@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.
  }
}