在等距结点上Lagrange插值多项式的发散性

The divergence of Lagrange interpolation at equidistant nodes

191 8年 ,Bernstein证明了对于函数 |x|,由闭区间 [-1 ,1 ]上的等距结点所构成的 Lagrange插值多项式序列 ,除 -1 ,0 ,1以外 ,在闭区间 [-1 ,1 ]上的其它任何点都发散 .在本文中考虑了函数f (x) =x2 ,当 0≤ x≤ 1时 ,-x2 , 当 -1≤ x≤ 0时 ,将证明函数 f (x)对于闭区间 [-1 ,1 ]上的等距结点所构成的Lagrange插值多项式 ,当增大时 ,除 -1 ,0 ,1以外 ,在闭区间 [-1 ,1 ]上的其它任何点处都不收敛于 f (x) .

In 1918 S.N.Bernstein proved that the sequence of Lagrange interpolation polynomials to |x| at equidistant nodes in \ diverges everywhere ,except at zero and the end-points. In this paper, we consider the function f(x)=x\+2,0≤x≤1, -x\+2,-1≤x≤0,, and prove that the sequence of Lagrange interpolation polynomials to f(x) at equally spaced nodes in \ diverges everywhere, except at zero and the end-points.