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International Journal of Mathematics and Mathematical Sciences
Volume 2009, Article ID 590589, 26 pages
Research Article

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

1Department of Mathematics, The University of Texas-Pan American, 1201 West University, Edinburg, TX 78539, USA
2Department 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.


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 𝐢 πœ” 1 √ ( 𝑓 ; 1 / 𝑛 ) (with an unexplicit absolute constant 𝐢 > 0 ) and the question of improving the order of approximation πœ” 1 √ ( 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 πœ” 1 √ ( 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 πœ” 1 ( 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.