A Convergent Family of Linear Hermite Barycentric Rational Interpolants
Received:December 17, 2019  Revised:April 23, 2020
Key Words: linear Hermite rational interpolation   convergence rate   Hermite interpolation   barycentric form   higher order derivative
Fund Project:Supported by the National Natural Science Foundation of China (Grant No.11601224), the Science Foundation of Ministry of Education of China (Grant No.18YJC790069), the Natural Science Foundation of Jiangsu Higher Education Institutions of China (Grant No.18KJD110007) and the National Statistical Science Research Project of China (Grant No.2018LY28).
 Author Name Affiliation Ke JING School of Applied Mathematics, Nanjing University of Finance and Economics, Jiangsu 210023, P. R. China Yezheng LIU School of Management, Hefei University of Technology, Anhui 230009, P. R. China Ning KANG School of Economics, Nanjing University of Finance and Economics, Jiangsu 210023, P. R. China Gongqin ZHU School of Mathematics, Hefei University of Technology, Anhui 230009, P. R. China
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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.