= Interpolation nodes from best uniform approximation
{title2=$\ell_n(f)=p_n^*$}
For a real <continuous function> on a compact interval, choose its <best uniform approximation> $p_n^*$ of <polynomial degree> at most $n$. If its error is nonzero, the <Chebyshev alternation theorem> supplies $n+2$ ordered alternating extrema. The <intermediate value theorem> gives a zero of $f-p_n^*$ in each intervening interval. At these $n+1$ distinct points, the <polynomial interpolant> is exactly $p_n^*$. If the best error is zero, arbitrary distinct nodes suffice. The <Weierstrass approximation theorem> shows that these function-dependent interpolants converge uniformly. The nodes depend on the target function; the conclusion does not provide one universal node sequence for every continuous target.
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