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

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Starting from the study of theShepard nonlinear operator of max-prod typeby Bede et al. (2006, 2008), in the book by Gal (2008), Open Problem 5.5.4, pages 324–326, theBernstein max-prod-type operatoris 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 (with an unexplicit absolute constant ) and the question of improving the order of approximation 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 and it can easily be extended to other max-prod operators of Bernstein type. However, for subclasses of functions including, for example, that of concave functions, we find the order of approximation , which for many functions is essentially better than the order of approximation obtained by the linear Bernstein operators. Finally, some shape-preserving properties are obtained.