If $f\in C^2[0,1]$, then
\begin{align*}
\lim_{n\to\infty} n\bigl(B_n f(x)-f(x)\bigr)=\frac{x(1-x)}{2}f''(x)
\end{align*}
for each $x\in[0,1]$, and the convergence is uniform when $f''$ is continuous.
Knowledge Status
Analysis
Discussion
A theorem in [approximation theory](/page/Approximation%20Theory) concerning Voronovskaya Formula, used to organise the analytic structure of approximation methods and convergence results.