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.