  Sampling theory is a discipline in communications engineering involved with
the exact reconstruction of continuous signals from discrete sets of sample
points. From a physics perspective, this is interesting in relation to the
question of whether spacetime is continuous or discrete at the Planck scale,
since in sampling theory we have functions which can be viewed as equivalently
residing on a continuous or discrete space. Further, it is possible to
formulate analogues of sampling which yield discreteness without disturbing
underlying spacetime symmetries. In particular, there is a proposal for how
this can be adapted for Minkowski spacetime. Here we will provide a detailed
examination of the extension of sampling theory to this context. We will also
discuss generally how spacetime symmetries manifest themselves in sampling
theory, which at the surface seems in conflict with the fact that the
discreteness of the sampling is not manifestly covariant. Specifically, we will
show how the symmetry of a function space with a sampling property is
equivalent to the existence of a family of possible sampling lattices related
by the symmetry transformations.
