A local simplex spline basis for C3 quartic splines on arbitrary triangulations

We deal with the problem of constructing, representing, and manipulating C3 quartic splines on a given arbitrary triangulation r, where every triangle of r is equipped with the quartic Wang-Shi macro-structure. The resulting C3 quartic spline space has a stable dimension and any function in the space can be locally built via Hermite interpolation on each of the macro-triangles separately, without any geometrical restriction on T. We provide a simplex spline basis for the space of C3 quartics defined on a single macro-triangle which behaves like a B-spline basis within the triangle and like a Bernstein basis for imposing smoothness across the edges of the triangle. The basis functions form a nonnegative partition of unity, inherit recurrence relations and differentiation formulas from the simplex spline construction, and enjoy a Marsden-like identity.