  The high efficiency of a recently proposed method for computing with Gaussian
processes relies on expanding a (translationally invariant) covariance kernel
into complex exponentials, with frequencies lying on a Cartesian equispaced
grid. Here we provide rigorous error bounds for this approximation for two
popular kernels -- Mat\'ern and squared exponential -- in terms of the grid
spacing and size. The kernel error bounds are uniform over a hypercube centered
at the origin. Our tools include a split into aliasing and truncation errors,
and bounds on sums of Gaussians or modified Bessel functions over various
lattices. For the Mat\'ern case, motivated by numerical study, we conjecture a
stronger Frobenius-norm bound on the covariance matrix error for
randomly-distributed data points. Lastly, we prove bounds on, and study
numerically, the ill-conditioning of the linear systems arising in such
regression problems.
