= Bernstein falling-factorial identity
{c}
{title2=$B_n(\prod_{r=0}^{m-1}(x-r/n))=\prod_{r=0}^{m-1}(1-r/n)x^m$}
At the sampling point $j/n$, the polynomial inside the <Bernstein polynomial> has value $n^{-m}j(j-1)\cdots(j-m+1)$. The <falling factorial> identity $j^{\underline m}\binom nj=n^{\underline m}\binom{n-m}{j-m}$ and the <binomial theorem> prove the displayed formula. For fixed $m$, both the input and its scalar factor tend uniformly to $x^m$ as $n$ grows. The <supremum norm> contraction of the <Bernstein polynomial> therefore gives $\|B_n(x^m)-x^m\|_\infty\le m(m-1)/n$ for $n\ge m$.
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