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

Barnabás Bede,Sorin G Gal,Lucian Coroianu

doi:10.1155/2009/590589

Barnabás Bede, Sorin G Gal + Show 1 more

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https://doi.org/10.1155/2009/590589

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Abstract

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.

Highlights

Starting from the study of the Shepard nonlinear operator of max-prod-type in 1, 2, by the Open Problem in a recent monograph 3, pages 324–326, 5.5.4, the following nonlinear Bernstein operator of max-prod type is introduced here means maximum : BnM f x n k0pn,k x f k/n n k 0pn,k xInternational Journal of Mathematics and Mathematical Sciences where pn,k x n k xk 1 − x n−k, for which by a very complicated method in √ 4, Theorem6, an upper estimate of the approximation error of the form Cω1 f; 1/ n with C > 0 unexplicit absolute constant is obtained
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
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

Summary

Introduction

Starting from the study of the Shepard nonlinear operator of max-prod-type in 1, 2 , by the Open Problem in a recent monograph 3, pages 324–326, 5.5.4 , the following nonlinear Bernstein operator of max-prod type is introduced here means maximum : BnM f x n k. By Remark 7, 2 in the same paper 4 , the question if this order of approximation could be improved is raised. The first aim of this note is to obtain the same order of approximation but by a simpler method, which in ad√dition presents, at least, two advantages: it produces an explicit constant in front of ω1 f; 1/ n , and it can be extended to other max-prod operators of Bernstein type. One proves by a counterexample that in a sense, for arbitrary f, this order of approximation with respect to ω1 f; · cannot be improved, giving a negative answer to a question raised in 4, Remark 7, 2.

Preliminaries

Auxiliary Results

Approximation Results

Shape-Preserving Properties

Comparisons with the Linear Bernstein Operator