Advertisement

Approximation by a Kantorovich–Shilkret Quasi-interpolation Neural Network Operator

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

Abstract

In this chapter we present multivariate basic approximation by a Kantorovich–Shilkret type quasi-interpolation neural network operator with respect to supremum norm. This is done with rates using the multivariate modulus of continuity. We approximate continuous and bounded functions on \( \mathbb {R}^{N}\), \(N\in \mathbb {N}\). When they are additionally uniformly continuous we derive pointwise and uniform convergences. It follows (Anastassiou, Quantitative approximation by a Kantorovich–Shilkret quasi-interpolation neural network operator (2018) [3]).

References

  1. 1.
    M. Abramowitz, I.A. Stegun (eds.), Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables (Dover Publication, New York, 1972)zbMATHGoogle Scholar
  2. 2.
    G.A. Anastassiou, Univariate error function based neural network approximation. Indian J. Math. 57(2), 243–291 (2015)MathSciNetzbMATHGoogle Scholar
  3. 3.
    G. Anastassiou, Quantitative approximation by a Kantorovich-Shilkret quasi-interpolation neural network operator (2018). submittedGoogle Scholar
  4. 4.
    L.C. Andrews, Special Functions of Mathematics for Engineers, 2nd edn. (Mc Graw-Hill, New York, 1992)Google Scholar
  5. 5.
    I.S. Haykin, Neural Networks: A Comprehensive Foundation, 2nd edn. (Prentice Hall, New York, 1998)zbMATHGoogle Scholar
  6. 6.
    W. McCulloch, W. Pitts, A logical calculus of the ideas immanent in nervous activity. Bull. Math. Biophys. 7, 115–133 (1943)MathSciNetCrossRefGoogle Scholar
  7. 7.
    T.M. Mitchell, Machine Learning (WCB-McGraw-Hill, New York, 1997)zbMATHGoogle Scholar
  8. 8.
    Niel Shilkret, Maxitive measure and integration. Indag. Math. 33, 109–116 (1971)MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  1. 1.Department of Mathematical SciencesUniversity of MemphisMemphisUSA

Personalised recommendations