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 $S:F\to l_{|T|}^2$ defined by

$$S:f\mapsto \left(\begin{array}{c}\hfill f(t_1)\hfill \\ \hfill f(t_2)\hfill \\ \hfill \mathrm{⋮}\hfill \end{array}\right)$$

is bounded (i.e. continuous) and bounded below; i.e. if

$$\exists A,B>0\text{ such that }\forall f\in F,A{\parallel f\parallel }^2\le \sum _{i=1}^{|T|}{|f(t_i)|}^2\le B{\parallel f\parallel }^2.$$

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 $F$:

$$S:f\mapsto \left(\begin{array}{c}\hfill f(t_1)\hfill \\ \hfill f(t_2)\hfill \\ \hfill \mathrm{⋮}\hfill \end{array}\right)=\left(\begin{array}{c}\hfill ⟨f,g_1⟩\hfill \\ \hfill ⟨f,g_2⟩\hfill \\ \hfill \mathrm{⋮}\hfill \end{array}\right)$$

then note that

$$\forall f\in F,A{\parallel f\parallel }^2\le \sum _i{\left|⟨f,g_i⟩\right|}^2\le B{\parallel f\parallel }^2,$$

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.