Frame of a vector space

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In mathematics, a frame of a vector space can mean any ordered basis for that vector space. It can also refer to a somewhat different concept discussed below.

A frame of a vector space V with an inner product can be seen as a generalization of the idea of a basis to sets which may be linearly dependent. More precisely, a frame is a set {ek} of elements of V which satisfy the so-called frame condition:

There exist two real numbers A and B such that 0 < A \leq B < \infty and
A \| \mathbf{v} \|^{2} \leq \sum_{k} |\langle \mathbf{v} | \mathbf{e}_{k} \rangle|^{2} \leq B \| \mathbf{v} \|^{2}
for all \mathbf{v} \in V. This means that the constants A and B can be chosen independently of v: they only depend on the set {ek}.

The numbers A and B are called lower and upper frame bounds.

It can be shown that to any set of vectors which form a frame a set of dual frame vectors \mathbf{\tilde{e}}_{k} can be derived which has the following property:

\sum_{k} \langle \mathbf{v} | \mathbf{\tilde{e}}_{k} \rangle \mathbf{e}_{k} = \sum_{k} \langle \mathbf{v} | \mathbf{e}_{k} \rangle \mathbf{\tilde{e}}_{k} = \mathbf{v}

for any \mathbf{v} \in V. This implies that a frame together with its dual frame has the same properties as a basis and its dual basis in terms of reconstructing a vector from scalar products.

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[edit] Relation to bases

If the set {ek} is a frame of V, it spans V. Otherwise there would exist at least one non-zero \mathbf{v} \in V which would be orthogonal to all ek. If we insert \mathbf{v} into the frame condition, we obtain

A \| \mathbf{v} \|^{2} \leq 0 \leq B \| \mathbf{v} \|^{2} ;

therefore A \leq 0, which is a violation of the initial assumptions on the lower frame bound.

If a set of vectors spans V, this is not a sufficient for calling the set a frame. As an example, consider V = R2 and the infinite set

\{ (1,0) , \, (0,1), \, (0,1) , \, (0,1) \ldots \}

This set spans V but does not satisfy the frame condition since we get B = \infty. Consequently, a frame is, informally, a set of vectors which:

  • spans V,
  • are allowed to be linearly dependent
  • should be limited in number.

[edit] Types of frames

[edit] Tight frames

A frame is tight if the frame bounds A and B are equal. This means that the frame obeys a generalized Parseval's identity. A frame is normalized if A = B = 1.

[edit] Uniform frames

A frame is uniform if each element has the same norm: \forall k\ \|\mathbf{e}_k\| = c where c is a constant independent of k. A uniform normalized tight frame (c = 1) is an orthonormal basis.

[edit] History

Frames were introduced by Duffin and Schaeffer in their study on nonharmonic Fourier series. They remained obscure until Mallat, Daubechies, and others used them to analyze wavelets in the 1980s. Some practical uses of frames today include robust coding and design and analysis of filter banks.

[edit] References

  • Ole Christensen (2003). An Introduction to Frames and Riesz Bases. Birkhäuser. 
  • R. J. Duffin and A. C. Schaeffer (1952). "A class of nonharmonic Fourier series". Trans. Amer. Math. Soc. vol. 72: 341--366. doi:10.2307/1990760. 
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