Possibilistic Approaches of the Max-Product Type Operators

Abstract

It is known that the first proof of the uniform convergence for the Bernstein polynomials to a continuous function interprets them as a mean value of a random variable based on the Bernoulli distribution and uses the Chebyshev’s inequality in probability theory (see [33], or the more available [111]). The first main aim of this chapter is to give a proof for the convergence of the max-product Bernstein operators by using the possibility theory, which is a mathematical theory dealing with certain types of uncertainties and is considered as an alternative to probability theory. This new approach, which interprets the max-product Bernstein operator as a possibilistic expectation of a fuzzy variable having a possibilistic Bernoulli distribution, does not offer only a natural justification for the max-product Bernstein operators, but also allows to extend the method to other discrete max-product Bernstein type operators, like the max-product Meyer-Konig and Zeller operators, max-product Favard–Szasz–Mirakjan operators, and max-product Baskakov operators.