I have two signals which depend on x and z, $a(x,z)$ and $b(x,z)$. Their Fourier transform along both directions is denoted as $A(k_x,k_z)$ and $B(k_x,k_z)$, respectively. I would like to compute the average on $x$ of the product of the two, this is
$$P=\langle AB \rangle_x$$
I feel the solution is the first component ($k_x=0$) of the product but I don't know how to prove it mathematically. Could you help me out?