Bases for non-homogeneous polynomial C k splines on the sphere

Abstract

We investigate the use of non-homogeneous spherical polynomials for the approximation of functions defined on the sphere S2. A spherical polynomial is the restriction to S2 of a polynomial in the three coordinates x,y,z of ℝ3. Let P d be the space of spherical polynomials with degree ≤ d. We show that P d is the direct sum of P 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 defined on a geodesic triangulation T of the sphere. Let P [T] denote the space of all functions f from S2 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 [T] denote the subspace of P [T] consisting of those functions that are H d within each triangle of T. We show that P [T]=H [T]⊕H [T]. Combined with results of Alfeld, Neamtu and Schumaker on bases of H [T] this decomposition provides an effective construction for a basis of P [T]. There has been considerable interest recently in the use of the homogeneous spherical splines H [T] as approximations for functions defined on S2. We argue that the non-homogeneous splines P [T] would be a more natural choice for that purpose.