So it's probably not at all helpful, but I'm preparing to stuff myself with turkey tomorrow and it's been long enough that I don't remember FTs and FFTs in detail, but yeah. A FT is unique for a particular function. A FFT is unique for a given function and frequency division thing. I don't remember about DFTs either and yeah, there's allsorts of discretization issues there. Now, to melt your brain, here's a site that can eventually answer all sort questions:
Mathworld: Fourier Transform