Non-Homogeneous Polynomial $C_k$ Splines on the Sphere $S^n$

A {\em homogeneous spherical polynomial} (HSP) is the restriction to the sphere $S^{n-1}$ of a homogeneous polynomial on the cartesian coordinates $x_1, x_2,\dots,x_n$ of $R^n$. A {\em homogeneous spherical spline} is a function that is an HSP within each element of a geodesic triangulation of $S^{n-1}$. \par There has been considerable interest recently in the use of such splines for approximation of functions defined on the sphere. In this paper we introduce the {\em general} (non-homogeneous) {\em spherical splines} and argue that they are a more natural approximating spaces for spherical functions than the homogeneous ones. It turns out that the space of general spherical polynomials of degree $d$ is the direct sum of the homogeneous spherical polynomials of degrees $d$ and $d-1$. We then generalize this decomposition result to polynomial splines defined on a geodesic triangulation (spherical simplicial decomposition) $T$ of the sphere $S^{n-1}$, of arbitrary degree $d$ and continuity order $k$. \par For the particular case $n=3$, the homogeneous spline spaces were extensively studied by Alfeld, Neamtu, and Schumaker, who showed how to construct explicit local bases when $d\geq 3k + 2$. Combining their construction with our decomposition theorem, we obtain an explicit construction for a local basis of the general polynomial splines when $d\geq 3k + 3$.

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