 Quadratic barrier estimate for positive approximation operators

For positive linear operators on continuous functions on $[0,1]$, put $e_j=U_n(x^j)-x^j$. Uniform continuity supplies the barriers $|f(x)-f(t)|\le \epsilon +\gamma (x-t{)}^2$. Applying positivity and evaluating at $t$ gives the displayed estimate because $U_n((x-t{)}^2)(t)=e_2(t)-2t{e}_1(t)+t^2{e}_0(t)$. 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.