Past exam of the mathematics course of the University of Cambridge / 2009 / ii / Paper 2 / 2F (Topics in Analysis)

The Chebyshev alternation theorem says that for $f\in C([a,b])$ a polynomial $q$ of degree at most $n-1$ is a best uniform approximation precisely when there are $n+1$ ordered points $a\le x_0<\cdots <x_n\le b$ such thatThe best polynomial is unique. These equally large alternating extrema are the equal-ripple criterion; the zero-error case is interpreted in the evident way.

Set $p_{\ast }(x)=2^{1-n}{T}_n(x)$, a monic Chebyshev polynomial. Its norm on $[-1,1]$ is $2^{1-n}$. At the ordered points $x_j=\cos ((n-j)\pi /n)$, its values alternate between the two extreme values $\pm 2^{1-n}$.

Approximate $f(x)=x^n$ by polynomials of degree at most $n-1$. The error corresponding to $q_{\ast }=x^n-p_{\ast }$ is exactly $p_{\ast }$. The Chebyshev alternation theorem therefore makes $q_{\ast }$ the unique best approximant. For any monic $p$, the polynomial $q=x^n-p$ is an admissible approximant, soEquality holds precisely for $p=p_{\ast }$. Alternatively, a smaller norm would force $p-p_{\ast }$ to change sign between each consecutive pair of alternating extrema, giving at least $n$ zeros to a polynomial of degree at most $n-1$.