Approximation with Rates by Shift Invariant Multivariate Sublinear-Choquet Operators

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


A very general multivariate positive sublinear Choquet integral type operator is given through a convolution-like iteration of another multivariate general positive sublinear operator with a multivariate 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 multivariate specialized operators are presented fulfilling all the above properties, the higher order of multivariate approximation of these operators is also studied. It follows [3].


  1. 1.
    G.A. Anastassiou, High order approximation by multivariate shift-invariant convolution type operators, in Computers and Mathematics with Applications, ed. by G.A. Anastassiou. Special issue on Computational Methods in Analysis, vol. 48 (2004), pp. 1245–1261MathSciNetCrossRefGoogle Scholar
  2. 2.
    G.A. Anastassiou, Intelligent Mathematics: Computational Analysis (Springer, Heidelberg, 2011)CrossRefGoogle Scholar
  3. 3.
    G.A. Anastassiou, Quantitative approximation by shift invariant multivariate sublinear-Choquet operators. J. Appl. Anal. (2018). AcceptedGoogle Scholar
  4. 4.
    G.A. Anastassiou, H.H. Gonska, On some shift invariant integral operators, multivariate case, in Proceedings of International Conference on Approximation, Probability, and Related Fields, U.C.S.B., Santa Barbara, CA, 20–22 May 1993, ed. by G.A. Anastassiou, S.T. Rachev (Plenum Press, New York, 1993), pp. 41–64CrossRefGoogle Scholar
  5. 5.
    G.A. Anastassiou, S. Gal, Approximation Theory (Birkhauser, Boston, 2000)CrossRefGoogle Scholar
  6. 6.
    G. Choquet, Theory of capacities. Ann. Inst. Fourier (Grenoble) 5, 131–295 (1954)MathSciNetCrossRefGoogle Scholar
  7. 7.
    D. Denneberg, Non-additive Measure and Integral (Kluwer, Dordrecht, 1994)CrossRefGoogle Scholar
  8. 8.
    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

Personalised recommendations