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Approximation by truncated Favard-Szász-Mirakjan operator of max-product kind

Bede B., Coroianu L., Gal S. G.

Demonstratio Mathematica

2011

Vol. 44, nr 1

105-122

Starting from the study of the Shepard nonlinear operator of max-prod type made by Bede et al. in 2006 and 2008, in the book of Gal in 2008 (Open Problem 5.5.4, pp. 324-326) the Favard-Szász-Mirakjan max-prod type operator is introduced and the question of the approximation order by this operator is raised. In the paper of Bede and Gal in 2010, an answer is given by obtaining a pointwise upper estimate of the approximation error of the form (...), with C > 0 unexplicit absolute constant. The aim of this note is to obtain the order of uniform approximation (...) (with C = 6) by another operator, much simpler and called truncated Favard-Szász-Mirakjan operator of max-product kind and to prove by a counterexample that in some sense, in general this type of order of approximation with respect to (...) cannot be improved. In addition, for some subclasses of functions including, for example, the nondecreasing concave functions, the essentially better order (...) is obtained. Finally, some shape preserving properties are proved.