What is the Fourier transform of $|\partial f(x,t)/\partial x|^{2}$?
To put the question in some context, I was trying to write the KPZ equation in Fourier space. The KPZ equation is given by $\partial h(x,t)/\partial t=\nabla^{2}h(x,t)+|\nabla h|^{2}+\eta(x,t)$. While the Fourier transform of linear parts is done, I was stuck at deriving the Fourier transform of the non-linear term. I referred to the literature on this and the Fourier transform was given to be
$-i\omega h(k,w)=-k^{2}h(k,w)-\int\int \frac{d\Omega \: dq}{4\pi^{2}}\; q(k-q)\;h(q,\Omega)\;h(k-q,\omega-\Omega)\; +\eta(k,\omega)$
As I have said, the linear parts can be obtained by multiplying with $e^{i(kx-\omega t)}$ throughout and then integrating over $x$ and $t$. I am unable to figure out how such an operation would generate the above given non-linear term.
Use the convolution theorem for Fourier transform!