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MC430 Approximation Theory

MC430 Approximation Theory

Credits: 20 Convenor: Jeremy Levesley Semester: 1

Prerequisites: essential: MC144, MC145, MC146, MC147, MC240, MC241 desirable: MC243
Assessment: Continual assessment: 10% Three hour exam: 90%

Lectures: 36 Problem Classes: 10
Tutorials: 0 Private Study: 104
Labs: none Seminars: none
Project: none Other: none
Surgeries: none Total: 150

Explanation of Pre-requisites

I will use some results from MC243 about best approximations. Also since the main space under study will be an inner product space, again material from MC243 will smooth the student's path to understanding. However, it is not essential to have done MC243, which deals with abstract inner product spaces, whereas we have a single concrete space in mind.

Course Description

The course will expound the elementary theory of wavelets. These objects are of great importance in signal processing, and in the compression (and thus speedy transmission of) signals. A major modern application is in the transmission of signals for high resolution TV.

A wavelet can be thought of as a small wave. You will already be familiar with the waves $t\mapsto \sin(kt)$ and $t\mapsto \cos(kt)$, where k is a whole number. These guys are pretty big waves in the sense of modern wavelet theory. When they are used in signal processing, they are naturally associated with the Fourier Series of the signal:

\begin{displaymath} f(t) \approx (1/2)a_0 + \sum_{k=1}^\infty \Big[a_k\cos(kt) + b_k\sin(kt)\Big]. \end{displaymath}

Over the years, mathematicians have struggled to understand in what senses the $\approx$ sign here can be replaced by =, and much famous theory has resulted from this question. But it's not the fundamental one which a wavelet course looks at. Just suppose we have = in some suitable sense. If we then ask, ``Does the signal contain significant amounts of frequency 136?", the answer is easy. The frequency of the waves in the Fourier series is encoded via the index k, and the amount of frequency 136 present in the signal f is governed by the size of the numbers a136 and b136. If these numbers are (relatively) large, frequency 136 will play a prominent role in the signal. A much harder question is the analysis of the behaviour of the signal with the variable t, which is usually indicating the progression of the signal over time. Are there times when the signal is small? Are there times when the signal is slowly varying? Are there times when the signal is large? rapidly varying? It's more or less because the $\cos$ and $\sin$ waves are somehow too big that these questions are tricky to answer. The course will show how to construct the most elementary wavelets, and the famous decomposition and reconstruction algorithms for signals. Our main aim will be to construct one of the famous wavelets due to Ingrid Daubechies from scratch. This turns out to be a very tricky job, calling for some quite sophisticated mathematics!

Aims

To understand the concept of a wavelet and appreciate how to construct the most elementary wavelets. To understand the principles behind the reconstruction and decomposition algorithms.

Objectives

Students will poseess the necessary theoretical understanding to construct the Daubechies compactly supported wavelets. They will be able to write a computer programme to implement the reconstruction and decomposition algorithms. They will have gained knowledge of and experience with working with the Fourier transform in a formal way.

Transferable Skills

Students will expand their ability to apply mathematics to practical situations, and will also have considerable exposure to the Fourier transform.

Syllabus

The basic space L2(R) and the basic facts we need to know about it. Results about the Fourier transform, including Parseval's identity. Brief discussion of Fourier series. Introduction to wavelets, the Haar wavelet as an elementary model example, the two-scale equation, multiresolution analysis, how the multiresolution analysis aids the construction of wavelets, stable bases in L2(R), the decomposition and reconstruction algorithms, the symbol, compactly supported wavelets, the cascade algorithm for constructing wavelets from the two scale equation, the Daubechies wavelets.

Reading list

Recommended:

C. Chui, An Introduction to Wavelets, Academic Press, 1992.

Background:

C. Chui, Chapter 1 in, Advances in Numerical Analysis, Volume II: Wavelets, subdivision algorithms and radial functions (ed W. Light), OUP 1992.

Details of Assessment

The coursework for the continual assessment consists of about 8 pieces of written work, consisting of problems from the problem sheets.

The written January examination lasts for three hours, and contains 6 questions. Students must answer four.

Next: MC440 Commutative Algebra Up: Year 4 Previous: Year 4

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Last updated: 10/4/2000
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