Deconvolution Question on Article "Deriving Intrinsic Images from Image Sequences" by Yair Weiss
I didn't have time to fully read the article yet I'll try to answer according to your question.

If you take the first equation and convolove it on the left by $ {f}_{i} $ and use the second identity you can see this specific $ g $ holds the equation.

Regarding how to get this $ g $, I think he uses "Deconvolution" in the Fourier domain.