Fourier analysis, multiresolution analysis and dilation equations

Abstract

This thesis has essentially two parts. The first two chapters are an introduction to the related areas of Fourier analysis, multiresolution analysis and wavelets. Dilation equations arise in the context of multi resolution analysis. The mathematics of these two chapters is informal, and is intended to provide a feeling for the general subject. This work is loosely based on two talks which I gave, one during the 1997 Inter-varsity Mathematics competition and the other at the 1997 Dublin Institute for Advanced Studies Easter Symposium. The second part, Chapters 3 and 4, contain original work. Chapter 3 provides a new formal construction of the Fourier transform on Lp(Rn) (1 ≤ p ≤ 2) based on the ideas introduced in Chapter 2. The idea is to take some basic properties of the Fourier transform and show we can construct a bounded operator on L2(R) with these properties. I do this by constructing an operator on each level of the Haar multiresolution analysis, which I then show is well enough behaved to be extended by a limiting process to all of L2(R). Some of the important properties of the Fourier transform are also derived in terms of this construction, and the generalisations to Lp(Rn) are explored. Chapter 4 builds on the work of Chapters 3 and provides a uniqueness result for the Fourier transform. While searching for this result I also establish a related result for dilation equations (a subject also introduced in Chapter 2). Here the exact set of properties which were used to define the Fourier transform are varied in an effort to discover which are merely consistent with the Fourier transform and which strong enough to define it. I end up examining sets of dilation equations and determining when these will have a unique solution.