@techreport{TR-IC-00-13,
number = {IC-00-13},
author = {Anamaria Gomide and Jorge Stolfi},
title = {Non-Homogeneous Polynomial {$C_k$} Splines on the Sphere {$S^n$}},
month = {July},
year = {2000},
institution = {Institute of Computing, University of Campinas},
note = {In English, 13 pages.
\par\selectlanguage{english}\textbf{Abstract}
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 $d\geq 3k + 3$.
}
}