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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 usingthe spherical harmonics up to degree $L \in \mathbb N$ as ansatz space yieldsfor 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 thesampling nodes.  In this paper we show that the dimension of the ansatz space can be almosthalved, by including only spherical harmonics of even or odd degree up to $L$,while preserving the same order of approximation. Numerical experimentsindicate that using the reduced ansatz space is essential to ensure thenumerical stability of the MLS approximation scheme as $h \to 0$. Finally, wecompare our approach with an MLS approximation scheme that uses polynomials onthe tangent space as ansatz space.

An Optimal Ansatz Space for Moving Least Squares Approximation on Spheres