The concept of a frame is a generalizationPlanetmathPlanetmath of the concept of an orthonormal basisMathworldPlanetmath: each vector in the space can be represented as a sum of the elements in the frame, but not necessarily uniquely. It is because of this redundancy in representation that frames have found important applications. Surpisingly, despite the fact that frames do not in general consist of orthonormal vectors, the frame representation of a vector may still satisfy the Parseval equationMathworldPlanetmath.


A frame in a Hilbert spaceMathworldPlanetmath H is a collectionMathworldPlanetmath (xi)i=1n of vectors in H such that there exist constants A,B>0 such that for all xH,


The constants A,B are called the lower and upper frame bounds respectively, or the frame constants. The optimal frame constants A and B are defined respectively as the supremum and infimumMathworldPlanetmath of all possible lower and upper frame bounds. When A=B, the frame is a tight frame, or an A-tight frame. A 1-tight frame is referred to as a Parseval frame, and the Parseval equation holds with respect to the frame elements:


for all xH.

Associated Operators (for finite frames)

Let dimH=n and {xi}i=1k be a frame in H (kn).

The analysis operator θ:Hk is the function defined such that θ:x(x,xi)i.

The synthesis operator τ:kH is the function defined such that (ci)iki=1kcixi.

τ=θ, that is, θx,yk=x,τyH for all xH and all yk.

θx,yk = (x,xiH)i=1k,(yi)i=1kk
= i=1kx,xiHyi¯=i=1kx,yixiH
= x,i=1kyixiH=x,τyH

The frame operator is defined as the n×n matrix θθ:HH such that θθ:xi=1kx,xixi for all xH.

The Grammian operator is defined as the composition θθ:kk.

The Grammian and frame operators have the same nonzero eigenvaluesMathworldPlanetmathPlanetmathPlanetmathPlanetmath.

Title frame
Canonical name Frame1
Date of creation 2013-03-22 14:25:37
Last modified on 2013-03-22 14:25:37
Owner swiftset (1337)
Last modified by swiftset (1337)
Numerical id 9
Author swiftset (1337)
Entry type Definition
Classification msc 46C99
Related topic RieszSequence
Related topic SetOfSampling
Defines Parseval frame
Defines tight frame
Defines frame constants
Defines frame bounds
Defines synthesis operator
Defines analysis operator
Defines frame operator
Defines Grammian