Introductory description

Fourier analysis lies at the heart of many areas in mathematics. This course is about the applications of Fourier analytic methods to various problems in mathematics and sciences.

Module web page

Module aims

Outline syllabus

This is an indicative module outline only to give an indication of the sort of topics that may be covered. Actual sessions held may differ.

Fourier analysis lies at the heart of many areas in mathematics. This course is about the applications of Fourier analytic methods to various problems in mathematics and sciences. The emphasis will be on developing the ability of using important tools and theorems to solve concrete problems, as well as getting a sense of doing formal calculations to predict/verify results. Topics will include:

1. Fourier series of periodic functions, Gibbs phenomenon, Fejer and Dirichlet kernels, convergence properties, etc.
2. Basic properties of the Fourier transform on R^d, including L^p theory.
3. Topics on the Fourier inversion formula, including the Gauss-Weierstrass and Abel Poisson kernels, and connections to PDE.
4. A selection of more advanced topics, including the Hilbert transform and an introduction to Singular Integrals.

Learning outcomes

By the end of the module, students should be able to:

• Derive basic properties of Fourier series and Fourier transforms; Use the Dirichlet kernel to prove pointwise convergence of Fourier series for regular functions (Holder, BV, etc); Apply results about convolutions with good kernels to partial differential equations; Understand maximal operators and interpolation theorems and their relationship to almost sure convergence results for good kernels; Understand how to use results about Hilbert transforms to prove norm convergence of Fourier transforms in L^p.

Subject specific skills

Fourier analysis is a central topic in mathematics, physics and engineering, including applications to signal processing and cryptography. Students taking this module will develop their knowledge of Fourier analysis and its application to partial differential equations, as well as the mathematical analysis skills required for a deeper understanding of the topic.

Transferable skills

See subject specific skills.