Polynomial-reproducing spline spaces from fine zonotopal tilings

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