> And for that you need a band-limited signal.

Luckily, we live in a physical universe, where such mathematical oddities, like infinite bandwidth signals, cannot exist, so this isn't an actual issue. Any signal that that contains infinite bandwidths only exists because it has sampling artifacts. You would, necessarily, be attempting to reconstruct errors. There are many "tricks" around dealing with such flawed signals. But yes, you can't fully reconstruct impossible signals with FFT.