On application of Gaussian kernels and Laplace functions combined with Kolmogorov's theorem for approximation of functions of several variables

A. V. Chernov

Izvestiya Instituta Matematiki i Informatiki Udmurtskogo Gosudarstvennogo Universiteta, Tome 63 (2024), pp. 114-131.

Résumé

A special class of approximations of measurable functions of several variables on the unit coordinate cube is investigated. The class is constructed on the base of Kolmogorov's theorem (in version by Sprecher–Golubkov) stating that a continuous function $f$ of several variables can be represented as a finite superposition of continuous single-variable functions — so called outer functions (which depend of $f$) and inner one $\Psi$ (which is independent of $f$ and is monotone). In the case of continuous functions $f$ the class under study is obtained with outer functions approximated by linear combinations of quadratic exponentials (also known as Gaussian functions or Gaussian kernels) and with the inner function $\Psi$ approximated by Laplace functions. As is known, a measurable function $f$ can be approximated by a continuous one (up to a set of small measure) with the help of classical Luzin's theorem. The effectiveness of such approach is based on assertions that, firstly, 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, and, secondly, that a continuous monotone function on such an interval can be approximated as accurately as desired by a linear combination of translations and dilations of the Laplace integral (in other words, Laplace 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 and piecewise continuous functions of two variables, numerical results are presented that confirm the effectiveness of approximations of the studied class.

Détail

Keywords: approximation of functions of several variables, Kolmogorov's theorem, Gaussian function, Laplace integral, quadratic exponential

Bibliographie

[1] Chernov A.V., “Gaussian functions combined with Kolmogorov’s theorem as applied to approximation of functions of several variables”, Computational Mathematics and Mathematical Physics, 60:5 (2020), 766–782 | DOI | DOI | MR | Zbl

[2] Kolmogorov A.N., “On the representation of continuous functions of many variables by superposition of continuous functions of one variable and addition”, Doklady Akademii Nauk SSSR, 114:5 (1957), 953–956 (in Russian) | Zbl

[3] Sprecher D.A., “On the structure of continuous functions of several variables”, Transactions of the American Mathematical Society, 115 (1965), 340–355 | DOI | MR | Zbl

[4] Golubkov A.Yu., “The tracing of external and internal representation functions of continuous functions of several variables by superposition of continuous functions of one variable”, Fundamental’naya i Prikladnaya Matematika, 8:1 (2002), 27–38 (in Russian) | MR | Zbl

[5] Butyrsky Eu.Yu., Kuvaldin I.A., Chalkin V.P., “Multidimensional functions’ approximation”, Nauchnoe Priborostroenie, 20:2 (2010), 82–92 (in Russian) | DOI

[6] Chernov A.V., “On the application of Gaussian functions for discretization of optimal control problems”, Vestnik Udmurtskogo Universiteta. Matematika. Mekhanika. Komp’yuternye Nauki, 27:4 (2017), 558–575 (in Russian) | DOI | MR | Zbl

[7] Chernov A.V., “On uniform monotone approximation of continuous monotone functions with the help of translations and dilations of the Laplace integral”, Computational Mathematics and Mathematical Physics, 62:4 (2022), 564–580 | DOI | MR | Zbl

[8] Chernov A.V., “On monotone approximation of piecewise continuous monotone functions with the help of translations and dilations of the Laplace integral”, Izvestiya Instituta Matematiki i Informatiki Udmurtskogo Gosudarstvennogo Universiteta, 61 (2023), 187–205 (in Russian) | DOI | Zbl

[9] Maz’ya V., Schmidt G., Approximate approximations, American Mathematical Society, Providence, RI, 2007 | DOI | MR | Zbl

[10] Riemenschneider S.D., Sivakumar N., “Cardinal interpolation by Gaussian functions: A survey”, The Journal of Analysis, 8 (2000), 157–178 | MR | Zbl

[11] Luh Lin-Tian, “The shape parameter in the Gaussian function”, Computers and Mathematics with Applications, 63:3 (2012), 687–694 | DOI | MR | Zbl

[12] Hangelbroek T., Madych W., Narcowich F., Ward J.D., “Cardinal interpolation with Gaussian kernels”, Journal of Fourier Analysis and Applications, 18:1 (2012), 67–86 | DOI | MR | Zbl

[13] Hamm K., “Approximation rates for interpolation of Sobolev functions via Gaussians and allied functions”, Journal of Approximation Theory, 189 (2015), 101–122 | DOI | MR | Zbl

[14] Griebel M., Schneider M., Zenger C., “A combination technique for the solution of sparse grid problems”, Iterative methods in linear algebra, Proceedings of the IMACS international symposium (Brussels, Belgium, 2–4 April, 1991), North-Holland, Amsterdam, 1992, 263–281 | MR | Zbl

[15] Georgoulis E.H., Levesley J., Subhan F., “Multilevel sparse kernel-based interpolation”, SIAM Journal on Scientific Computing, 35:2 (2013), A815–A831 | DOI | MR | Zbl

[16] Vulikh B.Z., A brief course of the theory of real variable functions (introduction to the theory of integral), Nauka, Moscow, 1973

[17] Moré J.J., Thuente D.J., “Line search algorithms with guaranteed sufficient decrease”, ACM Transactions on Mathematical Software, 20:3 (1994), 286–307 | DOI | MR | Zbl