MA5242 Wavelets Lecture 2 Euclidean and Unitary Spaces

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

Scalar Product for Euclidean Space Definition: A Euclidean space is a vector space V over R together with a scalar product that satisfies, for all Linear in the first argument Symmetric Positive Definite and =0 iff u = 0

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

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

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

Problem Set 1 1. Orthogonalize the following basis of 2. Continue the GS Orth. on previous page to obtain 3. Show that 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]

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

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

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

Haar Transform Haar Transform (one stage) on given by matrix

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

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

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

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

Problem Set 3 • Derive necessary and sufficient conditions on and for the discrete wavelet transform to be orthogonal 2. Show that Daubechies’ length 4 filters are good 3. Show that d has zero 0-th and 1-st order moments 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

MA5242 Wavelets Lecture 2 Euclidean and Unitary Spaces