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author | Lukaszogg <82384106+Lukaszogg@users.noreply.github.com> | 2021-07-08 20:10:11 +0200 |
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committer | Lukaszogg <82384106+Lukaszogg@users.noreply.github.com> | 2021-07-08 20:10:11 +0200 |
commit | 14033ca595b5c933caea3b214d2246529e6845b8 (patch) | |
tree | 0d6d2b2eb34e5ef5df3c517be5c1c9d803fa066c /vorlesungen/slides/8/wavelets/framekonstanten.tex | |
parent | Update teil1.tex (diff) | |
parent | Only include buch.ind if it exists. (diff) | |
download | SeminarMatrizen-14033ca595b5c933caea3b214d2246529e6845b8.tar.gz SeminarMatrizen-14033ca595b5c933caea3b214d2246529e6845b8.zip |
Merge remote-tracking branch 'upstream/master'
Diffstat (limited to 'vorlesungen/slides/8/wavelets/framekonstanten.tex')
-rw-r--r-- | vorlesungen/slides/8/wavelets/framekonstanten.tex | 71 |
1 files changed, 71 insertions, 0 deletions
diff --git a/vorlesungen/slides/8/wavelets/framekonstanten.tex b/vorlesungen/slides/8/wavelets/framekonstanten.tex new file mode 100644 index 0000000..a436536 --- /dev/null +++ b/vorlesungen/slides/8/wavelets/framekonstanten.tex @@ -0,0 +1,71 @@ +% +% template.tex -- slide template +% +% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule +% +\bgroup +\begin{frame}[t] +%\setlength{\abovedisplayskip}{5pt} +%\setlength{\belowdisplayskip}{5pt} +\frametitle{Framekonstanten} +\vspace{-20pt} +\begin{columns}[t,onlytextwidth] +\begin{column}{0.48\textwidth} +\begin{block}{Definition} +Eine Menge $\mathcal{F}$ von Vektoren heisst ein Frame, +falls es Konstanten $A$ und $B$ gibt derart, dass +\[ +A\|v\|^2 +\le +\|\mathcal{T}v\|^2 +\sum_{b\in\mathcal{F}} |\langle b,v\rangle|^2 +\le +B\|v\|^2 +\] +\uncover<2->{$A>0$ garantiert Invertierbarkeit} +\end{block} +\uncover<3->{% +\begin{block}{$\|\mathcal{T}v\|$ für Graph-Wavelets} +\begin{align*} +\|\mathcal{T}v\|^2 +&= +\sum_k |\langle D_he_k,v\rangle|^2 ++ +\sum_{i,k} |\langle D_{g_i}e_k, v\rangle|^2 +\\ +&\uncover<4->{= +\sum_k |h(\lambda_k) \hat{v}(k)|^2 ++ +\sum_{k,i} |g_i(\lambda_k) \hat{v}(k)|^2} +\end{align*} +\end{block}} +\end{column} +\begin{column}{0.48\textwidth} +\uncover<5->{% +\begin{block}{$A$ und $B$} +Frame-Norm-Funktion +\begin{align*} +f(\lambda) +&= +h(\lambda) ++ +\sum_i g_i(\lambda) +\\ +&\uncover<6->{= +h(\lambda) ++ +\sum_i g(a_i\lambda)} +\end{align*} +\uncover<7->{Abschätzung für Frame-Konstanten +\begin{align*} +A&\uncover<8->{= +\min_{i} f(\lambda_i)} +\\ +B&\uncover<9->{= +\max_{i} f(\lambda_i)} +\end{align*}} +\end{block}} +\end{column} +\end{columns} +\end{frame} +\egroup |