Given $F(t) = \mathcal{F}\{f(x)\}$ is the Fourier transform of $f$, how can one express $\mathcal{F}\left\{\dfrac{1}{f(x)}\right\}$ in terms of $F(t)$?
EDIT: To be more concrete, I want to compute $\mathcal{F}\left\{\dfrac{1}{f(x)}\dfrac{dg(x)}{dx}\right\}$. So far I have $\mathcal{F}\left\{\dfrac{1}{f(x)}\dfrac{dg(x)}{dx}\right\} = \mathcal{F}\left\{\dfrac{1}{f(x)}\right\}\star\mathcal{F}\left\{\dfrac{dg(x)}{dx}\right\}$, where $\star$ denotes convolution. The point of this computation is to use the resulting expression in a numerical method (spectral Galerkin), so $f$ and $g$ do not have a specific analytical form.