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Multivariate and Convex Quantitative Approximation by Choquet Integrals

  • George A. Anastassiou
Chapter
Part of the Studies in Systems, Decision and Control book series (SSDC, volume 190)

Abstract

Here we consider the quantitative approximation of positive sublinear operators to the unit operator. These are given a precise Choquet integral interpretation. Initially we start with the study of the rate of the convergence of the well-known Bernstein–Kantorovich–Choquet and Bernstein–Durrweyer–Choquet polynomial Choquet-integral operators. We introduce also their multivariate analogs. Then we study the very general comonotonic positive sublinear operators based on the representation theorem of Schmeidler [10]. We finish with the approximation by the very general direct Choquet-integral form positive sublinear operators. All approximations are given via inequalities involving the modulus of continuity of the approximated function or its higher order derivative. We derive univariate and multivariate results without or with convexity assumptions. In the latter case estimates become very elegant and brief. It follows [4].

References

  1. 1.
    G. Anastassiou, Approximation by multivariate sublinear and max-product operators, Revista De La real Academia De Ciencias exactas, Fisicas Y Naturales Serie A. Matematicas (RACSAM) (2017). Accepted for publicationGoogle Scholar
  2. 2.
    G. Anastassiou, Approximation by multivariate sublinear and max-product operators under convexity. Demonstr. Math. 51, 85–105 (2018)MathSciNetCrossRefGoogle Scholar
  3. 3.
    G. Anastassiou, Approximation by sublinear and max-product operators using convexity (2018). Submitted for publicationGoogle Scholar
  4. 4.
    G. Anastassiou, Multivariate and convex approximation by Choquet integrals, RACSAM (2018). Accepted for publicationGoogle Scholar
  5. 5.
    G. Choquet, Theory of capacities. Ann. Inst. Fourier (Grenoble) 5, 131–295 (1954)MathSciNetCrossRefGoogle Scholar
  6. 6.
    D. Denneberg, Non-additive Measure and Integral (Kluwer, Dordrecht, 1994)CrossRefGoogle Scholar
  7. 7.
    D. Dubois, H. Prade, Possibility Theory (Plenum Press, New York, 1988)CrossRefGoogle Scholar
  8. 8.
    S. Gal, Uniform and pointwise quantitative approximation by Kantorovich-Choquet type integral operators with respect to monotone and submodular set functions. Mediter. J. Math. 14(5), Art. 205 (2017), 12 ppGoogle Scholar
  9. 9.
    S. Gal, S. Trifa, Quantitative estimates in uniform and pointwise approximation by Bernstein-Durrmeyer-Choquet operators. Carpathian J. Math. 33(1), 49–58 (2017)MathSciNetzbMATHGoogle Scholar
  10. 10.
    D. Schmeidler, Integral representation without additivity. Proc. Am. Math. Soc. 97, 255–261 (1986)MathSciNetCrossRefGoogle Scholar
  11. 11.
    D. Schmeidler, Subjective probability and expected utility without additivity. Econometrica 57, 571–587 (1989)MathSciNetCrossRefGoogle Scholar
  12. 12.
    L.S. Shapley, A value for n-person games, in Contributions to the Theory of Games, ed. by H.W. Kuhn, A.W. Tucker. Annals of Mathematical Studies, vol. 28 (Princeton University Press, Princeton, 1953), pp. 307–317Google Scholar
  13. 13.
    Z. Wang, G.J. Klir, Generalized Measure Theory (Springer, New York, 2009)CrossRefGoogle Scholar

Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  1. 1.Department of Mathematical SciencesUniversity of MemphisMemphisUSA

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