# B4.4 Fourier Analysis - Material for the year 2020-2021

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Distribution Theory and Analysis of PDEs is a pre-requisite.

### Assessment type:

- Written Examination

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.

Students will become acquainted with the basic techniques that in many situations form the starting point for the modern treatment of PDEs.

The Fourier transform on $\mathbb{R}^n$: the Schwartz class $\mathcal{S}$ of test functions on $\mathbb{R}^n$ , properties of the Fourier transform on $\mathcal{S}$, the Fourier transform of a Gaussian and the inversion formula on $\mathcal{S}$. [4 lectures]

The class of tempered distributions $\mathcal{S}'$ and their calculus. Fourier transforms of tempered distributions: definitions and examples, convolutions with tempered distributions. The inversion formula on $\mathcal{S}'$. Fourier transfor in $L^2$ and ~Plancherel's theorem. [5 lectures]

Solving PDEs using Fourier transformation: the Laplace equation, the heat equation, the wave equation, Schrödinger's equation. Fundamental solutions, ellipticity and hypoellipticity. [3 lectures]

Fourier Analysis: the Riemann-Lebesgue lemma, Paley-Wiener theorems, the Poisson summation formula, the uncertainty principle.[4 lectures]

The main recommended book is

1) R.S. Strichartz, *A Guide to Distribution Theory and Fourier Transforms*

(World Scientific, 1994. Reprinted: 2008, 2015)

In particular, Chapters 3-5 and Sections 7.1, 7.2, 7.3 and 7.5.

2) L.C. Evans, *Partial Diļ¬erential Equations *(Amer. Math. Soc. 1998)

3) E.H. Lieb and M. Loss, *Analysis* (Amer. Math. Soc. 1997)

4) E.M. Stein and R. Shakarchi, *Fourier analysis. An introduction* (Princeton

Univ. Press 2003)