We introduce an extension of cone splines and box splines to fractional
and complex orders. These new families of multivariate splines are
defined in the Fourier domain along certain s-directional meshes and
include as special cases the three-directional box splines and hex splines
considered earlier in the literature. These cone and hex splines of
fractional and complex order also generalize the univariate fractional and
complex B-splines. Explicit time domain representations are derived for
these splines on 3-directional meshes and some properties of these two
multivariate spline families such as recurrence, decay and refinement
presented. It is also shown that a bivariate hex spline and its integer lattice
translates form a Riesz basis of its linear span.
