On a numerical-analytical method for constructing extremal polynomials of a complex argument
https://doi.org/10.29235/1561-2430-2023-59-1-18-36
Abstract
This article is devoted to the development of a numerical-analytical method for constructing extremes in the Chebyshev norm polynomials, given on the square of the complex plane. The studied polynomials are a generalization of the classical Chebyshev polynomials of the first kind. In the complex case there are no classical Chebyshev alternance conditions, and the Kolmogorov criterion along with the Ivanov – Remez criterion are difficult to prove for establishing the extremality property of specific polynomials. On the basis of the subdifferential construction developed by the authors of the article the extremal polinomials on the squares of the complex plane are calculated in an explicit way. The basic research methods are the methods of functional and complex mathematical analysis, as well as the Maple 2021 computer mathematics system. Methods of function theory and some general results of optimization theory are also used.
Keywords
extremal polynomial,
Chebyshev polynomials of the first kind,
Banach space,
linear functional,
subgradient,
optimality criterion,
numerical-analytical solution
Supporting agencies: The research was carried out within the framework of the State Program of Scientific Research “Convergence-2025” (no. 20210494).
About the Authors
Yu. V. Trubnikov
Vitebsk State University named after P. M. Masherov
Belarus
Belarus
Yurii V. Trubnikov – Dr. Sc. (Physics and Mathema- tics), Professor, Professor of the Department of Engineering Physics
Vitebsk State University named after P. M. Masherov
M. M. Chernyavsky
Vitebsk State University named after P. M. Masherov
Belarus
Belarus
Mikhail M. Chernyavsky – Lecturer at the Department of Engineering Physics
Vitebsk State University named after P. M. Masherov
References
1. Trubnikov Yu. V. On approximate and exact polynomials of Chebyshev type in the complex domain. Tavricheskii vestnik informatiki i matematiki = Tauride Bulletin of Informatics and Mathematics, 2003, no. 2, pp. 45–56 (in Russian).
2. Dzyadyk V. K. Introduction to the Theory of Uniform Approximation of Functions by Polynomials. Moscow, Nauka Publ., 1977. 512 p. (in Russian).
3. Trubnikov Yu. V. Extremal Constructions in Nonsmooth Analysis and Operator Equations with Accretive Nonlinearities. Moscow, Astropress-XXI Publ., 2002. 256 p. (in Russian).
4. Ioffe A. D., Tikhomirov V. M. Theory of Extremal Problems. Moscow, Nauka Publ., 1974. 480 p. (in Russian).
5. Trubnikov Yu. V. Subdifferential and extreme constructions. Doklady akademii nauk BSSR [Reports of the Academy of Sciences of the BSSR], 1990, vol. 5, no. 6, pp. 505–507 (in Russian).
6. Trubnikov Yu. V. On one method for finding Chebyshev iterative parameters. Vestsі Natsyianal’nai akademіі navuk Belarusі. Seryia fіzіka-matematychnykh navuk = Proceedings of the National Academy of Sciences of Belarus. Physics and Mathematics series, 1999, no. 4, pp. 5–9 (in Russian).
7. Trubnikov Yu. V., Orekhova I. A., Syn Baiyi. Movement of the roots of extreme polynomials. Vesnik Vicebskaga dzjarzhawnaga universitjeta = Bulletin of Vitebsk State University, 2012, no. 3, pp. 5–14 (in Russian).
8. Krasnoselʼskii M. A., Lifshits E. A., Sobolev A. V. Positive Linear Systems. Moscow, Nauka Publ., 1985. 255 p. (in Russian).
9. Kozjakin V. S., Krasnoselʼski M. A. Some remarks on the method of minimal residues. Numerical Functional Analysis and Optimization, 1994, vol. 4, no. 3, pp. 211–239. https://doi.org/10.1080/01630568208816115
10. Zabrejko A. P., Zabrejko P. P. Chebyshev Polynomial Iterations and Approximate Solutions of Linear Operator Equations. Zeitschrift für Analysis und ihre Anwendungen, 1994, vol. 13, no. 4, pp. 667–681. https://doi.org/10.4171/zaa/484
11. Krasnoselʼski M. A., Vainikko G. M., Zabrejko P. P., Rutitskii Ya. B., Stetsenko V. Ya. Approximate Solution of Operator Equations. Moscow, Nauka Publ., 1969. 455 p. (in Russian).
12. Lebedev V. I. Functional Analysis and Computational Mathematics. Moscow, Fizmatlit Publ., 2000. 296 p. (in Russian).
13. Pashkovsky S. Computational Applications of Polynomials and Chebyshev Series. Moscow, Nauka Publ., 1983. 384 p. (in Russian).
14. Collatz L., Krabs W. Approximationstheorie. Wiesbaden, Vieweg+Teubner Verlag, 1973. 209 p. https://doi.org/10.1007/978-3-322-94885-4
15. Chernyavsky М. М., Trubnikov Yu. V. On a numerical method for finding an extremal polynomial of the seventh degree defined on a square of the complex plan. Materialy 74 Regional’noi nauchno-prakticheskoi konferencii prepodavatelei, nauchnyh sotrudnikov i aspirantov «Nauka – obrazovaniyu, proizvodstvu, ekonomike» [Materials of the 74th Regional scientific and practical conference of teachers, researchers and postgraduates “Science for education, production, economics”]. Vitebsk, Vitebsk State University named after P. M. Masherov, 2022, pp. 50–52 (in Russian).
16. Trubnikov Yu. V., Chernyavsky М. М. Extremal polynomials of a complex argument of high degrees. Materialy XX Mezhdunarodnoi nauchnoi konferencii po differencial’nym uravneniyam «Eruginskie chteniya – 2022». T. 2 [Materials of the XX International scientific conference on differential equations “Eruginskie chteniya – 2022”. Vol. 2]. Novopolotsk, Euphrosyne Polotskaya State University of Polotsk, 2022, pp. 50–52 (in Russian).
17. Trubnikov Yu. V., Orekhova I. A., Syn Baiyi. Extreme third degree polynomials of complex argument. Vesnik Vicebskaga dzjarzhawnaga universitjeta = Bulletin of Vitebsk State University, 2012, no. 6 (72), pp. 13–18 (in Russian).
18. Trubnikov Yu. V., Orekhova I. A. Construction of the extreme third degree polynomial assigned on the rectangle of the complex plane. Vestsі Natsyianal’nai akademіі navuk Belarusі. Seryia fіzіka-matematychnykh navuk = Proceedings of the National Academy of Sciences of Belarus. Physics and Mathematics series, 2013, no. 1, pp. 13–21 (in Russian).
19. Trubnikov Yu. V., Orekhova I. A. On the extreme third-degree polynomials with complex argument. Vestsі Natsyianal’nai akademіі navuk Belarusі. Seryia fіzіka-matematychnykh navuk = Proceedings of the National Academy of Sciences of Belarus. Physics and Mathematics series, 2014, no. 2, pp. 40–47 (in Russian).
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