APPROXIMATION AND SHAPE PRESERVING PROPERTIES OF THE NONLINEAR FAVARD-SZASZ-MIRAKJAN OPERATOR OF MAX-PRODUCT KIND

作者：Bede Barnabas^*; Coroianu Lucian; Gal Sorin G

来源：Filomat, 2010, 24(3): 55-72.

• Nonlinear Favard-Szasz-Mirakjan 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 [6], [7], in the book [8], Open Problem 5.5.4, pp. 324-326, the Favard-Szasz-Mirakjan max-prod type operator is introduced and the question of the approximation order by this operator is raised. In the recent paper [1], by using a pretty complicated method to this open question an answer is given by obtaining an upper pointwise estimate of the approximation error of the form C omega(1)(f; root x / root n) (with an unexplicit absolute constant C > 0) and the question of improving the order of approximation omega(1)(f; root x / root n) is raised. The first aim of this note is to obtain the same order of approximation but by a simpler method, which in addition presents, at least, two advantages : it produces an explicit constant in front of omega(1)(f; root x / root n) and it can easily be extended to other max-prod operators of Bernstein type. Also, we prove by a counterexample that in some sense, in general this type of order of approximation with respect to omega(1)(f; .) cannot be improved. However, for some subclasses of functions, including for example the bounded, nondecreasing concave functions, the essentially better order omega(1)(f; 1/n) is obtained. Finally, some shape preserving properties are obtained.

• 出版日期2010-9