Page MenuHomec4science
Files F112123256

6_hilbert.tex
No OneTemporary
Actions

File Metadata

Created
Wed, May 7, 21:03

6_hilbert.tex
View Options

\documentclass[aspectratio=169]{beamer}
%\documentclass[aspectratio=169,handout]{beamer}
\def\stylepath{../styles}
\usepackage{\stylepath/com303}
\usepackage{pst-3dplot}
%\setbeameroption{show only notes}\def\logoEPFL{}
\setbeameroption{show notes}
\begin{document}
\begin{frame} \frametitle{Hilbert Space -- the ingredients:}
\begin{enumerate}[<+->]
\item a vector space: $H(V, \mathbb{C})$
\item an inner product: $\langle \cdot, \cdot \rangle \,:\, V \times V \rightarrow \mathbb{C}$
\item completeness
\end{enumerate}
\end{frame}
\def\tickA#1#2{\psline(#1,0.8)(#1,1.2)\rput[t]{0}(#1,0.7){#2}}
\def\tickB#1#2#3{\psline(#1,0.8)(#1,1.2)\rput[t]{0}(#1,#3){\rnode{A}{#2}}\pnode(#1,0.8){B}\nccurve[angleA=0,angleB=-90]{->}{A}{B}}
\begin{frame} \frametitle{Completeness}
\centering
\uncover<1->{
limiting operations must yield vector space elements \\
\vspace{2em}}
\uncover<2->{
Example of an {\em incomplete}\/ space: the set of rational numbers
\[
x_n = \sum_{k = 0}^{n} \frac{1}{k!} \in \mathbb{Q}
\qquad\mbox{but}\qquad
\lim_{n\rightarrow\infty}x_n = e \not\in \mathbb{Q}
\]}
\begin{center}
\psset{xunit=40mm,yunit=10mm}
\begin{pspicture}(0,0)(3,2)
\only<3->{
\psline(0,1)(3,1)
\tickA{0}{$0$}\tickA{1}{$1$}
\tickA{3}{$3$}
\tickA{2}{$x_1 = 2$}
\tickA{2.5}{$x_2$}
\tickA{2.6667}{$x_3$}
\tickB{2.7083}{$x_4$}{0}
\tickB{2.7167}{$x_5$}{-0.5}}
\end{pspicture}
\end{center}
\end{frame}
\begin{frame}
\frametitle{Signals in Hilbert Space}
Why did we do all this?
\begin{itemize}[<+->]
\item finite-length and periodic signals live in $\mathbb{C}^N$
\item infinite-length signals live in $\ell_2(\mathbb{Z})$
\vspace{1em}
\item different bases are different observation tools for signals
\item subspace projections are useful in filtering and compression
\end{itemize}
\end{frame}
\end{document}

Event Timeline

c4science · Help