Ukrainian Mathematical Journal

, Volume 68, Issue 12, pp 1874–1883 | Cite as

On the Uniqueness of Representation by Linear Superpositions

  • V. E. Ismailov

Let Q be a set such that every function on Q can be represented by linear superpositions. This representation is, in general, not unique. However, for some sets, it may be unique provided that the initial values of the representing functions are prescribed at some point of Q. We study the properties of these sets.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. k1.
    B. L. Fridman, “An improvement in the smoothness of functions in A. N. Kolmogorov’s theorem on superpositions,” Dokl. Akad. Nauk SSSR, 177, 1019–1022 (1967).MathSciNetGoogle Scholar
  2. 2.
    V. E. Ismailov, “On the representation by linear superpositions,” J. Approx. Theory, 151, 113–125 (2008).MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    S. Ya. Khavinson, “Best approximation by linear superpositions (approximate nomography),” Transl. Math. Monogr., Amer. Math. Soc., Providence, RI, 159 (1997).Google Scholar
  4. 4.
    A. Klopotowski, M. G. Nadkarni, and K. P. S. Bhaskara Rao, “When is f(x 1 ,x 2 , . . . ,x n) = u 1(x 1) + u 2(x 2) + . . . + u n(x n)?,” Proc. Indian Acad. Sci. Math. Sci., 113, 77–86 (2003).MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    A. Klopotowski, M. G. Nadkarni, and K. P. S. Bhaskara Rao, “Geometry of good sets in n-fold Cartesian product,” Proc. Indian Acad. Sci. Math. Sci., 114, 181–197 (2004).MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    A. N. Kolmogorov, “On the representation of continuous functions of many variables by the superposition of continuous functions of one variable and an addition,” Dokl. Akad. Nauk SSSR, 114, 953–956 (1957).MathSciNetzbMATHGoogle Scholar
  7. 7.
    G. G. Lorentz, “Metric entropy, widths, and superpositions of functions,” Amer. Math. Monthly, 69, 469–485 (1962).MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    D. E. Marshall and A. G. O’Farrell, “Approximation by a sum of two algebras. The lightning bolt principle,” J. Funct. Anal., 52, 353–368 (1983).MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    D. A. Sprecher, “An improvement in the superposition theorem of Kolmogorov,” J. Math. Anal. Appl., 38, 208–213 (1972).MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Y. Sternfeld, “Uniformly separating families of functions,” Israel J. Math., 29, 61–91 (1978).MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Y. Sternfeld, “Dimension, superposition of functions, and separation of points, in compact metric spaces,” Israel J. Math., 50, 13–53 (1985).MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    A. G. Vitushkin and G. M. Henkin, “Linear superpositions of functions,” Uspekhi Mat. Nauk, 22, No. 1, 77–124 (1967).MathSciNetzbMATHGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC 2017

Authors and Affiliations

  • V. E. Ismailov
    • 1
  1. 1.Institute of Mathematics and MechanicsAzerbaijan National Academy of SciencesBakuAzerbaijan

Personalised recommendations