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
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.