International Journal of Mathematics and Mathematical Sciences
Volume 2009 (2009), Article ID 590589, 26 pages
doi:10.1155/2009/590589

Approximation and shape preserving properties of the Bernstein operator of max-product kind

Abstract

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 approximation ω1(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.