Journal of Inequalities and Applications
Volume 3 (1999), Issue 3, Pages 267-277
doi:10.1155/S102558349900017X

On the fundamental polynomials for Hermite–Fejér interpolation of Lagrange type on the Chebyshev nodes

Simon J. Smith

Division of Mathematics, La Trobe University, P.O. Box 199, Bendigo 3552, Victoria, Australia

Received 30 June 1997; Revised 9 July 1998

Copyright © 1999 Simon J. Smith. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

For a fixed integer m≥0 and 1≤k≤n, let Ak,2m,n(T,x) denote the kth fundamental polynomial for (0,1⋯,2m) Hermite–Fejér interpolation on the Chebyshev nodes {xj,n=cos[(2j−1)π/(2n)]:1≤j≤n}. (So Ak,2m,n(T,x) is the unique polynomial of degree at most (2m+1)n−1 which satisfies Ak,2m,n(T,xj,n)=δk,j, and whose first 2m derivatives vanish at each xj,n.) In this paper it is established that |Ak,2m,n(T,x)|≤A1,2m,n(T,1),  1≤k≤n,  −1≤x≤1. It is also shown that A1,2m,n(T,1) is an increasing function of n, and the best possible bound Cm so that |Ak,2m,n(T,x)|<Cm for all k, n and x∈[−1,1] is obtained. The results generalise those for Lagrange interpolation, obtained by P. Erdős and G. Grünwald in 1938.