Interpolation polynomials which give best order of approximation among continuously differentiable functions of arbitrary fixed order on $[-1, +1]$

Abstract:

The object of this paper is to show that there exists a polynomial ${P_n}(x)$ of degree $\leqslant 2n - 1$ which interpolates a given function exactly at the zeros of $n$th Tchebycheff polynomial and for which $||f - {P_n}|| \leqslant {C_k}{w_k}(1/n,f)$ where ${w_k}(1/n,f)$ is the modulus of continuity of $f$ of $k$th order.