set of sampling


Let F be a Hilbert spaceMathworldPlanetmath of functions defined on a domain D. Let T={ti}iI be a finite or infiniteMathworldPlanetmathPlanetmath sequence of points in D. T is said to be a set of sampling for F if the sampling operator S:Fl|T|2 defined by


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

A,B>0 such that fF,Af2i=1|T||f(ti)|2Bf2.

Relation to Frames

Using the Riesz Representation TheoremMathworldPlanetmath, 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={ti} be a set of sampling with sampling operator St. Use the Riesz representation theorem to rewrite St in terms of vectors {gi} in F:


then note that


so the {gi} form a frame with bounds A,B, and St=θg.


Particularly nice sets of sampling are those that correspond to tight frames, because then θgθg=θgSt=AI, and it is possible to reconstruct the function f, given its values over the set of sampling:


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