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.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
References
Agratini, O.: Approximation by Linear Operators. University Press of Cluj, Cluj-Napoca (2000, in Romanian)
Altomare, F., Campiti, M.: Korovkin-Type Approximation Theory and its Applications. de Gruyter Studies in Mathematics, Walter de Gruyter & Co., vol. 17. Berlin/New York (1994)
Aral, A., Gupta, V., Agarwal, R.P.: Applications of q-Calculus in Approximation Theory. Springer, New York (2013)
Bede, B., Coroianu L., Gal, S.G.: Approximation and shape preserving properties of the Bernstein operator of max-product kind. Int. J. Math. Math. Sci. 2009. Article ID 590589, 26 pp. (2009). doi:10.1155/2009/590589
Bede, B., Coroianu, L., Gal, S.G.: Approximation and shape preserving properties of the nonlinear Favard-Szàsz-Mirakjan operators of max-product kind. Filomat 24 (3), 55–72 (2010)
Bede, B., Coroianu, L., Gal, S.G.: Approximation and shape preserving properties of the nonlinear Baskakov operator of max-product kind. Stud. Univ. Babes-Bolyai Math. LV (4), 193–218 (2010)
Bede, B., Coroianu, L., Gal, S.G.: Approximation and shape preserving properties of the nonlinear Meyer-König and Zeller operator of max-product kind. Numer. Funct. Anal. Optim. 31 (3), 232–253 (2010)
Bernstein, S.N.: Démonstration du théorém de Weierstrass fondeé sur le calcul des probabilités. Commun. Soc. Math. Kharkov 13, 1–2 (1912/1913)
Coroianu, L., Gal, S.G., Opris, B.D., Trifa, S.: Feller’s scheme in approximation by nonlinear possibilistic integral operators (2016, submitted)
Feller, W.: An Introduction to Probability Theory and Its Applications, vol. II. Wiley, New York (1966)
Gal, S.G.: Shape-Preserving Approximation by Real and Complex Polynomials. Birkhäuser, Boston/Basel/Berlin (2008)
Gal, S.G.: A possibilistic approach of the max-product Bernstein operators. Results Math. 65 (3–4), 453–462 (2014)
Gal, S.G.: Approximation by Choquet integral operators. Ann. Mat. Pura Appl. (online access). doi:10.1007/s10231-015-0495-x
Gupta, V., Agarwal, R.P.: Convergence Estimates in Approximation Theory. Springer, New York (2014)
Levasseur, K.N.: A probabilistic proof of the Weierstrass approximation theorem. Am. Math. Mon. 91 (4), 249–250 (1984)
Author information
Authors and Affiliations
Rights and permissions
Copyright information
© 2016 Springer International Publishing Switzerland
About this chapter
Cite this chapter
Bede, B., Coroianu, L., Gal, S.G. (2016). Possibilistic Approaches of the Max-Product Type Operators. In: Approximation by Max-Product Type Operators. Springer, Cham. https://doi.org/10.1007/978-3-319-34189-7_10
Download citation
DOI: https://doi.org/10.1007/978-3-319-34189-7_10
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-319-34188-0
Online ISBN: 978-3-319-34189-7
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)