“It can be difficult to justify picking up a textbook not specifically
related to your research area, but the summer school gives you an
opportunity to learn the basics of a new area in a relatively short
amount of time.”

Joshua Stevenson, University of Tasmania

Fourier Analysis and Distribution Theory

Lecturers

Dr Melissa Tacy, The University of Auckland, NZ
Dr Timothy Candy, The University of Otago, NZ

Synopsis

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.

Course Overview

  • Distributions
    • Motivation for distributions. Why can’t we just work with functions?
    • The Schwartz space and the space of tempered distributions.
    • Basic operations with distributions.
    • Limits in the distributional setting.
  • The Fourier transform
    • The Fourier transform on the Schwartz space and extension to integrable/square integrable functions.
    • The Fourier transform on the space of tempered distributions.
    • The uncertainty principle and applications.
    • Applications to PDE, in particular fundamental solutions.
    • Periodic distributions and the Poisson summation formula, from integrals to sums.
  • Fourier series
    • Convergence for suitably regular data.
    • Jumps around discontinuities, Gibb’s phenomena.
    • Broader notions of convergence, examples where convergence fails.
  • The Fourier restriction problem
    • When can we restrict the Fourier transform to lower dimensional sets in a sensible way?
    • Survey of known results and the current state of the art.

Prerequisites

  • Real analysis

Assessment

  • Quizzes 20% total
  • 2 assignments 15% each (30% total)
  • Final take home exam 50%

Attendance requirements

  • For those completing the subject for their own knowledge/interest, quizzes must be completed as an attendance requirement

Resources/pre-reading (if available)

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:

  • Introduction to the Theory of Distributions, by F. Friedlander,
  • A Guide to Distribution Theory and Fourier Transforms, by R. Strichartz.

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Dr Melissa Tacy, The University of Auckland, NZ

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.

Dr Timothy Candy, The University of Otago, NZ

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.