Joshua Stevenson, University of Tasmania
Dr Melissa Tacy, The University of Auckland, NZ
Dr Timothy Candy, The University of Otago, NZ
Fourier analysis is the study of representing functions as sums or integrals of simple waves. It has applications across a broad range of mathematical and physical sciences such as the analysis of solutions to partial differential equations, inverse problems and data processing. The natural setting for Fourier analysis is on the space of generalised functions, known as distributions.
In this course we introduce the space of tempered distributions and learn to compute with them. We then develop the Fourier transform first as an operator on functions, and then extend it to the space of tempered distributions. This extension gives a natural way to see that Fourier transforms and series are essentially the same object. As an application, we learn how to use distributions to study differential equations, and in particular we introduce the notion of a fundamental solution.
We next examine the various ways Fourier transforms/series can be understood to converge, and mention some perhaps surprising examples where convergence fails. Many issues regarding convergence of these oscillatory objects are still open and the focus of active research. We finish the course by discussing the Fourier restriction conjecture, which forms a major open problem in the field.
There is no set text book for this course, and full lecture notes will be provided. However similar material to that covered can be found in:
Take this quiz and look at some of the foundational skills required for this subject.
Melissa Tacy works in the areas of semiclassical and harmonic analysis. She completed her PhD from ANU in 2010 under the supervision of Andrew Hassell. After her studies she first worked in the US for 3.5 years (at the IAS and Northwestern University). Melissa returned to Australia for a further 3.5 years working at University of Adelaide and the ANU before moving to New Zealand to take up a position at the University of Otago. In 2020 she moved from Otago to the University of Auckland.
Timothy Candy completed his PhD at the University of Edinburgh, before moving to Imperial College as a Chapman Fellow in pure mathematics. After spending time as a postdoc at Johns Hopkins University and Bielefeld University, he become a lecturer in the Department of Mathematics and Statistics at the University of Otago. His research centers on problems in partial differential equations and harmonic analysis, and involves trying to understand the long time dynamics of nonlinear dispersive and wave equations using tools from harmonic analysis.