Derivatives of Bernstein polynomials ($B_n^{(r)}f=(n{)}_r{\sum }_{j=0}^{n-r}{\Delta }_{1/n}^{r}f(j/n)p_{n-r,j}$)

Differentiating the Bernstein basis $p_{n,j}(x)=\left(\genfrac{}{}{0ex}{}{n}{j}\right)x^j(1-x{)}^{n-j}$ gives $p_{n,j}^{\prime }=n(p_{n-1,j-1}-p_{n-1,j})$, with out-of-range terms zero. Telescoping coefficient differences and induction give the displayed formula, where $(n{)}_r=n!/(n-r)!$ and $0\le r\le n$. For $n>r$, set $g_{n,r}(t)=(n{)}_r{\Delta }_{1/n}^{r}f((n-r)t/n)$; then $B_n^{(r)}f=B_{n-r}{g}_{n,r}$. The rescaled argument compensates for the different sampling meshes.

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 $(n{)}_r/n^r\to 1$ imply $g_{n,r}\to f^{(r)}$ uniformly. Positivity and partition of unity make Bernstein polynomials contractions in the supremum norm. Consequently $\Vert B_{n-r}{g}_{n,r}-f^{(r)}{\Vert }_{\infty }\le \Vert g_{n,r}-f^{(r)}{\Vert }_{\infty }+\Vert B_{n-r}{f}^{(r)}-f^{(r)}{\Vert }_{\infty }\to 0$. The differentiability assumption is essential; continuity alone does not define the target derivative.