= Degree elevation of Bernstein coefficients
{title2=$\beta_{k,n+1}=\frac{k}{n+1}\beta_{k-1,n}+\left(1-\frac{k}{n+1}\right)\beta_{k,n}$}
Write $f=\sum_k\beta_{k,n}b_{k,n}$ in the <Bernstein basis>. Representing the same polynomial at degrees $n$ and $n+1$ gives the displayed convex combinations for internal indices; the endpoint coefficients are copied. Therefore the minimum coefficient cannot decrease under degree elevation. In the unnormalized basis $x^k(1-x)^{n-k}$, the elevated coefficients are $c'_k=c_{k-1}+c_k$ with the corresponding endpoint convention, so nonnegativity certificates are preserved.
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