Ordinary and Fractional Approximation by Non-additive Integrals: Choquet, Shilkret and Sugeno Integral Approximators pp 55-72 | Cite as

# Approximation with Rates by Shift Invariant Univariate Sublinear-Choquet Operators

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## Abstract

A very general positive sublinear Choquet integral type operator is given through a convolution-like iteration of another general positive sublinear operator with a scaling type function. For it sufficient conditions are given for shift invariance, preservation of global smoothness, convergence to the unit with rates. Furthermore, two examples of very general specialized operators are presented fulfilling all the above properties, the higher order of approximation of these operators is also studied. It follows [3].

## References

- 1.G.A. Anastassiou, High order approximation by univariate shift-invariant integral operators, in
*Nonlinear Analysis and Applications, 2 Volumes*, vol. I, ed. by R. Agarwal, D. O’Regan (Kluwer, Dordrecht, 2003), pp. 141–164Google Scholar - 2.G.A. Anastassiou,
*Intelligent Mathematics: Computational Analysis*(Springer, Heidelberg, 2011)CrossRefGoogle Scholar - 3.G.A. Anastassiou, Approximation by shift invariant univariate sublinear-Choquet operators. Indian J. Math. (2018). AcceptedGoogle Scholar
- 4.G.A. Anastassiou, H.H. Gonska, On some shift invariant integral operators, univariate case. Ann. Polon. Math.
**LXI**(3), 225–243 (1995)MathSciNetCrossRefGoogle Scholar - 5.G.A. Anastassiou, S. Gal,
*Approximation Theory*(Birkhauser, Boston, 2000)CrossRefGoogle Scholar - 6.G. Choquet, Theory of capacities. Ann. Inst. Fourier (Grenoble)
**5**, 131–295 (1954)MathSciNetCrossRefGoogle Scholar - 7.D. Denneberg,
*Non-additive Measure and Integral*(Kluwer, Dordrecht, 1994)CrossRefGoogle Scholar - 8.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 - 9.Z. Wang, G.J. Klir,
*Generalized Measure Theory*(Springer, New York, 2009)CrossRefGoogle Scholar

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