On the positivity of Fourier transform of the stretched Gau{}ian function

The stretched Gau{}ian function f(x)=exp (-\|x\|ˢ), as a real function defined on Rᵈ, has found numerous applications in mathematics and physics. For instance, to describe results from spectroscopy or inelastic scattering, the Fourier transform of the stretched Gau{}ian function is needed. For s ∈(0,2], we prove that the Fourier transform of f(x)=exp (-\|x\|ˢ) is everywhere positive on Rᵈ.