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

TL;DR

This paper proves that among functions with continuous derivatives up to order n+1, only degree-(n+1) polynomials are the hardest to approximate uniformly with degree-n polynomials, matching the known error bounds.

Contribution

It establishes a precise characterization of the functions that attain the worst-case approximation error, showing they are exactly degree-(n+1) polynomials.

Findings

Error bounds are tight only for degree-(n+1) polynomials.

Degree-(n+1) polynomials are the most difficult to approximate with degree-n polynomials.

The approximation error characterizes the degree of the polynomial being approximated.

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 $n + 1$.