### B4.4 Fourier Analysis (2023-24)

General Prerequisites:
Distribution Theory is a pre-requisite.
Course Term: Hilary
Course Weight: 1
Course Level: H
Assessment Type: Written Examination
Course Overview:
Distribution theory can be thought of as the completion of differential calculus, just as Lebesgue integration theory can be thought of as the completion of integral calculus. It was created by Laurent Schwartz in the 20th century, as was Lebesgue's integration theory.
Distribution theory is a powerful tool that works very well in conjunction with the theory of Fourier transforms. One of the main areas of applications is to the theory of partial differential equations. In this course we give an introduction to these three theories.
Learning Outcomes:
Students will become acquainted with the basic techniques that in many situations form the starting point for the modern treatment of PDEs.
Course Synopsis:
The Fourier transform on $$R^n$$: the Schwartz class $$S$$ of test functions on $$R^n$$, properties of the Fourier transform on $$S$$, the Fourier transform of a Gaussian and the inversion formula on $$S$$. [4 lectures]
The class of tempered distributions $$S'$$ and their calculus. Fourier transforms of tempered distributions: definitions and examples, convolutions with tempered distributions. The inversion formula on $$S'$$. Fourier transform in $$L2$$ and $$~Plancherel's$$ theorem. The Sobolev scale $$H^s$$. Elliptic PDEs and G$$\mathring{a}$$rdingâ€™s inequality. [5 lectures]
Fundamental solutions for elliptic PDEs and hypoellipticity. [3 lectures]
The Riemann-Lebesgue lemma, Paley-Wiener theorems, the Poisson summation formula, periodic distributions and Fourier series, the uncertainty principle. [4 lectures]