Consider a 3D scalar function $u(x,y,z)$, in a numerical implementation represented by discrete values $u_{i,j,k}$ on a Cartesian grid. If this scalar field is filtered by a 1D filter, either spatially as a convolution $$ \tilde{u}(x,y,z) = \int_{-\infty}^{+\infty} u(x') G(x-x')\ \mathrm{d}x', $$ or spectrally in the Fourier space $$ \hat{\tilde{u}}(k_x,y,z) = \hat{u}(k_x,y,z)\ \hat{G}(k_x). $$
The 1D power spectrum of the filtered field along the $x$ direction (and averaged over all $y$ and $z$) will then just be $$ S_{\tilde{u}}(k_x) = \|\hat{G}(k_x)\|^2 S_{u}(k_x). $$
When filtering the field with a 3D filter, the spectrum will change in a more complicated way. How can I derive the shape of the spectrum (1D or eventually 3D) after the application of a 3D filter $\hat{G}(k_x,k_y,k_z)$? A pointer to a relevant literature is sufficient.
I am mostly interested in a filter that is spherically symmetric ($\hat{G}(\|k\|)$) or separable ($\hat{G}(k_x,k_y,k_z) = \hat{G}_x(k_x) \hat{G}_y(k_y) \hat{G}_z(k_z)$) or both (gaussian).
I have some numerical examples for gridded values of a velocity component from a turbulent flow. The graph shows the ratio $S_{\tilde{u}}(k_x)/S_{u}(k_x)$.
