= Solution
The <Fourier coefficient> of the velocity-integrated density is its mixed transform at zero velocity frequency. Thus
$$
\widehat\rho_t(k)=\widehat f(t,k,0)=\widehat f_0(k,kt).
$$
The constant $\rho_\infty$ contributes only to the zero <Fourier mode>, where it equals $\widehat f_0(0,0)$. Therefore
$$
\boxed{\widehat r_t(k)=\widehat f_0(k,kt)-\mathbf1_{\{k=0\}}\widehat f_0(0,0)
=\begin{cases}\widehat f_0(k,kt),&k\ne0,\\0,&k=0.\end{cases}}
$$
It is important to remove the zero mode, which is conserved rather than mixed away.
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