5Fourier transform

II Probability and Measure

5 Fourier transform

We now turn to the exciting topic of the Fourier transform. There are two main

questions we want to ask — when does the Fourier transform exist, and when

we can recover a function from its Fourier transform.

Of course, not only do we want to know if the Fourier transform exists. We

also want to know if it lies in some nice space, e.g. L

2

.

It turns out that when we want to prove things about Fourier transforms,

it is often helpful to “smoothen” the function by doing what is known as a

Gaussian convolution. So after defining the Fourier transform and proving some

really basic properties, we are going to investigate convolutions and Gaussians

for a bit (convolutions are also useful on their own, since they correspond to

sums of independent random variables). After that, we can go and prove the

actual important properties of the Fourier transform.