Request for someone to make an intuitive explanation of why the Fourier Transform is (almost) it's own inverse. I know the math proof from taking analysis, but the formula is too pretty and symmetrical for the explanation to be so technical.
Same answer for both. It's an orthogonal transform, aka a change of basis. You're conceptually rotating the function/series to an equivalent one that's orthogonal to the original. The magnitude/energy hasn't changed (Parseval's theorem is a more succinct definition). And to perform the inverse transform you need to conceptually rotate it back to the original, which should mirror the original transform very nicely.
If it didn't preserve energy then there wouldn't necessarily be an inverse transform, since that implies information was lost.
I really appreciate this reply, since this is something I've always been curious about. If you have time, I would really appreciate it if you could elaborate on this point (maybe with some equations), but you've already given me a ton to think about, thank you!
Also, is the new basis orthogonal to the original, or just another orthonormal basis? I don't see why it would be orthogonal to the original.
It's an orthonormal basis. See my comment here[0] for more on why the forward and inverse transforms look similar (I've written e_w to mean e^iwt, but we want to think of it as a vector). The forward direction is doing a dot product with an exponential with a specific frequency to get the transform at that frequency (i.e. the function's projection/component at that frequency), which is a sum over the entire time basis. The inverse transform sums over all components/projections of the function at each frequency to rebuild the function. This is how you do an expansion in terms of orthogonal projections in any dimension.
This also explains why the forward transform has a minus and the inverse doesn't: a complex dot product sums over complexConjugate(v_i)*w_i, while the inverse/reconstruction just sums over the basis exponentials scaled by the Fourier coefficient for that frequency.
You can also prove that the sum over all imaginary exponentials is Dirac delta[1], so you could think of the inverse transform as being a dot product with delta to get the function at a specific time, and that dot product is a sum over the frequency basis.
Same for why it preserves L2 norm.