Consider the following known matrices, A, B, C and these unknown matrices X,Y, all of which comprise values in $\Bbb R$. Also consider $F(x)$ as the *Fast Fourier Transform function* (the discrete version) and $F_{inverse}(x)$ as the Inverse Fourier Transform function (which you might need for solving the equation below).
Can we solve for X and Y in this equation,
$$F(A) = F(B)*F(X)+F(C)*F(Y)$$
where ' $*$ ' is the element-wise multiplication.
Sure you can. First isolate $F(X)$: $$F(A) - F(C) * F(Y) = F(B) * F(X)$$ $$( 1.0 \text{./} F(B) ) * ( F(A) - F(C) * F(Y) ) = F(X)$$ where "./" represents elementwise division. Then apply the inverse transformation: $$F_{\text{inverse}}\left( ( 1.0 \text{./} F(B) ) * ( F(A) - F(C) * F(Y) ) \right) = X$$