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Possibilistic Approaches of the Max-Product Type Operators

In: Approximation by Max-Product Type Operators

Author

Listed:
  • 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)

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.

Suggested Citation

  • Barnabás Bede & Lucian Coroianu & Sorin G. Gal, 2016. "Possibilistic Approaches of the Max-Product Type Operators," Springer Books, in: Approximation by Max-Product Type Operators, chapter 0, pages 407-428, Springer.
    • Handle: RePEc:spr:sprchp:978-3-319-34189-7_10
    DOI: 10.1007/978-3-319-34189-7_10
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