I read a paper that assumes a prior distribution on the Fourier components of a 3D model--specifically that the components are independent and normally distributed: $$ p(\Theta) = \prod_{l=1}^L \frac{1}{2\pi\sigma_l^2} \exp\left(\frac{|V_l|^2}{-2\sigma_l^2}\right) $$ Where the $V_l$ are 3D Fourier components and $\sigma_l$ are hyper-parameters. As you might expect, the result of using this prior is to "regularize" the model, effectively making it smooth.
My question is, what does this prior indicate about the real spatial process?
What I have so far
The only thing I can think of is Bochner's Theorem, which (according to Gaussian Processes in Machine Learning, Rasmussen, page 82) says that in some cases, the spectral density and spatial covariance are Fourier duals of each other. Computing the covariance would be great because then I can write down the spatial process corresponding to the above distribution.
The problems are: 1) I think that only applies in stationary cases and I don't think the distribution above is stationary and 2) I don't know how to go from a distribution on component values to a spectral density.