A thesis submitted to the University of Tasmania

The thesis consists of two volumes. Volume I comprises work in various areas of mathematical analysis, especially abstract harmonic analysis, differential equations and moment problems. Volume II comprises work primarily in harmonic analysis, especially work on the Fourier transform, invariant linear forms, and associated phenomena. The ideas in this work have had a long period of gestation, although the fuller development and expression of them has occurred over a relatively short and intense period. The dominant ideas emerged in an endeavour to answer the following question: if f is a function in L2(IRn), how can the behaviour of its Fourier transform near the origin of IRn be described and characterized? The corresponding question for the circle group had been given a satisfactory answer by Gary Meisters and Wolfgang Schmidt in 1972, so the work also can be regarded as arising from an attempt to extend their result from the compact case of the circle group to the non-compact case of IRn. The answers presented to these and related questions have implications for other areas of analysis; the notable ones being the ranges of partial differential operators, and the behaviour of some of the singular integral operators of classical analysis.