Interpolating the mth power of x at the zeros of the \(n\)th Chebyshev polynomial yields an almost best Chebyshev approximation (Q579552)

Let f be real-valued and continuous on \(I=[-1,1]\). Let \(E_ n(f)\) denote the sup-norm distance of f from polynomials of degree \(\leq n\). Let \(E^*_ n(f)=\| f-L_ f\|\) where \(L_ f\) denotes the Lagrange interpolant to f on zeros of the Chebyshev polynomial \(T_{n+1}(x)\) of degree \(n+1\). \textit{D. J. Newman} and \textit{T. J. Rivlin} [Aequationes Math. 14, 451-455 (1976; Zbl 0327.41005)] had shown that \((4e)^{- 1}Q_{k,n}\leq E_{n-1}(x^{n+k})\leq Q_{k,n},\) \(k\in N_ 0\) where \(Q_{k,n}=2^{1-n-k}\sum^{k/2}_{j=0}\left( \begin{matrix} n+k\\ j\end{matrix} \right).\) Here the author proves that \(E^*_{n- 1}(x^{n+k})\leq 8eE_{n-1}(x^{n+k}),\) \(k\in N\), \(k\in N_ 0\).