If is square integrable over the real -axis, then any one of the following implies the other two:
1. The Fourier transform is 0 for .
2. Replacing by , the function is analytic in the complex plane for and approaches almost everywhere as . Furthermore, for some number and (i.e., the integral is bounded).
3. The real and imaginary parts of are Hilbert transforms of each other
(Bracewell 1999, Problem 8, p. 273).