Bases for non-homogeneous polynomial Ck splines on the sphere
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 k d [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 k d [T] denote the subspace of P k d [T] consisting of those functions that are H d within each triangle of T. We show that P k d [T]=H k d [T]⊕H k d−1 [T]. Combined with results of Alfeld, Neamtu and Schumaker on bases of H k d [T] this decomposition provides an effective construction for a basis of P k d [T].
There has been considerable interest recently in the use of the homogeneous spherical splines H k d [T] as approximations for functions defined on S2. We argue that the non-homogeneous splines P k d [T] would be a more natural choice for that purpose.
KeywordsSpherical Function Spline Space Spherical Triangle Piecewise Polynomial Function Spherical Spline
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