On the positivity of Fourier transform of the stretched Gaußian function

The stretched Gaußian function $f(\mathbf{x})=\exp (-\mathrm{\parallel }\mathbf{x}{\mathrm{\parallel }}^s)$, as a real function defined on ${\mathbb{R}}^d$, has found numerous applications in mathematics and physics. For instance, to describe results from spectroscopy or inelastic scattering, the Fourier transform of the stretched Gaußian function is needed. For $s\in (0,2]$, we prove that the Fourier transform of $f(\mathbf{x})=\exp (-\mathrm{\parallel }\mathbf{x}{\mathrm{\parallel }}^s)$ is everywhere positive on ${\mathbb{R}}^d$.