# Lecture 16: Finish Fourier Transform, Uncertainty Principle
###### tags: `224a 2020`

$\newcommand{\F}{\mathcal{F}}$

We showed that
$$\F e^{x^2\sigma^2/2} = \frac{1}{\sigma}e^{-x^2/2\sigma^2},$$
i.e., that the Fourier transform of a Gaussian is (up to the leading $1/\sigma$) a Gaussian of inverse variance. The proof involved completing the square in the Fourier integral and using a contour shifting argument. This was a key ingredient in the proof of the Fourier inversion formula.

This is a special case of a more general phenomenon that the Fourier transform maps localized functions to delocalized ones and vice versa.

We proved two manifestations of this: Heisenberg's uncertainty principle in one dimension, and the fact that both $f$ and $\hat{f}$ cannot be compactly supported. Details are in Teschl section 7.1

Many variants and higher dimensional generalizations are discussed in http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.475.5324&rep=rep1&type=pdf