How can I solve for $f(x)$ and $g(x)$:
$$e^{-x^2}=\int_{-\infty}^\infty g(t)e^{-f(t)(x-t)^2}\text{d}t\;\;\;\;?$$ Not necessarily an elementary solution, a numerical one would also suffice. I am trying to express the Gaussian as a sum of infinitely many Gaussians at all points with their amplitudes & variances depending upon their positions. (I know that the function is not exactly a Gaussian, but that does not matter.)