set of sampling SetOfSampling 2004-07-06 20:06:13 2004-07-06 20:06:13 Definition set of sampling sampling operator % this is the default PlanetMath preamble. as your knowledge % of TeX increases, you will probably want to edit this, but % it should be fine as is for beginners. % almost certainly you want these \usepackage{amssymb} \usepackage{amsmath} \usepackage{amsfonts} % used for TeXing text within eps files %\usepackage{psfrag} % need this for including graphics (\includegraphics) %\usepackage{graphicx} % for neatly defining theorems and propositions %\usepackage{amsthm} % making logically defined graphics %\usepackage{xypic} % there are many more packages, add them here as you need them % define commands here \paragraph{Definition} Let $F$ be a Hilbert space of functions defined on a domain $D$. Let $T = \{t_i\}_{i\in I}$ be a finite or infinite sequence of points in $D$. $T$ is said to be a \emph{set of sampling} for $F$ if the sampling operator $S: F \rightarrow l^2_{|T|}$ defined by $$ S: f \mapsto \begin{pmatrix} f(t_1) \\ f(t_2) \\ \vdots \end{pmatrix} $$ is bounded (i.e. continuous) and bounded below; i.e. if $$\exists A,B>0 \hbox{ such that } \forall f \in F, A\|f\|^2 \leq \sum_{i=1}^{|T|} |f(t_i)|^2 \leq B \|f\|^2.$$ \paragraph{Relation to Frames} Using the Riesz Representation Theorem, it is easy to show that every set of sampling determines a unique frame in such a way that the analysis operator of that frame is the sampling operator associated with the set of sampling. In fact, let $t=\{t_i\}$ be a set of sampling with sampling operator $S_t$. Use the Riesz representation theorem to rewrite $S_t$ in terms of vectors $\{g_i\}$ in $F$: $$ S: f \mapsto \begin{pmatrix} f(t_1) \\ f(t_2) \\ \vdots \end{pmatrix} = \begin{pmatrix} \langle f, g_1 \rangle \\ \langle f, g_2 \rangle \\ \vdots \end{pmatrix} $$ then note that $$ \forall f\in F, A\|f\|^2 \leq \sum_{i} \left| \langle f, g_i \rangle \right|^2 \leq B\|f\|^2, $$ so the $\{g_i\}$ form a frame with bounds $A, B$, and $S_t = \theta_g.$ \paragraph{Reconstruction} Particularly nice sets of sampling are those that correspond to tight frames, because then $\theta_g^\ast\theta_g=\theta_g^\ast S_t=AI$, and it is possible to reconstruct the function $f$, given its values over the set of sampling: $$ f = \frac{1}{A}\sum_i f(t_i) g_i.$$ Sets of sampling which correspond to tight frames are referred to as tight sets of sampling.