Say you have two functions, F(x,y), and G(x,y), and you want to expand them in finite fourier series. Let their coefficients be designated as F_ij and G_ij. When you multiply the two functions, you get X=FG, and this should also have its own fourier series, call its components X_mn. What is the relation between F_ij, G_ij, and X_mn? I was hoping you had something like X_ij = F_ij G_ij, but I've been looking at this for a little while and it seems you don't have any nice relation like that.