Past exam of the mathematics course of the University of Cambridge / 2012 / iii / Paper 70 / 3 / a

Use real-valued continuous functions and real trigonometric polynomials. The trigonometric Chebyshev alternation theorem says that $p\in {\mathcal{T}}_n$ is the unique best uniform approximation to $f$ if and only if there are distinct points, ordered within one period,Thus the required number of alternating extrema is $2n+2$. When the error is zero, the assertion is interpreted trivially and $p=f$ is unique. The count reflects $\dim {\mathcal{T}}_n=2n+1$ and the fact that a nonzero degree-$n$ trigonometric polynomial has at most $2n$ zeros in one period. In particular a strictly better candidate would force its difference from $p$ to have alternating signs at these points and hence at least $2n+2$ zeros, counting the closing interval around the circle. The trigonometric Chebyshev alternation theorem also gives necessity and uniqueness.