set of sampling

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 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

 $\exists A,B>0\hbox{ such that }\forall f\in F,A\|f\|^{2}\leq\sum_{i=1}^{|T|}|f% (t_{i})|^{2}\leq B\|f\|^{2}.$

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 $S_{t}=\theta_{g}.$

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.

Title set of sampling SetOfSampling 2013-03-22 14:27:50 2013-03-22 14:27:50 swiftset (1337) swiftset (1337) 4 swiftset (1337) Definition msc 46C99 sampling set Frame2 set of sampling sampling operator