Demystifying the Fourier Magic

I am a mathematician (not that it is relevant) but honestly, "demystifying" seems a bit exaggerated for a piece of commentary which simply makes it all more complicated than it is.

If the aim is to 'abstract' it, then call it 'abstraction' but demystifying is rather far-fetched.

Yes, the Fourier transform is no more than the expression of the elements of a Hilbert space in a complete basis: get over it and move on? Is it clear? You can generalize it even more.

But honestly, as Abhyankar has been quoted saying, the real question is... "What is a polynomial?" That is the really difficult question to answer.

Sorry for the rant but titles should really be at least honest (I agree they have to be attractive but...).