Homogeneous simplex splines

Homogeneous simplex splines, also known as cone splines or multivariate truncated power functions, are discussed from a perspective of homogeneous divided differences and polar forms. This makes it possible to derive the basic properties of these splines in a simple and economic way. In addition, a construction of spaces of homogeneous simplex splines is considered, which in the nonhomogeneous setting is due to Dahmen, Micchelli, and Seidel. A proof for this construction is presented, based on knot insertion. Restricting the homogeneous splines to a sphere gives rise to spaces of spherical simplex splines.