= Bernstein monomial recurrence
{c}
{title2=$B_n(x^m)$}
For $m\ge1$ and $n\ge2$, the <Bernstein polynomial> of the <monomial> $x^m$ satisfies
$$
B_n(x^m)=x\sum_{\ell=0}^{m-1}\binom{m-1}{\ell}\frac{(n-1)^{m-1-\ell}}{n^{m-1}}B_{n-1}(x^{m-1-\ell}).
$$
It follows by applying $k\binom nk=n\binom{n-1}{k-1}$ and expanding $k=(k-1)+1$ with the <binomial theorem>. The coefficients sum to one; only the first tends to one as $n\to\infty$. Induction on $m$ proves <uniform convergence> of these monomial approximants, starting from $B_n(1)=1$.
Back to article page
