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.

翻译：通过对局部支撑的平方可积向量场在Hardy空间$H_+(\mathbb{S})$和$H_-(\mathbb{S})$上分别进行正交投影所得到的子空间，因其刻画了底层源场的唯一可恢复分量，在各类反位势问题中具有重要作用。本文研究利用特定球面基函数集在这些子空间中的逼近问题。我们给出了误差界，并进一步考虑了满足给定局部化约束的范数最小化向量场。本文的新颖之处在于所使用的球面基函数自身即为所考察子空间的元素。