 Lipschitz error bound for Bernstein approximation

If $f$ is Lipschitz continuous on $[0,1]$ with Lipschitz constant $L$, then Cauchy-Schwarz inequality and the binomial distribution variance giveThis is an explicit error guarantee, rather than only an existence theorem. For the Bernstein approximation of a corner, choosing $n\ge 2500$ guarantees supremum norm error at most $0.01$; $n>2500$ guarantees error strictly less than $0.01$. The guarantee is sufficient, not optimal.

For this specific target, the error reaches its maximum at $x=1/2$. Indeed the derivative of $B_{n}f$ is a weighted average of the slopes $n[f((k+1)/n)-f(k/n)]$, all in $[-1,1]$: see derivatives of Bernstein polynomials. Subtracting the target makes the error nondecreasing to the left of the corner and nonincreasing to its right. For even $n\ge 2$, the symmetric binomial distribution at $x=1/2$ gives the exact valueThe last equality follows by pairing terms about $n/2$ and telescoping the binomial coefficients. It gives $1/4$ for $n=2$ and $3/16$ for $n=4$.