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.