Finite Tight Frames

566 Views

Download Presentation
## Finite Tight Frames

- - - - - - - - - - - - - - - - - - - - - - - - - - - E N D - - - - - - - - - - - - - - - - - - - - - - - - - - -

**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: