Connections between the Approximation Orders of Positive Linear Operators and Their Max-Product Counterparts

Abstract

In this study we establish some direct connections between arbitrary positive linear operators and their corresponding nonlinear (more exactly sublinear) max-product versions, with respect to uniform and Lp convergence. There are numerous concrete examples of approximation operators, such as Bernstein-type operators, neural network operators, sampling operators and others, where the linear and the max-product versions converge both uniformly. Here, from the quantitative uniform approximation result for an arbitrary sequence of positive linear operators, we deduce by a simple general method a quantitative uniform approximation result for its max-product counterpart. We also establish convergence with respect to the Lp -norm involving the well-known K-functionals, when the supremum of the kernel is bounded from below. Our results cover the cases of bounded and unbounded domains and the case of the Kantorovich variants of the considered operators. Applications to some max-product operators are presented.