Frame of A Vector Space

In linear algebra, 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. The key issue related to the construction of a frame appears when we have a sequence of vectors, with each and we want to express an arbitrary element as a linear combination of the vectors :

and want to determine the coefficients . If the set does not span, then these coefficients cannot be determined for all such . If spans and also is linearly independent, this set forms a basis of, and the coefficients are uniquely determined by : they are the coordinates of relative to this basis. If, however, spans but is not linearly independent, the question of how to determine the coefficients becomes less apparent, in particular if is of infinite dimension.

Given that spans and is linearly dependent, it may appear obvious that we should remove vectors from the set until it becomes linearly independent and forms a basis. There are some problems with this strategy:

  1. By removing vectors randomly from the set, it may lose its possibility to span before it becomes linearly independent.
  2. Even if it is possible to devise a specific way to remove vectors from the set until it becomes a basis, this approach may become infeasible in practice if the set is large or infinite.
  3. In some applications, it may be an advantage to use more vectors than necessary to represent . This means that we want to find the coefficients without removing elements in .

In 1952, Duffin and Schaeffer gave a solution to this problem, by describing a condition on the set that makes it possible to compute the coefficients in a simple way. More precisely, a frame is a set of elements of V which satisfy the so-called frame condition:

There exist two real numbers, A and B such that and
A \| \mathbf{v} \|^{2} \leq \sum_{k} |\langle \mathbf{v} | \mathbf{e}_{k} \rangle|^{2} \leq B \| \mathbf{v} \|^{2} \text{ 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 .

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

It can be shown that the frame condition is sufficient to be able to find a set of dual frame vectors with the property that

\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 . This implies that a frame together with its dual frame has the same property as a basis and its dual basis in terms of reconstructing a vector from scalar products.

Read more about Frame Of A Vector Space:  Relation To Bases, The Dual Frame, History

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