= Rounded Bernstein polynomial
{title2=$B_n^*(f,x)$}
The rounded <Bernstein polynomial> replaces each coefficient $\binom nk f(k/n)$ in the unnormalized <basis> $x^k(1-x)^{n-k}$ by its <floor function>. It therefore has <integer> <coefficients> in the ordinary monomial <basis>. If $f(0),f(1)\in\mathbb Z$, only interior rounding errors remain and
$$
\|B_n^*f-B_nf\|_\infty\le\max\{3/(2n),(n-1)(3/4)^n\}\longrightarrow0\qquad(n\ge2).
$$
To prove this, bound each interior rounding error by one. On $[0,1/4]$, sum the resulting powers as a <geometric series> of ratio at most $1/3$ and maximize $x(1-x)^{n-1}$; reflect for $[3/4,1]$. On the middle interval each degree-$n$ product is at most $(3/4)^n$. <integer> endpoints are necessary for this convergence: a noninteger endpoint produces a fixed nonzero rounding error there.
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