Finite Tight Frames Dejun Feng, Long Wang And Yang Wang Introduction Questions and Results An Algorithm Examples Introduction Frames and Tight Frames Let be a Hilbert space. A set of vectors in is called a frame if there exist
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Dejun Feng, Long Wang And Yang Wang
Frames and Tight Frames
Let be a Hilbert space. A set of vectors
in is called a frame if there exist
such that for any , we have
The constants are called lower frame bound and upper frame bound.
A frame is called a tight frame if .
A frame where all elements have the same norm is called an equal-norm frame.
We only consider finite dimensional Hilbert space
is a tight frame in .
----- Mercedes-Benz frame
is a frame matrix (resp. TFM) if and only if the column vectors of form a frame (resp. tight frame) of .
Let be a frame matrix. Let
be the maximal and minimal eigenvalues of , respectively. Then
is called the condition number of
is a TFM if and only if
For any , let
Suppose that are all the eigenvalues of
. Then for any vectors , the matrix
Furthermore, the equality can be attained by some
. In particular, at most vectors are needed to make a TF.
Theorem 2(D. J. Feng, W & Y. Wang)
For any with equal norm 1, let
. Suppose that
are all the eigenvalues of . Denote by q the smallest integer greater than or equal to Then we can find such that
form an equal-norm TF.
is it possible to find a TF such that
For example, is it possible to find a TF
Let Then there exists a TF
if and only if
This is called the fundamental inequality.
A Householder matrix is a matrix of the form
Theorem 4 (D. J. Feng, W & Y. Wang)
For a given sequence satisfying the fundamental inequality
Inductively, we can construct a sequence of matrices by using a sequence of Householder matrices
such that ,
with possibly some columns
interchanged and the matrices satisfies the following properties
Furthermore, for any
Then for any
we can find
In fact, can be found explicitly as follows:
we can construct a Householder matrix
Such that the column vectors of
Let be two positive integers and . Let
be given and satisfy the fundamental inequality
with prescribed norms.
Our algorithm yields the following TFM: