The assignments (and solutions) are on Ruth’s MA22S3 page (mirror) and are worth 20% of your grade.

Conor’s notes from 2012 and his 231.II notes might be helpful. Darran also has some excellent notes on Fourier analysis (§§1,2).

I tutored this course before and my teaching page for it has a couple of nice proofs. (The course used to be in x and k for position and momentum, but now we use t and ω.)

Here’s a derivation of the Fourier series from a Hilbert space perspective, which explains why there is a factor of a half in the a₀ term.

See my teaching pages from 2014/15 and 2015/16 for my solutions to old tutorials.

I also wrote a simple implementation of the discrete Fourier transformation and fast Fourier transformation (radix-2 Cooley–Tukey).

For fun applications of Fourier analysis, see SWIFFT and The Deviation of China Map as a Regression Problem.

This is the best tweet about Fourier analysis.
Update: this is the second best tweet.