  Given a point configuration A, we uncover a connection between
polynomial-reproducing spline spaces over subsets of conv(A) and fine zonotopal
tilings of the zonotope Z(V) associated to the corresponding vector
configuration. This link directly generalizes a known result on Delaunay
configurations and naturally encompasses, due to its combinatorial character,
the case of repeated and affinely dependent points in A. We prove the existence
of a general iterative construction process for such spaces. Finally, we turn
our attention to regular fine zonotopal tilings, specializing our previous
results and exploiting the adjacency graph of the tiling to propose a set of
practical algorithms for the construction and evaluation of the associated
spline functions.
