In this paper, we present a generalization of Lehmann’s approach for solving approximation problems on hypersurfaces to situations with arbitrary codimension. We show that as in the case of hypersurfaces, the method is able to transfer approximation orders from the ambient space to the submanifold. In particular, the resulting approximant is $${\mathrm {C}}^{m-2}$$ and the error decays at an optimal $$h^m$$ for tensor product B-splines of order m. Additionally, the method is easily implemented and comes with an optimal computational expense when applying quasi interpolation techniques. Applications include, in particular, surfaces embedded into $${{\mathbb {R}}}^4$$ but not into $${{\mathbb {R}}}^3$$.

中文翻译：

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嵌入式子流形上的环境逼近

在本文中，我们提出了 Lehmann 方法的推广，该方法用于解决超曲面上的逼近问题到具有任意码维的情况。我们表明，在超曲面的情况下，该方法能够将近似阶数从环境空间转移到子流形。特别是，结果近似值是 $${\mathrm {C}}^{m-2}$$ 并且对于 m 阶张量积 B 样条，误差以最优 $$h^m$$ 衰减。此外，该方法易于实现，并且在应用准插值技术时具有最佳的计算开销。应用程序特别包括嵌入到 $${{\mathbb {R}}}^4$$ 但不嵌入 $${{\mathbb {R}}}^3$$ 的表面。