[math-fun] Question on Fourier transform for positive probability densities

Suppose p: R -> R is a continuous mapping with p(x) > 0 everywhere, and integral{R} p(x) dx = 1. Question: Can the set of Fourier transforms P(t) := (1/sqrt(2pi)) integral{R} p(x) exp(ixt) dx of such p's be characterized explicitly? --Dan