If f(ω) is square integrable over the real ω-axis, then any one of the following implies the other two: 1. The Fourier transform F(t) = ℱ_ω[f(ω)](t) is 0 for t<0. 2. Replacing ω by z congruent x + i y, the function f(z) is analytic in the complex plane z for y>0 and approaches f(x) almost everywhere as y->0. Furthermore, integral_(-∞)^∞ ( left bracketing bar f(x + i y) right bracketing bar )^2 d x0 (i.e., the integral is bounded). 3. The real and imaginary parts of F(z) are Hilbert transforms of each other (Bracewell 1999, Problem 8, p. 273).

Fourier transform | Hilbert transform