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}
-- 
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