The limitations of the Fourier transform are intimately tied to the uncertainty principle: https://www.youtube.com/watch?v=MBnnXbOM5S4 Reality does appear to be limited by the limitations of the Fourier transform.
Not so fast. The uncertainty principle has narrow conditions outside which it doesn't apply. If you don't satisfy them, you can game the whole system and measure anything you want.
If you can "fix" what that video describes, you'd better start preparing your Fields medal acceptance speech. That you can "get around" it sometimes doesn't remove the underlying math.
