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Approximation and shape preserving properties of the truncated Baskakov operator of max-product kind

Starting from the study of the Shepard nonlinear operator of max-prod type in [2], [3], in the recent monograph [4], Open Problem 5.5.4, pp. 324-326, the Baskakov max-prod type operator is introduced and the question of the approximation order by this operator is raised. The aim of this note is to obtain the order of uniform approximation Cω1(f; 1/√n) (with the explicit constant C = 24) of another operator called the truncated max-prod Baskakov operator and to prove by a counterexample that in some sense, for arbitrary f this type of order of approximation with respect to ω1(f; √n) cannot be improved. However, for some subclasses of functions including for example the nondecreasing concave functions, the essentially better order of approximation ω1(f; 1/n) is obtained. Finally, some shape preserving properties are proved.