PlanetMath: set of sampling

(more info)

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set of sampling

(Definition)

Definition

Let

be a Hilbert space of functions defined on a domain

. Let $T = \{t_i\}_{i\in I}$ be a finite or infinite sequence of points in

is said to be a set of sampling for

if the sampling operator $S: F \rightarrow l^2_{\vert T\vert}$ defined by

$\displaystyle S: f \mapsto \begin{pmatrix} f(t_1) \ f(t_2) \ \vdots \end{pmatrix}$

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

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

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

. Use the Riesz representation theorem to rewrite

in terms of vectors $\{g_i\}$ in

$\displaystyle S: f \mapsto \begin{pmatrix} f(t_1) \ f(t_2) \ \vdots \end{pm... ...rix} \langle f, g_1 \rangle \ \langle f, g_2 \rangle \ \vdots \end{pmatrix}$

then note that

$\displaystyle \forall f\in F, A\Vert f\Vert^2 \leq \sum_{i} \left\vert \langle f, g_i \rangle \right\vert^2 \leq B\Vert f\Vert^2,$

so the $\{g_i\}$ form a frame with bounds

, 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

, given its values over the set of sampling:

$\displaystyle f = \frac{1}{A}\sum_i f(t_i) g_i.$

Sets of sampling which correspond to tight frames are referred to as tight sets of sampling.

"set of sampling" is owned by swiftset.

(view preamble)

Definition

Relation to Frames

Reconstruction