The uncertainty principle (for Fourier transform pairs) follows immediately from the scaling theorem (§B.4). It may be loosely stated as
Time Duration Frequency Bandwidth cwhere is some constant determined by the precise definitions of ``duration'' in the time domain and ``bandwidth'' in the frequency domain.
If duration and bandwidth are defined as the ``nonzero interval,'' then we obtain , which is not useful. This conclusion follows immediately from the definition of the Fourier transform and its inverse (§2.2).