# Average of product of Fourier transform of two signals

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?