@techreport{TR-IC-00-19, number = {IC-00-19}, author = {Anamaria Gomide and Jorge Stolfi}, title = {Non-Homogeneous Spline Bases for Approximation on the Sphere}, month = {December}, year = {2000}, institution = {Institute of Computing, University of Campinas}, note = {In English, 8 pages. \par\selectlanguage{english}\textbf{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. } }