Korovkin-type theorem and application

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