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Diffstat (limited to 'vorlesungen/slides/2/hilbertraum/plancherel.tex')
-rw-r--r-- | vorlesungen/slides/2/hilbertraum/plancherel.tex | 23 |
1 files changed, 22 insertions, 1 deletions
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} |