The <Chebyshev alternation theorem> says that $p^*\in\mathcal P_n$ is the unique <best uniform approximation> to $f\in C[-1,1]$ if and only if there are at least $n+2$ ordered points
$$
-1\leq t_1\lt t_2\lt\cdots\lt t_{n+2}\leq1
$$
and a sign $\varepsilon\in\{-1,1\}$ such that
$$
\boxed{f(t_i)-p^*(t_i)
=\varepsilon(-1)^i\|f-p^*\|_\infty,
\qquad i=1,\ldots,n+2}.
$$
Thus the extremal error has common magnitude and alternating sign at $n+2$ points.

Solution

Past exam of the mathematics course of the University of Cambridge / 2024 / iii / Paper 318 / 2 / a / Solution

 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2024/iii/paper-318/2/a/solution