Finite Tight Frames

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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

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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

More definitions

• 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: