  Subspaces obtained by the orthogonal projection of locally supported
square-integrable vector fields onto the Hardy spaces $H_+(\mathbb{S})$ and
$H_-(\mathbb{S})$, respectively, play a role in various inverse potential field
problems since they characterize the uniquely recoverable components of the
underlying sources. Here, we consider approximation in these subspaces by a
particular set of spherical basis functions. Error bounds are provided along
with further considerations on norm-minimizing vector fields that satisfy the
underlying localization constraint. The new aspect here is that the used
spherical basis functions are themselves members of the subspaces under
consideration.
