Strong Converse Inequality for Weighted Approximation of Functions by the Szász–Mirakjan–Kantorovich Operator

Abstract

We investigate the weighted approximation of functions in \(L_p\)-norm by Kantorovich modifications of the classical Szász-Mirakjan operator, with weights of type \((1+x)^{\alpha }\), \(\alpha \in {\mathbb {R}}\). By using an appropriate K-functional we prove a strong converse inequality for the weighted error of approximation and characterize it exactly. We prove a Voronovskaya and Bernstein-type inequalities for the Szász-Mirakjan–Kantorovich operator.