变分递减样条函数逼近的下界
Lower bounds for the approximation with variation-diminishing splines
摘要：
我们证明了下界的近似误差的变分递减Schoenberg算子的区间$[0,1]$在经典的光滑模取决于程度的样条基使用泛函分析为基础的框架。由此，我们刻画了Schoenberg算子的谱，并研究了其迭代的渐近性态。最后，我们证明了近似误差和经典的二阶光滑模之间的等价性，作为2002年公开猜想的改进版本。
英文摘要：
We prove lower bounds for the approximation error of the variation-diminishing Schoenberg operator on the interval $[0,1]$ in terms of classical moduli of smoothness depending on the degree of the spline basis using a functional analysis based framework. Thereby, we characterize the spectrum of the Schoenberg operator and investigate the asymptotic behavior of its iterates. Finally, we prove the equivalence between the approximation error and the classical second order modulus of smoothness as an improved version of an open conjecture from 2002.
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