Assuming $X$ and $Y$ are two Gaussians with parameters of $\mu_X,\Sigma_X$ and $\mu_Y,\Sigma_Y$ then for their convolution we know that (reference) :
$Z=X*Y$ is also a Gaussian with parameters of $\mu_Z=\mu_X+\mu_Y$ and $\Sigma_Z=\Sigma_X+\Sigma_Y$
what can we say about deconvolution of $Z$ and $Y$? Is the deconvolution of two Gaussians, also a Gaussian? Are the parameters of $Z$ can be similarly computed?
($F=Z/ Y => \mu_F=\mu_Z-\mu_Y$ and $\Sigma_F=\Sigma_Z-\Sigma_Y$ ?)
Best Answer
I think so.
The Fourier transform of a Gaussian is a Gaussian, so we can go to the f domain.
Furthermore, dividing a Gaussian by a Gaussian (deconvolution in the f domain) also yields a Gaussian, so we now have a Gaussian f domain quotient.
The inverse Fourier of a Gaussian is also a Gaussian.
Voila (I won't say QED as this is far from formal!!!)
Regarding the parameters I would say yes as well.