Frame of A Vector Space - The Dual Frame

The Dual Frame

The frame condition is both sufficient and necessary for allowing the construction of a dual or conjugate frame, relative the original frame, . The duality of this frame implies that

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

is satisfied for all . In order to construct the dual frame, we first need the linear mapping: defined as

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

From this definition of and linearity in the first argument of the inner product, it now follows that

$\langle \mathbf{S} \mathbf{v} | \mathbf{v} \rangle = \sum_{k} |\langle \mathbf{v} | \mathbf{e}_{k} \rangle|^{2}$

which can be inserted into the frame condition to get

$A \| \mathbf{v} \|^{2} \leq \langle \mathbf{S} \mathbf{v} | \mathbf{v} \rangle \leq B \| \mathbf{v} \|^{2} \text{ for all }\mathbf{v} \in V$

The properties of can be summarised as follows:

is self-adjoint, positive definite, and has positive upper and lower bounds. This leads to
the inverse of exists and it, too, is self-adjoint, positive definite, and has positive upper and lower bounds.

The dual frame is defined by mapping each element of the frame with :

$\tilde{\mathbf{e}}_{k} = \mathbf{S}^{-1} \mathbf{e}_{k}$

To see that this make sense, let be arbitrary and set

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

It is then the case that

$\mathbf{u} = \sum_{k} \langle \mathbf{v} | \mathbf{e}_{k} \rangle ( \mathbf{S}^{-1} \mathbf{e}_{k} ) = \mathbf{S}^{-1} \left ( \sum_{k} \langle \mathbf{v} | \mathbf{e}_{k} \rangle \mathbf{e}_{k} \right ) = \mathbf{S}^{-1} \mathbf{S} \mathbf{v} = \mathbf{v}$

which proves that

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

Alternatively, we can set

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

By inserting the above definition of and applying known properties of and its inverse, we get

$\mathbf{u} = \sum_{k} \langle \mathbf{v} | \mathbf{S}^{-1} \mathbf{e}_{k} \rangle \mathbf{e}_{k} = \sum_{k} \langle \mathbf{S}^{-1} \mathbf{v} | \mathbf{e}_{k} \rangle \mathbf{e}_{k} = \mathbf{S} (\mathbf{S}^{-1} \mathbf{v}) = \mathbf{v}$

which shows that

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

This derivation of the dual frame is a summary of section 3 in the article by Duffin and Schaeffer. They use the term conjugate frame for what here is called dual frame.

Read more about this topic: Frame Of A Vector Space

Famous quotes containing the words dual and/or frame:

“Thee for my recitative,
Thee in the driving storm even as now, the snow, the winter-day
declining,
Thee in thy panoply, thy measur’d dual throbbing and thy beat
convulsive,
Thy black cylindric body, golden brass and silvery steel,”
—Walt Whitman (1819–1892)

“From Harmony, from heavenly Harmony
This universal Frame began:
From Harmony to Harmony
Through all the Compass of the Notes it ran,
The Diapason closing full in Man.”
—John Dryden (1631–1700)