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set of sampling
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(Definition)
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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 $$\exists A,B>0 \hbox{ such that } \forall f \in F, A\|f\|^2 \leq \sum_{i=1}^{|T|} |f(t_i)|^2 \leq B \|f\|^2.$$
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 $S_t = \theta_g.$
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.
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"set of sampling" is owned by swiftset.
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See Also: frame
| Other names: |
sampling set |
| Also defines: |
set of sampling, sampling operator |
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Cross-references: tight, tight frames, bounds, vectors, terms, analysis operator, frame, Riesz representation theorem, continuous, bounded, points, sequence, infinite, finite, domain, functions, Hilbert space
This is version 1 of set of sampling, born on 2004-07-06.
Object id is 5984, canonical name is SetOfSampling.
Accessed 4013 times total.
Classification:
| AMS MSC: | 46C99 (Functional analysis :: Inner product spaces and their generalizations, Hilbert spaces :: Miscellaneous) |
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Pending Errata and Addenda
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