Approximation of fuzzy numbers by max-product Bernstein operators

Abstract

In this paper firstly we extend to an arbitrary compact interval the definition of the nonlinear Bernstein operators of max-product kind, by proving that their order of uniform approximation is the same as in the particular case of the unit interval. Then, similarly to the particular case of the unit interval, we will prove that these operators preserve the quasi-concavity too. Since these properties will help us to generate in a simple way fuzzy numbers of the same support, it turns out that these results are very suitable in the approximation of fuzzy numbers. Moreover, these operators approximate the (non-degenerate) segment core with a good order of approximation. In addition, in the case when the fuzzy numbers are given in parametric form, the Bernstein max-product operator generates a sequence of fuzzy numbers whose widths, expected intervals, ambiguities and values, approximate with a convergent rate the width, the expected interval, the ambiguity and the value of the approximated fuzzy number. Finally, we obtain a quantitative approximation with respect to a well-known integral type metric for some subclasses of functions, leading to applications to the approximation of some subclasses of fuzzy numbers.