@techreport{gom-sto-00-appr-tr,
  author = {Anamaria Gomide and Jorge Stolfi},
  title = {Non-Homogeneous Spline Bases for Approximation on the Sphere},
  institution = {Institute of Computing, Univ. of Campinas},
  number = {IC-00-19},
  pages = {8},
  year = 2000,
  month = dec,
  abstract = {A {\em spherical polynomial} is the restriction to the sphere ${\bf S}^2$ of a polynomial in the three coordinates $x,y,z$ of ${\bf R}^3$. Let $T$ be an arbitrary triangulation on the sphere, and let ${\cal P}^d_k[T]/{\bf S}^2$ (resp ${\cal H}^d_k[T]/{\bf S}^2$) be the space of all $C_k$-continuous functions $f$ from ${\bf S}^2$ to ${\bf R}$ such that the restriction of $f$ to each triangle of $T$ is a spherical polynomial (resp. homogeneous). These are the {\em spherical polynomial} (resp {\em homogeneous}) {\em splines} of degree ${}\leq d$ (resp. exactly $d$) and continuity $k$. \par In a previous paper, we have shown that ${\bf P}^d_k[T]/{\bf S}^2 = {\cal H}^{d}_k[T]/{\bf S}^2 \oplus {\cal H}^{d-1}_k[T]/{\bf S}^2$. Alfeld, Neamtu and Schumaker have recently constructed explicit bases for the spaces ${\cal H}^{d}_k[T]/{\bf S}^2$. Combining these two results, we obtain explicit constructions for bases of ${\bf P}^d_k[T]/{\bf S}^2$. \par We believe that the general spline spaces ${\bf P}^d_k[T]/{\bf S}^2$ provide better approximations than the homogeneous spaces ${\cal H}^d_k[T]/{\bf S}^2$ when used over the relatively large regions (radius $10^{-1}$ to $10^{-2}$) that are likely to occur in pratice. In this paper we report numerical experiments in least squares approximation which offer some evidence for this claim.}
}