Possibilistic Approaches of the Max-Product Type Operators

Barnabás Bede, Lucian Coroianu and Sorin G. Gal
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Barnabás Bede: DigiPen Institute of Technology, Department of Mathematics
Lucian Coroianu: University of Oradea, Department of Mathematics and Computer Science
Sorin G. Gal: University of Oradea, Department of Mathematics and Computer Science

Chapter Chapter 10 in Approximation by Max-Product Type Operators, 2016, pp 407-428 from Springer

Abstract: 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-König and Zeller operators, max-product Favard–Szász–Mirakjan operators, and max-product Baskakov operators.

Keywords: Approximation Operator; Fuzzy Variable; Bernstein Polynomial; Possibility Theory; Possibilistic Approach (search for similar items in EconPapers)
Date: 2016
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Persistent link: https://EconPapers.repec.org/RePEc:spr:sprchp:978-3-319-34189-7_10

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DOI: 10.1007/978-3-319-34189-7_10

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