Approximation and Shape Preserving Properties of the Nonlinear Meyer–König and Zeller Operator of Max-Product Kind

Starting from the study of the Shepard nonlinear operator of max-prod type in [2, 3; 6, Open Problem 5.5.4], the Meyer–König and Zeller max-product type operator is introduced and the question of the approximation order by this operator is raised. The first aim of this article is to obtain the order of pointwise approximation for these operators. Also, we prove by a counterexample that in some sense, in general this type of order of approximation with respect to ω1(f; ·) cannot be improved. However, for some subclasses of functions, including for example the continuous nondecreasing concave functions, the essentially better order (of uniform approximation) ω1(f; 1/n) is obtained. Several shape preserving properties are obtained including the preservation of quasi-convexity.