= Uniform convergence of Bernstein polynomial derivatives
{title2=$\|B_n^{(r)}f-f^{(r)}\|_\infty\to0$}
For fixed $r$ and $f\in C^r[0,1]$, uniform convergence of normalized <finite differences> to $f^{(r)}$, uniform continuity of $f^{(r)}$, and $(n)_r/n^r\to1$ imply $g_{n,r}\to f^{(r)}$ uniformly. Positivity and partition of unity make <Bernstein polynomials> contractions in the <supremum norm>. Consequently $\|B_{n-r}g_{n,r}-f^{(r)}\|_\infty\le\|g_{n,r}-f^{(r)}\|_\infty+\|B_{n-r}f^{(r)}-f^{(r)}\|_\infty\to0$. The differentiability assumption is essential; continuity alone does not define the target derivative.
Back to article page
