Advertisement

Fractional Approximation by Riemann–Liouville Fractional Derivatives

  • George A. AnastassiouEmail author
Chapter
Part of the Studies in Computational Intelligence book series (SCI, volume 886)

Abstract

In this chapter we study quantitatively with rates the pointwise convergence of a sequence of positive sublinear operators to the unit operator over continuous functions. This takes place under low order smoothness, less than one, of the approximated function and it is expressed via the left and right Riemann–Liouville fractional derivatives of it. The derived related inequalities in their right hand sides contain the moduli of continuity of these fractional derivatives and they are of Shisha-Mond type.

References

  1. 1.
    F.B. Adda, J. Cresson, Fractional differentiation equations and the Schrödinger equation. Appl. Math. Comput. 161, 323–345 (2005)MathSciNetzbMATHGoogle Scholar
  2. 2.
    G.A. Anastassiou, Moments in Probability and Approximation Theory, vol. 287, Pitman Research Notes in Mathematics (Longman Scientific & Technical, Harlow, 1993)zbMATHGoogle Scholar
  3. 3.
    G.A. Anastassiou, Quantitative Approximations (CRC Press, Boca Raton, 2001)zbMATHGoogle Scholar
  4. 4.
    G.A. Anastassiou, Intelligent Mathematics: Computational Analysis (Springer, Heidelberg, 2011)CrossRefGoogle Scholar
  5. 5.
    G.A. Anastassiou, Nonlinearity: Ordinary and Fractional Approximations by Sublinear and Max-Product Operators (Springer, Heidelberg, 2018)CrossRefGoogle Scholar
  6. 6.
    G.A. Anastassiou, Approximation with Riemann-Liouville Fractional Derivatives, Studia Mathematica Babes Bolyai (2019)Google Scholar
  7. 7.
    B. Bede, L. Coroianu, S. Gal, Approximation by Max-Product Type Operators (Springer, Heidelberg, 2016)CrossRefGoogle Scholar
  8. 8.
    G. Choquet, Theory of capacities. Ann. Inst. Fourier (Grenoble) 5, 131–295 (1954)MathSciNetCrossRefGoogle Scholar
  9. 9.
    D. Denneberg, Non-additive Measure and Integral (Kluwer, Dordrecht, 1994)CrossRefGoogle Scholar
  10. 10.
    P.P. Korovkin, Linear Operators and Approximation Theory (Hindustan Publishing Corporation, Delhi, 1960)Google Scholar
  11. 11.
    I. Podlubny, Fractional Differentiation Equations (Academic, San Diego, 1999)zbMATHGoogle Scholar
  12. 12.
    D. Schmeidler, Integral representation without additivity. Proc. Am. Math. Soc. 97, 255–261 (1986)MathSciNetCrossRefGoogle Scholar
  13. 13.
    O. Shisha, B. Mond, The degree of convergence of sequences of linear positive operators. Nat. Acad. of Sci. U.S. 60, 1196–1200 (1968)MathSciNetCrossRefGoogle Scholar
  14. 14.
    Z. Wang, G.J. Klir, Generalized Measure Theory (Springer, New York, 2009)CrossRefGoogle Scholar

Copyright information

© The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Switzerland AG 2020

Authors and Affiliations

  1. 1.Department of Mathematical SciencesUniversity of MemphisMemphisUSA

Personalised recommendations