Given functions $g$ and $u$, what can we say about solutions of the equation: $$ f \star g = u $$ where $\star$ is the convolution and $f$ is the unknown function. When does it exist? Do we know an explicit solution? All solutions?
I am not specifying smoothness or domain here as I am also interested in answer in specific cases on $g$ and $u$.
Let me re-state the problem. I am assuming the question is:
Given
$\int_{R^d} f(\tau)g(t - \tau)d\tau = u(t)$,
assuming $g(t)$ and $u(t)$ are known, how can we determine $f(t)$? This is an integral equation of the first kind, see
https://en.wikipedia.org/wiki/Fredholm_integral_equation
The response in the comments above reflects the solution given in the above link. What I don't know is under what conditions the inverse Fourier transform will work. Probably only when the Fourier transform of $f(t)$ and $g(t)$ both exist.