Intuitive explication of Fourier Transformation
The author wrote "This formula, as anyone can see, makes no sense at all. I decided that Fourier must have been speaking to aliens, because if you gave me all the time and paper in the world, I would not have been able to come up with that." That sounds like a predictable symptom of trying to understand Fourier analysis while avoiding linear algebra. And that seems like unnecessary masochism, because basic linear algebra is very useful and pretty easy. And once you have it, (elementary) Fourier analysis becomes trivial to understand as a change of basis by recognizing the supposed "no sense at all" formula as a perfectly sensible change of basis to a basis of sinusoidal functions.<p>Then you just need to understand that the sines and cosines are a complete basis. So think about the sines and cosines for a while until you can say "yeah, they're orthogonal, and I can believe they're a complete basis for the kind of functions under consideration." Then to promote this from "I can believe" to "obviously," for the discrete FT (the orthogonality and) counting/dimensionality arguments suffice, and for the continuous FT you can look at Gaussians, say "obviously Gaussians are a complete basis for the kinds of functions under consideration" and then do the easy integrals to show that any Gaussian can be expressed as a linear combination of sines and cosines.<p>(This assumes you're interested in transforming reasonably smooth things like wavefunctions in chemistry, as opposed to trying to see how far you can push Fourier analysis into the netherworld of bizarre jagged twisted functions shown to exist by invoking the Axiom of Choice. If you want to do that, feel free to take a course from Terence Tao studying theorems whose prerequisites involve concepts like "countable.")