一类具有优良性质的插值基函数

A class of interpolation basis functions with excellent properties

本文构造了一类具有不同光滑性与支撑区间的插值基函数列,获得了这类插值算子精确的Lebesgue常数,证明其逼近连续函数总是收敛的,并给出其逼近连续函数的收敛速度.将其与已有的3个重心有理插值算子进行比较,结果表明本文的插值算子在逼近精确性、算法稳定性和计算复杂性等方面均具有很大优势,尤其逼近精度比Berrut的有理插值算子和其他有理插值算子均有极大的提高.本文将这类插值基函数列用于构造插值曲线和曲面,大量例子佐证了结论的正确性.

In this paper,we construct a series of interpolation basis functions with different smoothness and support intervals,obtain the exact Lebesgue constants of these interpolation operators,prove that their approximation to continuous functions is always convergent,and give the convergence rate of their approximation to continuous functions.Compared with three barycentric rational interpolation operators,the results show that the interpolation operator in this paper has great advantages in approximation accuracy,algorithm stability,computational complexity,and so on.Especially,the approximation accuracy is greatly improved compared with Berrut’s rational interpolation operator and other rational interpolation operators.This kind of interpolation basis function sequence is used to construct interpolation curves and surfaces,and a large number of examples prove the correctness of the conclusion.