Solutions to dilation equations

Abstract

This thesis aims to explore part of the wonderful world of dilation equations. Dilation equations have a convoluted history, having reared their heads in various mathematical fields. One of the early appearances was in the construction of continuous but nowhere differentiable functions. More recently dilation equations have played a significant role in the study of subdivision schemes and in the construction of wavelets. The intention here is to study dilatione quations as entities of interest in their own right, just as the similar subjects of differential and difference equations are often studied. It will often be Lp(R) properties we are interested in and we will often use Fourier Analysis as a tool. This is probably due to the author’s original introduction to dilation equations through wavelets. A short introduction to the subject of dilation equations is given in Chapter 1. The introduction is fleeting, but references to further material are given in the conclusion. Chapter 2 considers the problem of finding all solutions of the equation which arises when the Fourier transform is applied to a dilation equation. Applying this result to the Haar dilation equation allows us first to catalogue the L2(R) solutions of this equation and then to produce some nice operator results regarding shift and dilation operators. We then consider the same problem in Rn where, unfortunately, techniques using dilation equations are not as easy to apply. However, the operator results are retrieved using traditional multiplier techniques. In Chapter 3 we attempt to do some hands-on calculations using the results of Chapter 2. We discover a simple ‘factorisation’ of the solutions of the Haar dilation equation. Using this factorisation we produce many solutions of the Haar dilation equation. We then examine how all these results might be applied to the solutions of other dilation equations. A technique which I have not seen exploited elsewhere is developed in Chapter 4. This technique examines a left-hand or right-hand ‘end’ of a dilation equation. It is initially developed to search for refinable characteristic functions and leads to a characterisation of refinable functions which are constant on intervals of the form [n, n +1). This left-hand end method is then applied successfully to the problem of 2- and 3- refinable functions and used to obtain bounds on smoothness and boundedness. Chapter 5 is a collection of smaller results regarding dilation equations. The relatively simple problem of polynomial solutions of dilation equations is covered, as are some methods for producing new solutions and equations from known solutions and equations. Results regarding when self-similar tiles can be of a simple form are also presented.