## “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

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.