An FFT transforms a function to another function so that nothing is lost.
But what if we are okay with losing considerable portions of our function?
If we have a weight function, saying how important a fit shall be.
$$\min_{x}\|W(F^{-1}x-d)\|^2_2$$ with some extra regularization.
We can see that if $W = I$, we get that the unregularized solution is given by $$F^{-1}x = d \Leftrightarrow x = Fd$$
, the straight-forward Fourier transform.
But what if W is not the identity, but perhaps a binary function $\{0,1\}$ or lies in $[0,1]$.