From 9aa06203d62e6d9092597fc7f89a0a8e3a6636c9 Mon Sep 17 00:00:00 2001
From: =?UTF-8?q?Andreas=20M=C3=BCller?= <andreas.mueller@ost.ch>
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?= <andreas.mueller@ost.ch>
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?= <andreas.mueller@ost.ch>
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

