Caputo Fractional Approximation Using Positive Sublinear Operators

  • George A. Anastassiou
Part of the Studies in Systems, Decision and Control book series (SSDC, volume 147)


Here we consider the approximation of functions by sublinear positive operators with applications to a big variety of Max-Product operators under Caputo fractional differentiability. Our study is based on our general fractional results about positive sublinear operators. We produce Jackson type inequalities under simple initial conditions. So our approach is quantitative by producing inequalities with their right hand sides involving the modulus of continuity of fractional derivative of the function under approximation. It follows Anastassiou, Caputo fractional approximation by sublinear operators (2017, submitted) [4].


  1. 1.
    G. Anastassiou, On right fractional calculus. Chaos, Solitons Fractals 42, 365–376 (2009)MathSciNetCrossRefGoogle Scholar
  2. 2.
    G. Anastassiou, Fractional Korovkin theory. Chaos, Solitons Fractals 42, 2080–2094 (2009)MathSciNetCrossRefGoogle Scholar
  3. 3.
    G. Anastassiou, Approximation by Sublinear Operators (2017, submitted)Google Scholar
  4. 4.
    G. Anastassiou, Caputo Fractional Approximation by Sublinear Operators (2017, submitted)Google Scholar
  5. 5.
    G. Anastassiou, L. Coroianu, S. Gal, Approximation by a nonlinear Cardaliaguet-Euvrard neural network operator of max-product kind. J. Comput. Anal. Appl. 12(2), 396–406 (2010)MathSciNetzbMATHGoogle Scholar
  6. 6.
    B. Bede, L. Coroianu, S. Gal, Approximation by Max-Product Type Operators (Springer, Heidelberg, 2016)CrossRefGoogle Scholar
  7. 7.
    R.A. DeVore, G.G. Lorentz, Constructive Approximation (Springer, Berlin, 1993)CrossRefGoogle Scholar
  8. 8.
    K. Diethelm, The Analysis of Fractional Differential Equations (Springer, Heidelberg, 2010)CrossRefGoogle Scholar
  9. 9.
    A.M.A. El-Sayed, M. Gaber, On the finite Caputo and finite Riesz derivatives. Electron. J. Theor. Phys. 3(12), 81–95 (2006)Google Scholar
  10. 10.
    L. Fejér, Über Interpolation, Göttingen Nachrichten, (1916), pp. 66–91Google Scholar
  11. 11.
    G.S. Frederico, D.F.M. Torres, Fractional optimal control in the sense of Caputo and the fractional Noether’s theorem. Int. Math. Forum 3(10), 479–493 (2008)MathSciNetzbMATHGoogle Scholar
  12. 12.
    G.G. Lorentz, Bernstein Polynomials, 2nd edn. (Chelsea Publishing Company, New York, 1986)zbMATHGoogle Scholar
  13. 13.
    T. Popoviciu, Sur l’approximation de fonctions convexes d’order superieur. Mathematica (Cluj) 10, 49–54 (1935)zbMATHGoogle Scholar
  14. 14.
    S.G. Samko, A.A. Kilbas and O.I. Marichev, Fractional Integrals and Derivatives, Theory and Applications, (Gordon and Breach, Amsterdam, 1993) [English translation from the Russian, Integrals and Derivatives of Fractional Order and Some of Their Applications (Nauka i Tekhnika, Minsk, 1987)]Google Scholar

Copyright information

© Springer International Publishing AG, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Department of Mathematical SciencesUniversity of MemphisMemphisUSA

Personalised recommendations