Finite Tight Frames

1
Finite Tight Frames

• Dejun Feng, Long Wang And Yang Wang
• Introduction
• Questions and Results
• An Algorithm
• Examples

2
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.
  

3
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
  
  

4
Examples

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

5

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 .

6
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

7
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?

8
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

9
Theorem 1(Continued)

• Furthermore, the equality can be attained by some
  
  
• . In particular, at most
  vectors are needed to make
  a TF.

10
Questions

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

11
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.

12
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

13
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.

14
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.
  

15
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
  ,
  

16
Main Results (Cont.)

• with
  possibly some columns
  
• interchanged and the matrices
  satisfies the following properties
  
• If we denote
  then
  
• Furthermore, for any

17
Lemmas

• Lemma 4.2
• Let
• Then for any
• we can find
• In fact, can be found
  explicitly as follows
  

18
Lemmas
19
Lemmas

• Lemma 4.3
• Let and
• For any
• we can construct a Householder matrix
• Such that the column vectors
  of
  
• satisfy

20
Algorithm

• Let be two positive integers and
  . Let
  
• 
  be given and satisfy the
  fundamental inequality
  
• Let

21
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 (k1)-th column.
  
• then skip.
• then search
  for a column with norm less than or equal to
  and then swap it with (k1)-th column.
  
• , where
• 
  

22
Algorithm

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

23
Example

• For
• Our algorithm yields the following TFM