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].
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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 publication
G. Anastassiou, Approximation by multivariate sublinear and max-product operators under convexity. Demonstr. Math. 51, 85–105 (2018)
G. Anastassiou, Approximation by sublinear and max-product operators using convexity (2018). Submitted for publication
G. Anastassiou, Multivariate and convex approximation by Choquet integrals, RACSAM (2018). Accepted for publication
G. Choquet, Theory of capacities. Ann. Inst. Fourier (Grenoble) 5, 131–295 (1954)
D. Denneberg, Non-additive Measure and Integral (Kluwer, Dordrecht, 1994)
D. Dubois, H. Prade, Possibility Theory (Plenum Press, New York, 1988)
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 pp
S. Gal, S. Trifa, Quantitative estimates in uniform and pointwise approximation by Bernstein-Durrmeyer-Choquet operators. Carpathian J. Math. 33(1), 49–58 (2017)
D. Schmeidler, Integral representation without additivity. Proc. Am. Math. Soc. 97, 255–261 (1986)
D. Schmeidler, Subjective probability and expected utility without additivity. Econometrica 57, 571–587 (1989)
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. 307–317
Z. Wang, G.J. Klir, Generalized Measure Theory (Springer, New York, 2009)
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Anastassiou, G.A. (2019). Multivariate and Convex Quantitative Approximation by Choquet Integrals. In: Ordinary and Fractional Approximation by Non-additive Integrals: Choquet, Shilkret and Sugeno Integral Approximators. Studies in Systems, Decision and Control, vol 190. Springer, Cham. https://doi.org/10.1007/978-3-030-04287-5_8
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DOI: https://doi.org/10.1007/978-3-030-04287-5_8
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