Korovkin-type theorems and applications

Abstract

Let {T (n) } be a sequence of linear operators on C[0,1], satisfying that {T (n) (e (i) )} converge in C[0,1] (not necessarily to e (i) ) for i = 0,1,2, where e (i) = t (i) . We prove Korovkin-type theorem and give quantitative results on C (2)[0,1] and C[0,1] for such sequences. Furthermore, we define King's type q-Bernstein operator and give quantitative results for the approximation properties of such operators.