= Quadratic barrier estimate for positive approximation operators
{title2=$\|U_nf-f\|_\infty\le\varepsilon\|U_n1\|_\infty+\|f\|_\infty\|e_0\|_\infty+\gamma(\|e_2\|_\infty+2\|e_1\|_\infty+\|e_0\|_\infty)$}
For <positive linear operators on continuous functions> on $[0,1]$, put $e_j=U_n(x^j)-x^j$. Uniform continuity supplies the barriers $|f(x)-f(t)|\le\varepsilon+\gamma(x-t)^2$. Applying positivity and evaluating at $t$ gives the displayed estimate because $U_n((x-t)^2)(t)=e_2(t)-2te_1(t)+t^2e_0(t)$. The constant $\gamma$ is independent of $t$ and $n$. Thus uniform reproduction of $1,x,x^2$ proves <Korovkin theorem> convergence on every continuous function; exact reproduction forces the operator to be the identity.
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