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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 l.jpg
Finite Tight Frames

Dejun Feng, Long Wang And Yang Wang

  • Introduction

  • Questions and Results

  • An Algorithm

  • Examples


Introduction l.jpg
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.


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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


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Examples

  • The union of two orthonormal bases is a tight frame with frame bound 2

    is a tight frame in .

    ----- Mercedes-Benz frame


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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 .


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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


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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?


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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


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Theorem 1(Continued)

Furthermore, the equality can be attained by some

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


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Questions

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


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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.


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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


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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.


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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.


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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 ,


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Main Results (Cont.)

with possibly some columns

interchanged and the matrices satisfies the following properties

  • If we denote then

    Furthermore, for any


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Lemmas

  • Lemma 4.2

    Let

    Then for any

    we can find

    In fact, can be found explicitly as follows:



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Lemmas

  • Lemma 4.3

    Let and

    For any

    we can construct a Householder matrix

    Such that the column vectors of

    satisfy


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Algorithm

Let be two positive integers and . Let

be given and satisfy the fundamental inequality

Let


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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


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Algorithm

  • , the result will be the TFM

    with prescribed norms.


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Example

  • For

    Our algorithm yields the following TFM:


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