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nonlinear Bleimann-Butzer-Hahn operator of max-product kind; degree of approximation; shape preserving properties
Starting from the study of the 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 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.
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