We consider the approximation of manifold-valued functions by embedding the manifold into a higher dimensional space, applying a vector-valued approximation operator and projecting the resulting vector back to the manifold. It is well known that the approximation error for manifold-valued functions is close to the approximation error for vector-valued functions. This is not true anymore if we consider the derivatives of such functions. In our paper we give pre-asymptotic error bounds for the approximation of the derivative of manifold-valued function. In particular, we provide explicit constants that depend on the reach of the embedded manifold.

中文翻译：

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流形值函数的逼近

我们通过将流形嵌入更高维的空间，应用矢量值的近似算子并将结果矢量投影回流形来考虑流形值函数的逼近。众所周知，流形值函数的近似误差接近矢量值函数的近似误差。如果我们考虑这些函数的导数，那就不再是事实了。在我们的论文中，我们给出了流形值函数的导数的逼近前渐近误差界。特别是，我们提供了取决于嵌入式歧管范围的显式常量。