Barnabás Bede, Lucian Coroianu, Sorin G. Gal
Approximation and shape preserving properties of the nonlinear Bleimann-Butzer-Hahn operators of max-product kind

Comment.Math.Univ.Carolin. 51,3 (2010) 397-415.

Abstract:Starting from the study of the {\it Shepard nonlinear operator of max-prod type\/} in (Bede, Nobuhara et al., 2006, 2008), in the book (Gal, 2008), Open Problem 5.5.4, pp.\,324--326, the {\it Bleimann-Butzer-Hahn max-prod type operator\/} is introduced and the question of the approximation order by this operator is raised. In this paper firstly we obtain an upper estimate of the approximation error of the form $\omega_{1}(f;(1+x)^{\frac{3}{2}}\sqrt{x/n})$. A consequence of this result is that for each compact subinterval $[0,a]$, with arbitrary $a>0$, the order of uniform approximation by the Bleimann-Butzer-Hahn operator is less than ${\mathcal{O}}(1/\sqrt{n})$. Then, one proves by a counterexample that in a sense, for arbitrary $f$ this order of uniform approximation cannot be improved. Also, for some subclasses of functions, including for example the bounded, nondecreasing concave functions, the essentially better order $\omega_{1}(f;(x+1)^{2}/n)$ is obtained. Shape preserving properties are also investigated.

Keywords: nonlinear Bleimann-Butzer-Hahn operator of max-product kind, degree of approximation, shape preserving properties
AMS Subject Classification: 41A30 41A25 41A29