Suppose there is a function $z=h(x)$. Its Fourier transform is $$h_q = \frac{1}{\sqrt{2\pi}}\int h(x)e^{iqx}dx.$$
1- What is $|h_q|^2=\frac{1}{2\pi}\int h(x)h(y)e^{iq(x-y)}dxdy$ equal to? What would happen if this were a discrete Fourier transform?
2- Furthermore, consider the following integral $$\int\int Dh_qDh_q^*e^{-\frac{1}{2}q^2|h_q|^2}.$$
We want to integrate over $|h_q|^2$ -- why is a double integration necessary, with separate integration variables $Dh_q$ and $Dh_q^*$, as $|h_q|^2$ is already a real number?