# 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

$$S:f\mapsto \left(\begin{array}{c}\hfill f({t}_{1})\hfill \\ \hfill f({t}_{2})\hfill \\ \hfill \mathrm{\vdots}\hfill \end{array}\right)$$ |

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 $F$:

$$S:f\mapsto \left(\begin{array}{c}\hfill f({t}_{1})\hfill \\ \hfill f({t}_{2})\hfill \\ \hfill \mathrm{\vdots}\hfill \end{array}\right)=\left(\begin{array}{c}\hfill \u27e8f,{g}_{1}\u27e9\hfill \\ \hfill \u27e8f,{g}_{2}\u27e9\hfill \\ \hfill \mathrm{\vdots}\hfill \end{array}\right)$$ |

then note that

$$\forall f\in F,A{\parallel f\parallel}^{2}\le \sum _{i}{\left|\u27e8f,{g}_{i}\u27e9\right|}^{2}\le B{\parallel f\parallel}^{2},$$ |

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.

Title | set of sampling |
---|---|

Canonical name | SetOfSampling |

Date of creation | 2013-03-22 14:27:50 |

Last modified on | 2013-03-22 14:27:50 |

Owner | swiftset (1337) |

Last modified by | swiftset (1337) |

Numerical id | 4 |

Author | swiftset (1337) |

Entry type | Definition |

Classification | msc 46C99 |

Synonym | sampling set |

Related topic | Frame2 |

Defines | set of sampling |

Defines | sampling operator |