Kantorovich Operators of Order k

This article is concerned with the k-th order Kantorovich modification of the classical Bernstein operators Bn, namely, [image omitted], where Dkf is the derivative of order k and Ikf is an antiderivative of order k of the function f. These operators are most useful in simultaneous approximation. We give detailed expressions of the moments up to order four of [image omitted] and estimates of ratios of moments. Next, they are used to refine Voronovskaya's result for derivative approximation by Bn. The properties of [image omitted] as approximation operators are also investigated.