Conformable Fractional Approximation by Choquet Integrals
Here we present the conformable fractional 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 conformable fractional rate of the convergence of the well-known Bernstein–Kantorovich–Choquet and Bernstein–Durrweyer–Choquet polynomial Choquet-integral operators. Then we study in the fractional sense the very general comonotonic positive sublinear operators based on the representation theorem of Schmeidler . We continue with the conformable fractional approximation by the very general direct Choquet-integral form positive sublinear operators. The case of convexity is also studied thoroughly and the estimates become much simpler. All approximations are given via inequalities involving the modulus of continuity of the approximated function’s higher order conformable fractional derivative. It follows .
- 1.M. Abu Hammad, R. Khalil, Abel’s formula and Wronskian for conformable fractional differential equations. Int. J. Differ. Equ. Appl. 13(3), 177–183 (2014)Google Scholar
- 2.G. Anastassiou, Moments in Probability and Approximation Theory. Pitman Research Notes in Mathematics Series (Longman Group UK, New York, 1993)Google Scholar
- 4.G. Anastassiou, Conformable fractional approximations by max-product operators using convexity. Arab. J. Math. (2018). Accepted for publicationGoogle Scholar
- 5.G.A. Anastassiou, Conformable fractional approximation by Choquet integrals. J. Comput. Anal. Appl. (2018). AcceptedGoogle Scholar
- 9.S. Gal, Uniform and pointwise quantitative approximation by Kantorovich–Choquet type integral operators with respect to monotone and submodular set functions. Mediterr. J. Math. 14(5), 12 pp. (2017). Art. 205Google Scholar
- 14.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