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The aim of this course is to give a rigorous introduction to Fourier series and Fourier transforms including some applications to the theory of ODEs and PDEs.

2 hours of lectures a week, over 11 weeks.

1 hour tutorial a week over 10 weeks (or equivalent)

Mandatory Entry Requirements

Level 3 Metric Spaces and Basic Topology.

Recommended Entry Requirements

Level 4 Functional Analysis

Assessment

100% Final Exam

Reassessment

In accordance with the University's Code of Assessment reassessments are normally set for all courses which do not contribute to the honours classifications. For non honours courses, students are offered reassessment in all or any of the components of assessment if the satisfactory (threshold) grade for the overall course is not achieved at the first attempt. This is normally grade D3 for undergraduate students, and grade C3 for postgraduate students. Exceptionally it may not be possible to offer reassessment of some coursework items, in which case the mark achieved at the first attempt will be counted towards the final course grade. Any such exceptions are listed below in this box.

Reassessments are normally available for all courses, except those which contribute to the Honours classification. For non Honours courses, students are offered reassessment in all or any of the components of assessment if the satisfactory (threshold) grade for the overall course is not achieved at the first attempt. This is normally grade D3 for undergraduate students and grade C3 for postgraduate students. Exceptionally it may not be possible to offer reassessment of some coursework items, in which case the mark achieved at the first attempt will be counted towards the final course grade. Any such exceptions for this course are described below. 

The aim of this course is to give a rigorous introduction to Fourier series and Fourier transforms including some applications to ODEs and PDEs assuming only minimal background in Measure Theory.

By the end of this course students will be able to:

- Demonstrate knowledge and the ability to apply central definitions and facts in the theory of Fourier series and Fourier transforms and use these to solve problems of a numerical or logical nature. Specific topics include:

1. Fourier series of (Riemann) integrable functions on the circle (or intervals), convolutions and kernels, basic results on convergence of Fourier series;

(e.g. l^2 convergence, Cesaro summability; Fejer's Theorem, and applications (Weierstrass approximation Theorem), Riemann-Lebesgue Lemma and Riemann localisation, smoothness properties, possible applications like construction of nowhere differentiable continuous functions, heat equation on the circle, Fourier series of continuous functions)

2. Basic theory of Fourier transforms on R, Fourier transforms of integrable functions on R Riemann-Lebesgue Lemma and Riemann localisation in this setting, Schwartz space and Riemann inversion formula and Plancherel formula for Schwartz functions, connection to Fourier series (Poisson summation formula).

3. Further possible applications include PDEs, Heisenberg uncertainty principle, finite Fourier Analysis and its applications to Analytic Number Theory (Dirichlet's Theorem).

Students must submit at least 75% by weight of the components (including examinations) of the course's summative assessment.