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 (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

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

23. Example • For Our algorithm yields the following TFM: