Chebyshev approximations by least squares method

V. I. Zorkaltsev ; E. Gubiy

The Bulletin of Irkutsk State University. Series Mathematics, Tome 33 (2020), pp. 3-19

Résumé

We consider the problem of linear approximation in the form of the minimization problem of the weighted Chebyshev norm, and that in the form of the minimization problem of the weighted Euclidean norm of the residual vector. We give an algorithm for the unambiguous calculation in all cases of the Chebyshev approximation that does not require the Haar condition. The theorem obtained indicates that any approximation by the method of least squares (for any set of positive weight coefficients in the minimized Euclidean norm) can be represented as the Chebyshev approximation based on the choice of weight coefficients in the Chebyshev norm. As an example we consider the approximation of the reduced fuel supply costs of a settlement based on an energy plantation as a quadratic dependence on volumes of reserved funds.

Détail

Keywords: Chebyshev approximation, least squares method, reliability of fuel supply from energy plantations.
Mots-clés : Haar condition

Bibliographie

[1] Burov V. N., “Some effective methods of solving the Chebyshev problem on best approximation”, Izv. Vyssh. Uchebn. Zaved. Mat., 1957, no. 1, 67–79 (in Russian) | Zbl

[2] Gubiy E. V., Zorkaltsev V. I., Energy plantation efficiency, Nauka Publ., Novosibirsk, 2018, 96 pp. (in Russian)

[3] Dem'yanov V. F., Malozemov V. N., Introduction to Minimax, Nauka Publ, M., 1972, 368 pp. (in Russian) | MR

[4] Dolganov R. L., “Chebyshev approximation by asymptotically convex families of functions”, Izv. Vyssh. Uchebn. Zaved. Mat., 1972, no. 7, 35–41 (in Russian) | MR | Zbl

[5] Zorkal'tsev V. I., “Interior point method: history and prospects”, Computational Mathematics and Mathematical Physics, 59 (2019), 1597–1612 | DOI | DOI | MR | Zbl

[6] Zorkal'tsev V. I., “Octahedral and Euclidean projections of a point to a linear manifold”, Proceedings of the Steklov Institute of Mathematics, 284 (2014), 185–197 | DOI | MR

[7] Zorkaltsev V. I., “Chebyshev approximations can dispense with the Haar condition”, Proceedings of the International Symposium dedicated to the 100th anniversary of mathematical education in Eastern Siberia and the 80th anniversary of the birth of Professor O. V. Vasiliev “Dynamical systems, optimal control and mathematical modeling”, Irkutsk State University Publ., Irkutsk, 2019, 29–33 (in Russian)

[8] Collatz L., Krabs W., Approximationstheorie. Tschebyscheffsche Approximation mit Anwendungen, B.G. Teubner, Stuttgart, 1973, 208 pp. | MR | Zbl

[9] Kolmogorov A. N., “A remark on the polynomials of P.L. Chebyshev deviating the least from a given function”, Uspekhi Mat. Nauk, 3:1(23) (1948), 216–221 (in Russian) | MR | Zbl

[10] Malozemov V. N., “The best uniform approximation of functions of several arguments”, USSR Computational Mathematics and Mathematical Physics, 10:3 (1970), 28–43 | DOI | MR | Zbl

[11] Nikol'skii S. M., Approximation of functions of several variables and imbedding theorems, Springer, Berlin–Heidelberg, 1975, 420 pp. | DOI | MR

[12] Rockafellar R., Convex analysis, Princeton University Press, Princeton–New Jersey, 1970, 472 pp. | MR | Zbl

[13] Tikhomirov V. M., “Some problems in approximation theory”, Mathematical Notes of the Academy of Sciences of the USSR, 9 (1971), 343–350 | DOI | MR

[14] Chebyshev P. L., “Questions on smallest quantities connected with the approximate representation of functions”, Collected works, v. 2, AN SSSP Publ., M.–L., 1947, 151–235 (in Russian)

[15] Haare A., “Die Minkowskische Geometrie und die Annäherung an stetige Funktionen”, Math. Ann., 78:36 (1918), 415–127 | MR