We investigate the use of non-homogeneous spherical polynomials for the approximation of functions defined on the sphere S
2. A
spherical polynomial is the restriction to S
2 of a polynomial in the three coordinates
x,y,z of ℝ
3. Let
P
d
be the space of spherical polynomials with degree ≤
d. We show that
P
d
is the direct sum of
P
d
and
H
d−1, where
H
d
denotes the space of
homogeneous degree-
d polynomials in
x,y,z.
We also generalize this result to splines defined on a geodesic triangulation T of the sphere. Let P
k
d
[T] denote the space of all functions f from S2 to ℝ such that (1) the restriction of f to each triangle of T belongs to P
d
; and (2) the function f has order-k continuity across the edges of T. Analogously, let H
k
d
[T] denote the subspace of P
k
d
[T] consisting of those functions that are H
d
within each triangle of T. We show that P
k
d
[T]=H
k
d
[T]⊕H
k
d−1
[T]. Combined with results of Alfeld, Neamtu and Schumaker on bases of H
k
d
[T] this decomposition provides an effective construction for a basis of P
k
d
[T].
There has been considerable interest recently in the use of the homogeneous spherical splines H
k
d
[T] as approximations for functions defined on S2. We argue that the non-homogeneous splines P
k
d
[T] would be a more natural choice for that purpose.
This research was partly funded by CNPq grant 301016/92-5