Submission for the degree of Doctor of Science

My research lies in harmonic analysis, overlapping the areas of pure and applied mathematics, and over the last twenty years has developed along two major themes: (a) Abstract approximation and multiplier theory (b) Hypergroups and probability theory In addition I have been active in: (c) Mathematics education and other areas My work in abstract approximation and multiplier theory began with my doctoral studies on Bernstein's inequality for locally compact groups. An important breakthrough was to prove versions of this inequality based purely on properties of group translation, and without any reference to a differential structure. This led to versions of the Jackson and Bernstein theorems estimating the degree of approximation of smooth functions by partial sums of their Fourier series. An integral part of these developments was a study of multiplier theory for various related spaces of functions. During the past ten years my research has been in hypergroups and probability theory on these spaces. Much of this work has been done in collaboration with Herbert Heyer, and covers a study of the properties of the Fourier transform and convolution, through to the development of properties of positive definite and negative definite functions. As part of this work a major activity of the past five years has been the preparation (jointly with Herbert Heyer) of a substantial research monograph Harmonic analysis of probability measures on hypergroups which will be published by Walter de Gruyter publishers this year. This thesis covers those publications listed under (a) and (b) that have appeared in print