Degree elevation of Bernstein coefficients (${\beta }_{k,n+1}=\frac{k}{n+1}{\beta }_{k-1,n}+\left(1-\frac{k}{n+1}\right){\beta }_{k,n}$)

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 $c_k^{\prime }=c_{k-1}+c_k$ with the corresponding endpoint convention, so nonnegativity certificates are preserved.