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 ^d _k [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 ^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 effective 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 defined on S2. We argue that the non-homogeneous splines P ^d _k [T] would be a more natural choice for that purpose.

This research was partly funded by CNPq grant 301016/92-5