@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. } }