Inverses of Matrn covariances on grids

We conduct a study of the aliased spectral densities of Matrn covariance functions on a regular grid of points, elucidating the properties of a popular approximation based on stochastic partial differential equations. While other researchers have shown that this approximation can work well for the covariance function, we find that it assigns too much power at high frequencies and does not provide increasingly accurate approximations to the inverse as the grid spacing goes to zero, except in the one-dimensional exponential covariance case.