This question is more a reference request, I have never learnt a method (other than by visual inspection) to solve "convolution equations" and wish to know how they are usually dealt with, formally:
Given a real valued function $h$ and a real valued function $f$ what is the mathematical procedure to find a function $g$ s.t.
$$h(t) = \int_{- \infty}^{\infty} f(t- \tau) g(\tau) \ d \tau$$