I encountered a question I'm not entirely sure how to address.
Suppose $f,g \in L^1(\mathbb{R})$ and further that $f*g = h$.
For known $f$ and known $h$, when is the unknown solution $g$ unique?
The Fourier and inverse-Fourier transform can be taken of both sides to solve for $g,$ as $F^{-1}(F(h)/F(f))$ but without establishing uniqueness, there still may multiple or infinite solutions.