Fourier transforms with only real zeros

The class of even, nonnegative, finite measures ρ ρ on the real line such that for any b > 0 b > 0 the Fourier transform of exp ⁡ ( − b t 2 ) d ρ ( t ) exp ( - b{t^2})dρ (t) has only real zeros is completely determined. This result is then applied to the Riemann hypothesis.