1. MA5242 Wavelets Lecture 3 Discrete Wavelet Transform Wayne M. Lawton Department of Mathematics National University of Singapore 2 Science Drive 2 Singapore 117543 Email matwml@nus.edu.sg Tel (65) 6874-2749

2. Riesz Representation Theorem. If is a finite dimensional unitary space there exists an antilinear isomorphism such that Proof. Let and define Then be an ONB for

3. Adjoint Transformations Theorem: Given unitary spacesand a linear there exists a uniquetransformation linear transformation with (adjoint of T) Proof. Defineby composition let be the Riesz Rep. transformations and define

4. Problem Set 1 and that 1. Assume that are ONB for unitary spaces between the matrices is linear. Derive the relationship that represent with respect to these bases. 3. Derive the Riesz Representation, Adjoint and matrix representations, and characterization for orthogonal transformations for Euclidean spaces. 2. Prove that a transformation is unitary iff

5. General Discrete Wavelet Transform

6. Convolution Representation where a,b,c,d are infinite sequences that extend the finite sequences

7. Orthogonality Conditions Theorem. The wavelet transform matrix is unitary iff for all

8. Laurent Polynomials Definition: A Laurent polynomial is a function that admits a representation where c is a finitely supported sequence. Theorem: For seq. a, b, Definition: For a sequence c let and define the unit circle

9. Conjugate Quadrature Filters Definition: A sequence c that satisfies the quadratic equations necessary for a wavelet transform matrix to me unitary is called a Conjugate Quadrature Filter Theorem. A sequence c is a CQF iff it satisfies Theorem: Prove that if c is a CQF and if d is related to c by the equation on the previous page then d is also a CQF and the WT is unitary Theorem: If c is a CQF then the WT is unitary if

10. Problem Set 2 1. Derive the conditions for a WT to be unitary. 2. Prove the theorems about Laurent polynomials and the two theorems on the preceding page. 4. Prove that d on the previous page is the same as 3. Prove that c, d form a unitary WT iff (the modulation matrix) is unitary for all

11. Moment Conditions Definition. d has -1< p vanishing moments if Theorem. If c,d gives a unitary WT then d has -1< p vanishing moments has a factoriff has a factor iff

12. Moment Consequences Theorem. If d has -1< p vanishing moments and is supported on the set {0,1,…,2N-1} then can be represented by a polynomial having degree < N Proof. if the finite sequence d(k),d(k-1),…,d(k-2N+1) by the binomial theorem and vanishing moments.

13. Riesz-Fejer Spectral Factorization Theorem. A Laurent polynomial N ison iff there exists a LP P such that Proof. Let where be the set of roots of is the multiplicity of Since N is real-valued furthermore, since N is non-negative the are even hence paired, now choose P to contain one root from each pair and the result easily follows.

14. Daubechies Wavelets Theorem. If c is a CQF supported on 0,1,…,2N-1 and then satisfies and is uniquely determined by the equations has a factor Furthermore, and c can be chosen by the R.-F. Theorem.

15. Problem Set 3 1. Prove all of the Theorems after Problem Set 2.