  Splines can be constructed by convolving the indicator function of the
Voronoi cell of a lattice. This paper presents simple criteria that imply that
only a small subset of such spline families can be refined: essentially the
well-known box splines and tensor-product splines. Among the many non-refinable
constructions are hex-splines and their generalization to non-Cartesian
lattices. An example shows how non-refinable splines can exhibit increased
approximation error upon refinement of the lattice.
