Approximation and Shape Preserving Properties of the Bernstein Operator of Max-Product Kind

Starting from the study of the Shepard nonlinear operator of max-prod type by Bede et al. 2006, 2008 , in the book by Gal 2008 , Open Problem 5.5.4, pages 324–326, the Bernstein max-prod-type operator is introduced and the question of the approximation order by this operator is raised. In recent paper, Bede and Gal by using a very complicated method to this open question an answer is given by obtaining an upper estimate of the approximation error of the form Cω1 f ; 1/ √ n with an unexplicit absolute constant C > 0 and the question of improving the order of approximation ω1 f ; 1/ √ n is raised. The first aim of this note is to obtain this order of approximation but by a simpler method, which in addition presents, at least, two advantages: it produces an explicit constant in front of ω1 f ; 1/ √ n and it can easily be extended to other max-prod operators of Bernstein type. However, for subclasses of functions f including, for example, that of concave functions, we find the order of approximationω1 f ; 1/n , which for many functions f is essentially better than the order of approximation obtained by the linear Bernstein operators. Finally, some shape-preserving properties are obtained.