1. MA5242 Wavelets Lecture 2 Euclidean and Unitary Spaces 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. Scalar Product for Euclidean Space Definition: A Euclidean space is a vector space V over R together with a scalar product Linear in the first argument Symmetric that satisfies, for all Positive Definiteand =0 iff u = 0

3. Examples Example 1. Example 2. where with is symmetric and positive definite with Example 3. Euclidean space V with dot product Example 4.

4. Gramm-Schmidt Orthogonalization Theorem: Given a linearly independent set of a Euclidean space V, there exists an orthogonal set whenever i < j ) and satisfies(so for allProof. Let and where

5. Example of GS Orthogonalisation Example Applying GS Orthogonalisation to yields where

6. Problem Set 1 2. Continue the GS Orth. on previous page to obtain 3. Show that 1. Orthogonalize the following basis of contains only even powers of x for k odd and odd powers of x for k even 4. Show that all zeros of are in [-1,1]

7. Gramm Matrices Definition: The Gramm matrix of a set is defined by its entries is the Hilbert matrix Example 2. The Gramm matrix for the monomial basis ofwith scalar product Example 1. The Gramm matrix for a basis of that consists of the columns of a matrix B is

8. Orthogonal Transforms Definition: A linear transformation A : V  V is orthogonal if Definition: The norm ||. || : V  R on a ES is Theorem A linear transformation A : V  V is ort. iff Proof. Follows from the polarization identity

9. Gramm Orthonormalization Theorem: Given a linearly independent set the set of vectors where A is a matrix is orthonormal iff defined by Proof. with Gramm matrix G iff

10. Haar Transform Haar Transform (one stage) ongiven by matrix

11. Discrete Wavelet Transform For suitable real entries this matrix is orthogonal.

12. Scalar Product for Unitary Space Definition: A Unitary space is a vector space V over C together with a scalar product Linear in the first argument Hermitian Symmetric that satisfies, for all Positive Definiteand =0 iff u = 0

13. Unitary Transforms Definition: A linear transformation A : V  V is unitary if Definition: The norm ||. || : V  R on a US is Theorem A linear transformation A : V  V is uni. iff Proof. Follows from the unitary polarization identity

14. Problem Set 2 2. Derive the polarization identities and theorems 1.Prove that a set of vectors in a Euclidean or Unitary space is linearly independent iff its Gramm matrix is positive definite 4. State & derive the Schwarz inequality for ES & US 5. State & derive the triangle inequality for ES & US 3. Show that the columns of a square matrix having complex entries are orthogonal iff its rows are.

15. Problem Set 3 for the discrete wavelet transform to be orthogonal 1.Derive necessary and sufficient conditions on 2. Show that Daubechies’ length 4 filters are good 3. Show that d has zero 0-th and 1-st order moments and 4. Write a MATLAB program to make wavelet mat. 5. Use it to compute & plot the Daubecheis length-4 WT of the vector v = abs([1:40 – 21.5]’) & analyse