Bases for Non-homogeneous Polynomial Ck Splines on the Sphere

We investigate the use of non-homogeneous spherical poly-nomials for the approximation of functions deened on the sphere S 2. A spherical polynomial is the restriction to S 2 of a polynomial in the three coordinates x; y; z of R 3. Let P d be the space of spherical polynomials with degree d. We show that P d is the direct sum of H d and H d?1 , where H d denotes the space of homogeneous degree-d polynomials in x; y; z. We also generalize this result to splines deened on a geodesic triangula-tion T of the sphere. Let P d k T ] denote the space of all functions f from S 2 to R such that (1) the restriction of f to each triangle of T belongs to P d ; and (2) the function f has order-k continuity across the edges of T. Analogously, let H d k T ] denote the subspace of P d k T ] consisting of those functions that are H d within each triangle of T. We show that P d k T ] = H d k T ]H d?1 k T ]. Combined with results of Alfeld, Neamtu and Schumaker on bases of H d k T ] this decomposition provides an eeective construction for a basis of P d k T ]. There has been considerable interest recently in the use of the homogeneous spherical splines H d k T ] as approximations for functions deened on S 2. We argue that the non-homogeneous splines P d k T ] would be a more natural choice for that purpose.