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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

such that for any , we have

The constants are called lower frame bound and upper frame bound.

Introduction

A frame is called a tight frame if .

- Equal-norm Frames
A frame where all elements have the same norm is called an equal-norm frame.

We only consider finite dimensional Hilbert space

Examples

- The union of two orthonormal bases is a tight frame with frame bound 2
is a tight frame in .

----- Mercedes-Benz frame

- Frame matrix
- A matrix is called a frame matrix (FM) if rank ( ) =
- is called a tight frame matrix (TFM) if
- for some
is a frame matrix (resp. TFM) if and only if the column vectors of form a frame (resp. tight frame) of .

More Definitions

- Condition number
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

Questions

- Given vectors , how many vectors do we need to add in order to obtain a tight frame?
- If only a fixed number of vectors are allowed to be added, how small can we make the condition number to be?

Main Results

- Theorem 1 (D.J. Feng, W &Y. Wang)
For any , let

Suppose that are all the eigenvalues of

. Then for any vectors , the matrix

satisfies

where

Theorem 1(Continued)

Furthermore, the equality can be attained by some

. In particular, at most vectors are needed to make a TF.

Questions

- Given vectors with equal norm 1, how many vectors do we need to add in order to obtain an equal-norm tight frame?

Main Results

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.

Questions

- For a given sequence
is it possible to find a TF such that

?

- If so, how to construct such a TF?
For example, is it possible to find a TF

More Known Results

- Theorem 3 (P. Casazza, M. Leon & J. C. Tremain)
Let Then there exists a TF

such that

if and only if

This is called the fundamental inequality.

More Definitions

A Householder matrix is a matrix of the form

- Any Householder matrix is unitary.
- If A is a TFM and U is a unitary matrix, then AU is a TFM.

Main Results

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 ,

Main Results (Cont.)

with possibly some columns

interchanged and the matrices satisfies the following properties

- If we denote then
Furthermore, for any

Lemmas

- Lemma 4.2
Let

Then for any

we can find

In fact, can be found explicitly as follows:

Lemmas

- Lemma 4.3
Let and

For any

we can construct a Householder matrix

Such that the column vectors of

satisfy

Algorithm

Let be two positive integers and . Let

be given and satisfy the fundamental inequality

Let

Algorithm

Algorithm

- Multiply and let
- Repeat the following for
- Calculate the norm of the
- Compare
- then search for a column with norm great than or equal to and then swap it with (k+1)-th column.
- then skip.
- then search for a column with norm less than or equal to and then swap it with (k+1)-th column.

- , where

Algorithm

- , the result will be the TFM
with prescribed norms.

Example

- For
Our algorithm yields the following TFM:

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