int_{-\infty}^{\infty}  F^*(\omega) G(\nu) \, e^{i(\nu-\omega) \tau+ i\omega t} d \tau \, d\omega \, d\nu   \\
=\frac{1}{2\pi}\int_{-\infty}^{\infty}\int_{-\infty}^{\infty}  F^*(\omega) G(\nu) \, e^{i\omega t}\left[ \frac{1}{2\pi}\int_{-\infty}^{\infty} e^{i(\nu-\omega) \tau} d\tau \right] d\omega \, d\nu \\
=\frac{1}{2\pi}\int_{-\infty}^{\infty} F^*(\omega) e^{i\omega t}\left[\int_{-\infty}^{\infty}  G(\nu) \, \delta(\nu-\omega)  d\nu \right] d\omega \\
  =\frac{1}{2\pi}\int_{-\infty}^{\infty} F^*(\omega) 