关于Lagrange插值导数的误差估计

DERIVATIVE ERROR APPROXIMATION FOR LAGRANGE INTERPOLATION

关于Ｌａｇｒａｎｇｅ插值导数的误差估计曾提出过以下猜想：对任意ｎ＋１次连续可微的函数ｆ和任意ｎ＋１个连续点，ｎ次ｌａｇｒａｎｇｅ插值多项式Ｌ满足：‖ｆ（ｊ）－Ｌ（ｊ）‖≤‖ｗ（ｊ）‖‖ｆ（ｎ＋１）‖（ｎ＋１）！，其中ｊ是所有的整数，‖·‖表示上确界范数．ＧａｒｙＷＨｏｗｅｌ在文献［１］中证明了ｊ＝ｎ时猜想成立，ｊ＝１，２，…，ｎ－１猜想没被解决，而仅得到了‖ｆ（ｊ）－Ｌ（ｊ）‖≤‖ｗ（ｊ）‖‖ｆ（ｎ＋１）‖ｊ！（ｎ＋１－ｊ）！这样的估计式，本文主要研究ｊ＝ｎ－１时。

In 《An Extension of Cauchy′s Bound for the Error of Lagrange Interpolation》, Gary W Howell gave that for any n+1 times continuously differentiable funtion f and any choice of n+1 knots, the lagrange interpolation polynomial L of degree n satisfies: ‖ f (j) -L (j) ‖≤‖w (j) ‖‖f (n+1) ‖(n+1)! , it is also shown that for j=1,2,…,n-1, ‖f (j) -L (j) ‖≤‖w (j) ‖‖f (n+1) ‖j!(n+1-j)! , where j is integer and ‖·‖ denotes the supremum norm, when j=n-1 . We′ll get a better approximation formula.