What if you take a function that has bounded support in the time domain, and then turn it into a periodic function? Might the resulting function have bounded support in the frequency domain even though the original function did not? I suppose doing this would force the Fourier transform to have discrete support? But under what conditions would it have bounded support?…
I guess technically a low-pass filter applied to a signal with finite support in the time domain, would result in a function which has infinite support in the time domain.
I suppose sinc(f t + c) doesn’t have bounded support, and it is unsurprising that a non-trivial linear combination of finitely many terms of this form would also not have finite support.
Still, such a linear combination could decay rather quickly, I imagine. (Idk if asymptotically faster than (1/t) , but (1/(f t)) is still pretty fast I think, for large f.)
Soon enough the decay should be enough that the amplitude should be smaller than the smallest that the speaker hardware is capable of producing, I suppose.