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.

翻译：我们考虑多种价值函数的近似值,方法是将元件嵌入一个更高维度的空间,应用一个矢量估值近似值操作器,并将由此产生的矢量投射回元件。众所周知,多重价值函数的近似误差接近矢量估值函数的近似误差。如果我们考虑这些函数的衍生物,就不再如此了。在我们的文件里,我们给出了多种价值函数衍生物近似值前的误差界限。特别是,我们提供了取决于嵌入的元件范围的明确的常数。