Aldaz–Kounchev–Render Operators and Their Approximation Properties

Abstract

The approximation properties of the Aldaz–Kounchev–Render (AKR) operators where investigated in several papers. We improve some existing quantitative results concerning these approximation properties. Moreover, we describe classes of functions for which these operators approximate better than the classical Bernstein operators and classes of functions for which Bernstein operators approximate better than AKR operators. The new results, in particular involving monotonic convergence and Voronovskaja type formulas, are then extended to the bivariate case on the square \([0,1]^2\) and compared with other existing results. Several numerical examples, illustrating the relevance and supporting the theoretical findings, are presented.