Discrete Fourier transform of complex time series
Proposed answer to my own question:

We have that:

$$F_m = \frac{1}{2}(H_m + H^*_m)$$

and

$$G_m = \frac{1}{2i}(H_m - H^*_m)$$

where $H^*_m$ represents the complex conjugate of $H_m$.