{% +\begin{block}{Plancherel-Gleichung} +\begin{align*} +\|\tilde{x}\|^2 +&= +\langle \tilde{x},\tilde{x}\rangle += +\biggl\langle +\sum_k \hat{x}_kb_k, +\sum_l \hat{x}_lb_l +\biggr\rangle +\\ +&\uncover<12->{= +\sum_{k,l} \overline{\hat{x}}_k\hat{x}_l\langle b_k,b_l\rangle} +\uncover<13->{= +\sum_{k,l} \overline{\hat{x}}_k\hat{x}_l\delta_{kl}} +\\ +\uncover<14->{ +\|\tilde{x}\|^2 +&= +\sum_k |\hat{x}_k|^2} +\uncover<15->{= +\|\hat{x}\|_{l^2}^2} +\uncover<16->{= +\|\mathcal{F}x\|_{l^2}^2} +\end{align*} +\end{block}} +\vspace{-12pt} +\uncover<17->{% +\begin{block}{Isometrie} +\begin{align*} +\mathcal{F} +\colon +H \to l^2 +\colon +x\mapsto \hat{x} +\end{align*} +\uncover<18->{Alle separablen Hilberträume sind isometrisch zu $l^2$ via +%Fourier-Transformation +$\mathcal{F}$} +\end{block}} +\end{column} +\end{columns} +\end{frame} +\egroup |