Consider a Fourier convolution
$f(x) = (g * h)(x)$,
where $g$ and $h$ are arbitrary but known functions with reasonable properties. Is there any possibility to determine the inverse function of this relation in an analytic way so that
$x = f^{-1}( f(x) ) = \dots ?$
holds true? If yes, how does this relation look like? If no, why is it not possible?
When looking for this subject all I find is the deconvolution, which describes how $h$ can be determined from knowledge about $f$ and $g$. But this is certainly not my problem at hand.