Extracting a Function from Inside an Integral

Question

I have an imaging system that produces a signal in a pixel, in part, by responding to incident radiance $L(\lambda)$ that passes through a filter with transmission $T(\lambda)$. Lumping all the other parameters under a generic constant I can write the signal: $S = k\cdot\int L(\lambda)\ast T(\lambda)d\lambda\,$. I'm trying to find a way to determine, or even estimate $L(\lambda)$. The system is such that I know $S, k$, and $T(\lambda)$. Is there a way to recover the incident radiance? Note, I am not necessarily looking for a closed form solution, approximation is fine.