The strong converse inequality for Bernstein-Kantorovich operators

We give a solution to a problem posed by Totik at the 1992 Texas conference concerning the strong converse inequality for approximation by Bernstein-Kantorovich operators. The approximation behaviour of these operators is characterized for 1 ≤ p ≤ ∞ by using an appropriate K-functional which, for 1 < p ≤ ∞, is equivalent to a second order modulus and an extra term. Crucial in our approach are estimates for the derivatives of iterated Kantorovich operators.