= Trigonometric Chebyshev alternation theorem
{title2=$2n+2\text{ alternating extrema}$}
For real <continuous> $2\pi$-periodic $f$, a degree-at-most-$n$ real <trigonometric polynomial> $p$ is its unique <best uniform approximation> exactly when $f-p$ has $2n+2$ alternating extrema of magnitude $\|f-p\|_\infty$ at distinct ordered points in one period. The closing interval around the circle also counts. A strictly improving approximant would force its difference from $p$ to have at least $2n+2$ zeros, whereas a nonzero degree-$n$ <trigonometric polynomial> has at most $2n$.
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