Gaussian Functions Combined with Kolmogorov’s Theorem as Applied to Approximation of Functions of Several Variables

Abstract

A special class of approximations of continuous functions of several variables on the unit coordinate cube is investigated. The class is constructed using Kolmogorov’s theorem stating that functions of the indicated type can be represented as a finite superposition of continuous single-variable functions and another result on the approximation of such functions by linear combinations of quadratic exponentials (also known as Gaussian functions). The effectiveness of such a representation is based on the author’s previously proved assertion that the Mexican hat mother wavelet on any fixed bounded interval can be approximated as accurately as desired by a linear combination of two Gaussian functions. It is proved that the class of approximations under study is dense everywhere in the class of continuous multivariable functions on the coordinate cube. For the case of continuous functions of two variables, numerical results are presented that confirm the effectiveness of approximations of the studied class.

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REFERENCES

  1. 1

    K. L. Teo, C. J. Goh, and K. H. Wong, A Unified Computational Approach to Optimal Control Problems (Longman Scientific and Technical, New York, 1991).

    Google Scholar 

  2. 2

    A. V. Chernov, “On approximate solution of free-time optimal control problems,” Vestn. Nizhegorod. Gos. Univ. im. N.I. Lobachevskogo, No. 6 (1), 107–114 (2012).

    Google Scholar 

  3. 3

    A. V. Chernov, “Smooth finite-dimensional approximations of distributed optimization problems via control discretization,” Comput. Math. Math. Phys. 53 (12), 1839–1852 (2013).

    MathSciNet  Article  Google Scholar 

  4. 4

    A. V. Chernov, “On applicability of control parametrization technique to solving distributed optimization problems,” Vestn. Udmurt. Univ. Mat. Mekh. Komp’yut. Nauki 24 (1), 102–117 (2014).

    Article  Google Scholar 

  5. 5

    A. V. Chernov, “On the smoothness of an approximated optimization problem for a Goursat–Darboux system on a varied domain,” Tr. Inst. Mat. Mekh. Ural. Otd. Ross. Akad. Nauk 20 (1), 305–321 (2014).

    Google Scholar 

  6. 6

    A. V. Chernov, “On piecewise constant approximation in distributed optimization problems,” Tr. Inst. Mat. Mekh. Ural. Otd. Ross. Akad. Nauk 21 (1), 264–279 (2015).

    MathSciNet  Google Scholar 

  7. 7

    A. N. Kolmogorov, “On the representation of continuous functions of several variables by superposition of continuous functions of one variable and addition,” Dokl. Akad. Nauk SSSR 114 (5), 953–956 (1957).

    MathSciNet  MATH  Google Scholar 

  8. 8

    D. A. Sprecher, “On the structure of continuous functions of several variables,” Trans. Am. Math. Soc. 115, 340–355 (1965).

    MathSciNet  Article  Google Scholar 

  9. 9

    A. Yu. Golubkov, “The tracing of external and internal representation functions of continuous functions of several variables by superposition of continuous functions of one variable,” Fundam. Prikl. Mat. 8 (1), 27–38 (2002).

    MathSciNet  MATH  Google Scholar 

  10. 10

    E. Yu. Butyrskii, I. A. Kuvaldin, and V. P. Chalkin, “Approximation of multidimensional functions,” Nauchn. Priborostr. 20 (2), 82–92 (2010).

    Google Scholar 

  11. 11

    R. Hecht-Nielsen, “Kolmogorov’s mapping neural network existence theorem,” in Proceedings of the First IEEE Annual International Joint Conference on Neural Networks, San-Diego (New York, NY, USA, 1987), Vol. 3, pp. 11–13.

  12. 12

    D. V. Alexeev, “Neural network approximation of several variable functions,” J. Math. Sci. 168 (1), 5–13 (2010).

    MathSciNet  Article  Google Scholar 

  13. 13

    A. N. Gorban’, “A generalized approximation theorem and an exact representation of polynomials of several variables by superpositions of polynomials of one variable,” Russ. Math. 42 (5), 4–7 (1998).

    MathSciNet  MATH  Google Scholar 

  14. 14

    V. E. Ismailov, “Approximation by sums of ridge functions with fixed directions,” St. Petersburg Math. J. 28 (6), 741–772 (2017).

    MathSciNet  Article  Google Scholar 

  15. 15

    Yu. N. Kul’chin, I. V. Denisov, A. V. Panov, and N. A. Rybal’chenko, “Application of perceptrons for nonlinear reconstruction tomography,” Probl. Upr., No. 4, 59–63 (2006).

  16. 16

    A. V. Chernov, “On using Gaussian functions with varied parameters for approximation of functions of one variable on a finite segment,” Vestn. Udmurt. Univ. Mat. Mekh. Komp’yut. Nauki 27 (2), 267–282 (2017).

    Article  Google Scholar 

  17. 17

    A. V. Chernov, “On the application of Gaussian functions for discretization of optimal control problems,” Vestn. Udmurt. Univ. Mat. Mekh. Komp’yut. Nauki 27 (4), 558–575 (2017).

    Article  Google Scholar 

  18. 18

    V. Maz’ya and G. Schmidt, Approximate Approximations (Am. Math. Soc., Providence, R.I., 2007).

    Google Scholar 

  19. 19

    S. D. Riemenschneider and N. Sivakumar, “Cardinal interpolation by Gaussian functions: A survey,” J. Anal. 8, 157–178 (2000).

    MathSciNet  MATH  Google Scholar 

  20. 20

    Luh Lin-Tian, “The shape parameter in the Gaussian function,” Comput. Math. Appl. 63, 687–694 (2012).

    MathSciNet  Article  Google Scholar 

  21. 21

    T. Hangelbroek, W. Madych, F. Narcowich, and J. D. Ward, “Cardinal interpolation with Gaussian kernels,” J. Fourier Anal. Appl. 18 (1), 67–86 (2012).

    MathSciNet  Article  Google Scholar 

  22. 22

    K. Hamm, “Approximation rates for interpolation of Sobolev functions via Gaussians and allied functions,” J. Approx. Theory 189, 101–122 (2015).

    MathSciNet  Article  Google Scholar 

  23. 23

    M. Griebel, M. Schneider, and C. Zenger, “A combination technique for the solution of sparse grid problems,” Proceedings of the IMACS International Symposium on Iterative Methods in Linear Algebra, Belgium, April 2–4,1991 (North-Holland, Amsterdam, 1992), pp. 263–281.

  24. 24

    E. Georgoulis, J. Levesley, and F. Subhan, “Multilevel sparse kernel-based interpolation,” SIAM J. Sci. Comput. 35 (2), A815–A831 (2013).

    MathSciNet  Article  Google Scholar 

  25. 25

    J. J. More and D. J. Thuente, “Line search algorithms with guaranteed sufficient decrease,” ACM Trans. Math. Software 20, 286–307 (1994).

    MathSciNet  Article  Google Scholar 

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Correspondence to A. V. Chernov.

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Translated by I. Ruzanova

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Chernov, A.V. Gaussian Functions Combined with Kolmogorov’s Theorem as Applied to Approximation of Functions of Several Variables. Comput. Math. and Math. Phys. 60, 766–782 (2020). https://doi.org/10.1134/S0965542520050073

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Keywords:

  • approximation of continuous functions of several variables
  • Gaussian functions
  • quadratic exponentials
  • Kolmogorov’s theorem