Dimension and local bases of homogeneous spline spaces

Recently, we have introduced spaces of splines defined on triangulations lying on the sphere or on sphere-like surfaces. These spaces arose out of a new kind of Bernstein–Bezier theory on such surfaces. The purpose of this paper is to contribute to the development of a constructive theory for such spline spaces analogous to the well-known theory of polynomial splines on planar triangulations. Rather than working with splines on sphere-like surfaces directly, we instead investigate more general spaces of homogeneous splines in $\mathbb{R}^3 $. In particular, we present formulas for the dimensions of such spline spaces, and construct locally supported bases for them.