Module description
Syllabus
Series expansions. Definition of Fourier series. Related expansions. Bessel's inequality. Pointwise and uniform convergence of Fourier series. Periodic solutions of differential equations. The vibrating string. Convolution equations. Mean square convergence. Schwartz space S. Fourier transform in S. Inverse Fourier transform. Parseval's formula. Solutions of differential equations with constant coefficients.
Prerequisites
You are advised to have taken 5CCM221A Real Analysis or similar analysis courses using normed spaces. You cannot take this module if you will take 6CCM318A
Assessment details
2 hr written examination or alternative assessment
Educational aims & objectives
The purpose of the module is to introduce the notions of Fourier series and Fourier transform and to study their basic properties. The main part of the module will be devoted to the one dimensional case in order to simplify the definitions and proofs. Many multidimensional results are obtained in the same manner, and those results may also be stated. The Fourier technique is important in various fields, in particular, in the theory of (partial) differential equations. It will be explained how one can solve some integral and differential equations and study the properties of their solutions using this technique.
Teaching pattern
Three hours of lectures and one hour of tutorial per week throughout the term
Suggested reading list
Indicative reading list - link to Leganto system where you can search with module code for lists