What Is the Fourier Transform?

I’ve dabbled in explaining the FT some, and I think there’s an important trait of that video that needs to be highlighted: it’s a superb demonstration of what the Fourier transform does, mechanically, but it does not at all try to explain why it works in the places we usually apply it or how (having, admittedly, a thoroughly anachronistic mathematical background) you could have invented it.

To be extra clear: it’s a very good video and you should watch it if you don’t have a feel for the Fourier transform. I’m just trying to proactively instil a tiny bit of dissatisfaction with what you will know at the end of it, so that you will then go looking for more.