Rod Nillsen BSc Tas, MSc PhD Flin, DSc Tas Tel: +61 (02) 4221 3835 Fax: +61 (02) 4221 4845 Email: rodney_nillsen@uow.edu.au Research and ideas website Postal Address: Department of Mathematics University of Wollongong Northfields Ave WOLLONGONG NSW 2522 Australia

I was born in Tasmania and my undergraduate studies were in Hobart at the University of Tasmania, where I studied science, mathematics and literature. My postgraduate work was at The Flinders University of South Australia, where I was studied under Igor Kluvánek. I spent two years at the Royal University of Malta, and another two years at the University College of Swansea in Wales. As well, I was for a period a tutor with the Open University in Britain. I have been at Wollongong since 1974.

My interests are in analysis, especially the Fourier transform and harmonic analysis, but I have also carried out research in differential equations, measure theory and functional analysis. My main work on the Fourier transform characterizes the functions whose Fourier transforms have a prescribed behaviour near the origin on, say, the real line. The research provides an alternative description of such functions as being precisely those which can be expressed as a finite sum of finite differences of functions. For a given prescribed behaviour, such functions form a Hilbert space in a natural norm, and it turns out that these spaces are related to the ranges of differential operators. For example, a function is the derivative of some square integrable function (in the sense of Schwartz distributions) if and only if it equals a finite sum of three first order differences of functions. The admissible wavelets in the wavelet theory are those whose Fourier transforms have a certain behaviour near the origin, so these results are related to characterization of the admissible wavelets.

These ideas are developed in the research monograph Difference spaces and invariant linear forms (Lecture Notes in Mathematics volume 1586, Springer 1994). The original motivation for this research was to extend work of Gary Meisters and Wolfgang Schmidt for the compact case of the circle group to the non-compact case of the real line. Their work was primarily concerned with the problem of when a translation invariant linear form on some given space of functions or distributions is necessarily continuous, and this connection with invariant forms extends to the non-compact case.

As a result of my research on the behaviour of the Fourier transform, I received a Doctor of Science degree from the University of Tasmania in 2000.

After taking lectures in a third year undergraduate course "Topology and chaos", I became interested in chaos theory and randomness, and am currently writing a book on this theme. I have an interest in making advanced mathematical ideas and results more accessible to a wider audience, and in finding applications of school mathematics to current problems and issues in society. I also have a serious interest in education, universities and political philosophy, and have publications in, or touching upon, these areas.

Details of the above research, and descriptions of further research and ideas, including downloads, may be found on my research and ideas website.

Updated: Oct 05