In mathematics, a frame of a vector space V with a scalar product can be seen as a generalization of the idea of a basis to sets which are linearly dependent. More precisely, a frame is a set {e_k} 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}

for all $\backslash mathbf\{v\}\; \backslash in\; V$ and for all k. This means that the constants A and B can be chosen independently of v: they only depend on the set {e_k}.

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 $\backslash mathbf\{\backslash tilde\{e\}\}\_\{k\}$ 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 $\backslash mathbf\{v\}\; \backslash in\; V$. 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.

Relation to bases

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

$$

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

therefore $A\; \backslash leq\; 0$, 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 $V\; =\; R^\{2\}$ and the infinite set

$\backslash \{\; (1,0)\; ,\; \backslash ,\; (0,1),\; \backslash ,\; (0,1)\; ,\; \backslash ,\; (0,1)\; \backslash ldots\; \backslash \}$

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

• spans V,
• are allowed to be linearly independent
• cannot be any arbitrary set of vectors which spans V.

History

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Frames were introduced by Duffin and Schaeffer in their study on nonharmonic Fourier series.

References

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Linear algebra