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 ð ¶ ð œ” 1 √ ( ð ‘“ ; 1 / ð ‘› ) (with an unexplicit absolute constant ð ¶ > 0 ) and the question of improving the order of approximation 𠜔 1 √ ( ð ‘“ ; 1 / ð ‘› ) 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 𠜔 1 √ ( ð ‘“ ; 1 / ð ‘› ) and it can easily be extended to other max-prod operators of Bernstein type. However, for subclasses of functions ð ‘“ including, for example, that of concave functions, we find the order of approximation 𠜔 1 ( ð ‘“ ; 1 / ð ‘› ) , which for many functions ð ‘“ is essentially better than the order of approximation obtained by the linear Bernstein operators. Finally, some shape-preserving properties are obtained.