Bases for non-homogeneous polynomial ck splines on the sphere

Abstract

We investigate the use of non-homogeneous spherical polynomials for the approximation of functions defined on the sphere $2. A spherical polynomial is the restriction to $2 of a polynomial in the three coordinates x, y, z IR3. Let pd be the space of spherical polynomials with degree ≤d. We show that pd is the direct sum of Hd and Hd-1 , where Hd denotes the space of homogeneous degree-d polynomials in x,y,z. We also generalize this result to splines defined on a geodesic triangulation T of the sphere. Let Pdk[T] denote the space of all functions f from S2 to IR such that (1) the restriction of f to each triangle of T belongs to pd; and (2) the function f has order-k continuity across the edges of T. Analogously, let Hdk[T] denote the subspace of Pdk[T] consisting of those functions that are Hd within each triangle of T. We show that :Pdk[T] = Hdk[T] Hdk-1 [T]. Combined with results of Alfeld, Seamtu and Schumaker on bases of pdk[T] this decomposition provides an effective construction for a basis of Pdk[T]. There has been considerable interest recently in the use of the homogeneous spherical splines Hdk IT] as approximations for functions defined on S2. We argue that the non-homogeneous splines Pdk[T] would be a more natural choice for that purpose.