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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
Finite Tight Frames

Dejun Feng, Long Wang And Yang Wang

  • Introduction
  • Questions and Results
  • An Algorithm
  • Examples
introduction
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.

introduction3
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
Examples
  • The union of two orthonormal bases is a tight frame with frame bound 2

is a tight frame in .

----- Mercedes-Benz frame

slide5

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
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
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
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
Theorem 1(Continued)

Furthermore, the equality can be attained by some

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

questions10
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 results11
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.

questions12
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
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 definitions14
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 results15
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
Main Results (Cont.)

with possibly some columns

interchanged and the matrices satisfies the following properties

  • If we denote then

Furthermore, for any

lemmas
Lemmas
  • Lemma 4.2

Let

Then for any

we can find

In fact, can be found explicitly as follows:

lemmas19
Lemmas
  • Lemma 4.3

Let and

For any

we can construct a Householder matrix

Such that the column vectors of

satisfy

algorithm
Algorithm

Let be two positive integers and . Let

be given and satisfy the fundamental inequality

Let

algorithm21
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
algorithm22
Algorithm
  • , the result will be the TFM

with prescribed norms.

example
Example
  • For

Our algorithm yields the following TFM: