The degree-$(n+1)$ polynomials are the most difficult $C^{n+1}$ functions to uniformly approximate with degree-$n$ polynomials

Abstract

There exist well-known tight bounds on the error between a function $f\in C^{n+1}([-1,1])$ and its best polynomial approximation of degree $n$. We show that the error meets these bounds when and only when $f$ is a polynomial of degree $n+1$.