I learned about periodic functions & Fourier transforms while an undergraduate. In particular, I learned that the Fourier transform of a periodic function was a countable set of complex numbers representing the Fourier coefficients of evenly spaced impulse "functions". These coefficients represented the amplitude and phase of each of the "overtones" of the fundamental frequency of the periodic function. The only problem with this analysis is that we have taken an *uncountable* set of real numbers -- i.e., the periodic function values over one period -- and represented them by a *countable* set of complex numbers -- the Fourier coefficients. I would guess that the problem is that nearly all such periodic functions *can't* be represented by such a Fourier transform, perhaps because "almost all" such functions have too many *discontinuities*; i.e., they aren't sufficiently "smooth". If there were a countable number of discontinuities, that function could still conceivably be represented by a Fourier series. But once the number of discontinuities becomes uncountable, we're out of luck? Would this explanation be a reasonable intuition? (No wonder Cantor went mad!)