Fourier transform of $F(x)$ $$F(x)=\int^{\infty}_{-\infty} e^{-i \xi x}f(x)\ dx$$
Fourier transform of $F$:
$$\int^{\infty}_{-\infty}\left[\int^{\infty}_{-\infty} f(\xi)e^{-x^2\xi^2}\ d\xi\right]\ dx$$
Is that the way to proceed? If yes, how do I continue and if No, show me how I should proceed.

$=2 \pi [\int^{\infty}_{-\infty} F(x) e^{ix(-\xi)}$ $dx]=2\pi$ $f(-\xi)$
