= Interpolation property of weighted least-squares polynomial approximation
{title2=$\#\{x:f(x)=p_n(x)\}\geq n+1$}
Let $f$ be a real <continuous function> on a nondegenerate compact interval and let $p_n$ be its <best approximation> among <polynomials> of <polynomial degree> at most $n$ for a positive integrable <weighted inner product>. Then $f-p_n$ has at least $n+1$ distinct zeros. Indeed, if it had at most $n$, multiply it by the <polynomial> with simple zeros at precisely its interior sign changes. The product has constant nonzero sign away from finitely many zeros, so its weighted integral is nonzero, contradicting <orthogonality> of $f-p_n$ to all <polynomials> of <polynomial degree> at most $n$.
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