Korovkin-type theorem and application

Let (L"n) be a sequence of positive linear operators on C[0,1], satisfying that (L"n(e"i)) converge in C[0,1] (not necessarily to e"i) for i=0,1,2, where e"i(x)=x^i. We prove that the conditions that (L"n) is monotonicity-preserving, convexity-preserving and variation diminishing do not suffice to insure the convergence of (L"n(f)) for all [email protected]?C[0,1]. We obtain the Korovkin-type theorem and give quantitative results for the approximation properties of the q-Bernstein operators B"n","q as an application.