We consider the so-called simplest formula for local approximation by polynomial splines of order $ n $ (Schoenberg splines). The spline itself and all derivatives except that of the highest order, approximate a given function and its corresponding derivatives with the second order. We show that the jump of the highest derivative of order $ n-1 $ ; i. e., the value of discontinuity, divided by the meshsize, approximates the $ n $ th derivative of the original function. We found an asymptotic expansion of the jump.

中文翻译：

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样条局部逼近的误差估计

我们考虑通过 $n$ 阶多项式样条（勋伯格样条）进行局部逼近的所谓最简单公式。样条本身和除最高阶的导数之外的所有导数都用二阶近似给定函数及其相应的导数。我们证明了 $ n-1 $ 阶最高导数的跳跃；一世。即，不连续性的值除以网格大小，近似于原始函数的第 n 次导数。我们发现了跳跃的渐近扩展。