一类收敛的线性Hermite重心权有理插值

众所周知, Hermite有理插值比Hermite多项式插值具有更好的逼近性, 特别是对于插值点序列较大时, 但很难解决收敛性问题和控制实极点的出现. 本文建立了一类线性Hermite重心有理插值函数$r(x)$,并证明其具有以下优良性质: 第一, 在实数范围内无极点; 第二, 当$k=0,1,2$时,无论插值节点如何分布, 函数$r^{(k)}(x)$具有$O(h^{3d+3-k})$的收敛速度; 第三, 插值函数$r(x)$仅仅线性依赖于插值数据.

A Convergent Family of Linear Hermite Barycentric Rational Interpolants

It is well-known that Hermite rational interpolation gives a better approximation than Hermite polynomial interpolation, especially for large sequences of interpolation points, but it is difficult to solve the problem of convergence and control the occurrence of real poles. In this paper, we establish a family of linear Hermite barycentric rational interpolants $r$ that has no real poles on any interval and in the case $k=0,1,2,$ the function $r^{(k)}(x)$ converges to $f^{(k)}(x)$ at the rate of $O(h^{3d+3-k})$ as $h\rightarrow{0}$ on any real interpolation interval, regardless of the distribution of the interpolation points. Also, the function $r(x)$ is linear in data.