From 9aa06203d62e6d9092597fc7f89a0a8e3a6636c9 Mon Sep 17 00:00:00 2001 From: =?UTF-8?q?Andreas=20M=C3=BCller?= Date: Thu, 3 Jun 2021 08:28:20 +0200 Subject: new slides --- vorlesungen/slides/2/hilbertraum/plancherel.tex | 75 +++++++++++++++++++++++++ 1 file changed, 75 insertions(+) create mode 100644 vorlesungen/slides/2/hilbertraum/plancherel.tex (limited to 'vorlesungen/slides/2/hilbertraum/plancherel.tex') diff --git a/vorlesungen/slides/2/hilbertraum/plancherel.tex b/vorlesungen/slides/2/hilbertraum/plancherel.tex new file mode 100644 index 0000000..3caa54d --- /dev/null +++ b/vorlesungen/slides/2/hilbertraum/plancherel.tex @@ -0,0 +1,75 @@ +% +% 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} +\begin{block}{Analyse: Fourier-Koeffizienten} +\begin{align*} +a_k &= \hat{x}_k=\langle b_k, x\rangle +\end{align*} +\end{block} +\begin{block}{Synthese: Fourier-Reihe} +\begin{align*} +\tilde{x} +&= +\sum_k a_k b_k += +\sum_k \langle x,b_k\rangle b_k +\end{align*} +\end{block} +\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 += +\sum_k \langle b_k,x\rangle\langle b_l,b_k\rangle += +\sum_k \langle b_k,x\rangle\delta_{kl} += +\langle b_l,x\rangle += +\hat{x}_l +\end{align*} +\end{block} +\end{column} +\begin{column}{0.48\textwidth} +\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 +\\ +&= +\sum_{k,l} \overline{\hat{x}}_k\hat{x}_l\langle b_k,b_l\rangle += +\sum_{k,l} \overline{\hat{x}}_k\hat{x}_l\delta_{kl} +\\ +\|\tilde{x}\|^2 +&= +\sum_k |\hat{x}_k|^2 +\end{align*} +\end{block} +\end{column} +\end{columns} +\end{frame} +\egroup -- cgit v1.2.1 From 9e7524c25a0ba5a643fbb7555d01311f69aa603e Mon Sep 17 00:00:00 2001 From: =?UTF-8?q?Andreas=20M=C3=BCller?= Date: Thu, 3 Jun 2021 17:18:58 +0200 Subject: add slides --- vorlesungen/slides/2/hilbertraum/plancherel.tex | 23 ++++++++++++++++++++++- 1 file changed, 22 insertions(+), 1 deletion(-) (limited to 'vorlesungen/slides/2/hilbertraum/plancherel.tex') diff --git a/vorlesungen/slides/2/hilbertraum/plancherel.tex b/vorlesungen/slides/2/hilbertraum/plancherel.tex index 3caa54d..eaf8aaa 100644 --- a/vorlesungen/slides/2/hilbertraum/plancherel.tex +++ b/vorlesungen/slides/2/hilbertraum/plancherel.tex @@ -17,9 +17,12 @@ $\mathcal{B}=\{b_k\;|\; k>0\}$, $x\in H$ \end{block} \begin{block}{Analyse: Fourier-Koeffizienten} \begin{align*} -a_k &= \hat{x}_k=\langle b_k, x\rangle +a_k = \hat{x}_k &=\langle b_k, x\rangle +\\ +\hat{x}&=\mathcal{F}x \end{align*} \end{block} +\vspace{-10pt} \begin{block}{Synthese: Fourier-Reihe} \begin{align*} \tilde{x} @@ -29,6 +32,7 @@ a_k &= \hat{x}_k=\langle b_k, x\rangle \sum_k \langle x,b_k\rangle b_k \end{align*} \end{block} +\vspace{-6pt} \begin{block}{Analyse von $\tilde{x}$} \begin{align*} \langle b_l,\tilde{x}\rangle @@ -67,7 +71,24 @@ b_l,\sum_{k}\langle b_k,x\rangle b_k \|\tilde{x}\|^2 &= \sum_k |\hat{x}_k|^2 += +\|\hat{x}\|_{l^2}^2 += +\|\mathcal{F}x\|_{l^2}^2 +\end{align*} +\end{block} +\vspace{-12pt} +\begin{block}{Isometrie} +\begin{align*} +\mathcal{F} +\colon +H \to l^2 +\colon +x\mapsto \hat{x} \end{align*} +Alle separablen Hilberträume sind isometrisch zu $l^2$ via +%Fourier-Transformation +$\mathcal{F}$ \end{block} \end{column} \end{columns} -- cgit v1.2.1 From 680e1e763b8d899b3601b5ab0cf6f1fc2a114e1d Mon Sep 17 00:00:00 2001 From: =?UTF-8?q?Andreas=20M=C3=BCller?= Date: Thu, 3 Jun 2021 18:51:36 +0200 Subject: phases --- vorlesungen/slides/2/hilbertraum/plancherel.tex | 60 ++++++++++++++----------- 1 file changed, 33 insertions(+), 27 deletions(-) (limited to 'vorlesungen/slides/2/hilbertraum/plancherel.tex') diff --git a/vorlesungen/slides/2/hilbertraum/plancherel.tex b/vorlesungen/slides/2/hilbertraum/plancherel.tex index eaf8aaa..73dd46b 100644 --- a/vorlesungen/slides/2/hilbertraum/plancherel.tex +++ b/vorlesungen/slides/2/hilbertraum/plancherel.tex @@ -15,24 +15,27 @@ $H$ 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 \\ -\hat{x}&=\mathcal{F}x +\uncover<3->{\hat{x}&=\mathcal{F}x} \end{align*} -\end{block} +\end{block}} \vspace{-10pt} +\uncover<4->{% \begin{block}{Synthese: Fourier-Reihe} \begin{align*} \tilde{x} &= \sum_k a_k b_k -= -\sum_k \langle x,b_k\rangle b_k +\uncover<5->{= +\sum_k \langle x,b_k\rangle b_k} \end{align*} -\end{block} +\end{block}} \vspace{-6pt} +\uncover<6->{% \begin{block}{Analyse von $\tilde{x}$} \begin{align*} \langle b_l,\tilde{x}\rangle @@ -40,18 +43,19 @@ a_k = \hat{x}_k &=\langle b_k, x\rangle \biggl\langle b_l,\sum_{k}\langle b_k,x\rangle b_k \biggr\rangle -= -\sum_k \langle b_k,x\rangle\langle b_l,b_k\rangle -= -\sum_k \langle b_k,x\rangle\delta_{kl} -= -\langle b_l,x\rangle -= -\hat{x}_l +\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{block}} \end{column} \begin{column}{0.48\textwidth} +\uncover<11->{% \begin{block}{Plancherel-Gleichung} \begin{align*} \|\tilde{x}\|^2 @@ -63,21 +67,23 @@ b_l,\sum_{k}\langle b_k,x\rangle b_k \sum_l \hat{x}_lb_l \biggr\rangle \\ -&= -\sum_{k,l} \overline{\hat{x}}_k\hat{x}_l\langle b_k,b_l\rangle -= -\sum_{k,l} \overline{\hat{x}}_k\hat{x}_l\delta_{kl} +&\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 -= -\|\hat{x}\|_{l^2}^2 -= -\|\mathcal{F}x\|_{l^2}^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} +\end{block}} \vspace{-12pt} +\uncover<17->{% \begin{block}{Isometrie} \begin{align*} \mathcal{F} @@ -86,10 +92,10 @@ H \to l^2 \colon x\mapsto \hat{x} \end{align*} -Alle separablen Hilberträume sind isometrisch zu $l^2$ via +\uncover<18->{Alle separablen Hilberträume sind isometrisch zu $l^2$ via %Fourier-Transformation -$\mathcal{F}$ -\end{block} +$\mathcal{F}$} +\end{block}} \end{column} \end{columns} \end{frame} -- cgit v1.2.1