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

^{1}Department of Mathematics, The University of Texas-Pan American, 1201 West University, Edinburg, TX 78539, USA^{2}Department of Mathematics and Computer Science, The University of Oradea, Official Postal nr. 1, C.P. nr. 114, Universitatii 1, 410087 Oradea, Romania

Received 31 August 2009; Accepted 25 October 2009

Academic Editor: Narendra Kumar Govil

Copyright © 2009 Barnabás Bede et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### 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