@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.
}
}