@techreport{TR-IC-97-10,
number = {IC-97-10},
author = {Gomide, Anamaria and Stolfi, Jorge},
title = {Bases para Splines Polinomiais Não Homogêneos {${\bf C}_k$} na Esfera},
month = {August},
year = {1997},
institution = {Institute of Computing, University of Campinas},
note = {In Portuguese, 9 pages.
\par\selectlanguage{english}\textbf{Abstract}
We investigate the use of non-homogeneous spherical polynomials
for the approximation of functions defined on the sphere
${\bf S}^2$. A {\em spherical polynomial} is the restriction to
${\bf S}^2$ of a polynomial in the three coordinates $x,y,z$ of
${\bf R}^3$. Let ${\cal P}^d$ be the space of spherical polynomials
with degree ${}\leq d$. We show that ${\cal P}^d$ is the direct sum of
${\cal H}^d$ and ${\cal H}^{d-1}$, where ${\cal H}^d$ denotes the space of {\em
homogeneous} degree-$d$ polynomials in $x,y,z$.
\par
We also generalize this result to splines defined on a geodesic
triangulation $T$ of the sphere. Let ${\cal P}^d_k[T]$ denote the
space of all functions $f$ from ${\bf S}^2$ to ${\bf R}$ such
that (1) the restriction of $f$ to each triangle of $T$ belongs
to ${\cal P}^d$; and (2) the function $f$ has order-$k$
continuity across the edges of $T$. Analogously, let ${\cal
H}^{d}_k[T]$ denote the subspace of ${\cal P}^d_k[T]$ consisting
of those functions that are ${\cal H}^d$ within each triangle of
$T$. We show that ${\cal P}^d_k[T] = {\cal H}^{d}_k[T]\oplus
{\cal H}^{d-1}_k[T]$. Combined with results of Alfeld, Neamtu
and Schumaker on bases of ${\cal H}^{d}_k[T]$ this decomposition
provides on effective characterization of the bases of ${\cal
P}^d_k[T]$.
\par
There has been considerable interest recently in the use of the
homogeneous spherical splines ${\cal H}^{d}_k[T]$ as approximations for
functions defined on ${\bf S}^2$. We argue that the non-homogeneous
splines ${\cal P}^d_k[T]$ would be a more natural choice for that purpose.
}
}