We revisit the moving least squares (MLS) approximation scheme on the sphere $\mathbb S^{d-1} \subset \mathbb R^d$, where $d>1$. It is well known that using the spherical harmonics up to degree $L \in \mathbb N$ as ansatz space yields for functions in $\mathcal C^{L+1}(\mathbb S^{d-1})$ the approximation order $\mathcal O \left( h^{L+1} \right)$, where $h$ denotes the fill distance of the sampling nodes. In this paper we show that the dimension of the ansatz space can be almost halved, by including only spherical harmonics of even or odd degree up to $L$, while preserving the same order of approximation. Numerical experiments indicate that using the reduced ansatz space is essential to ensure the numerical stability of the MLS approximation scheme as $h \to 0$. Finally, we compare our approach with an MLS approximation scheme that uses polynomials on the tangent space as ansatz space.

翻译：本文重新审视了球面 $\mathbb S^{d-1} \subset \mathbb R^d$（其中 $d>1$）上的移动最小二乘（MLS）逼近方案。众所周知，采用次数不超过 $L \in \mathbb N$ 的球谐函数作为试验函数空间，对于 $\mathcal C^{L+1}(\mathbb S^{d-1})$ 类函数可获得 $\mathcal O \left( h^{L+1} \right)$ 的逼近阶，其中 $h$ 表示采样节点的填充距离。本文证明，通过仅包含偶次或奇次且次数不超过 $L$ 的球谐函数，可将试验函数空间的维数缩减近一半，同时保持相同的逼近阶。数值实验表明，当 $h \to 0$ 时，采用缩减后的试验函数空间对确保 MLS 逼近方案的数值稳定性至关重要。最后，我们将本文方法与采用切空间多项式作为试验函数空间的 MLS 逼近方案进行了比较。