# Lecture 15: Fourier Transform
###### tags: `224a 2020`
$\newcommand{\F}{\mathcal{F}}$
$\newcommand{\R}{\mathbb{R}}$
So far in the course, we studied differential operators on bounded intervals. We began with the Laplacian $-d^2/dx^2$ on $[a,b]$, which we "solved" using Fourier series, and then generalized this to Schrodinger operators of type $L=-d^2/dx^2+q(x)$ on $[a,b]$. The generalization involved developing the theory of compact operators, and then observing that the Green's function $(z-L)^{-1}$ (which is really just the resolvent at some point not in $\sigma(L)$) is compact, so $L$ inherits the (complete) set of eigenvectors of the compact case.

In the next three lectures we will generalize this to differential operators on $\R$. The analogue of Fourier series is the Fourier transform, which will give us a spectral representation of the momentum and Laplacian operators on $\R$. We will study Schrodinger operators on $\R$ by observing that $(z-L)^{-1}$ is *bounded* for a judicious choice of $z$.

In this lecture we develop the basic theory of Schwartz spaces and the Fourier transform, covering the following topics. The treatment is very similar to Teschl 7.1 so I will not (re)write notes.

Schwartz Space
Fourier Transform
Multiplication and Differentiation
Inversion Theorem, Plancharel
Extension to a unitary operator on L^2

A key fact which enabled these proofs is that the Fourier transform of a Gaussian is a Gaussian, which we will prove in the next lecture.