THE GENERALIZATION OF THE UNIVERSAL SERIES IN CHEBYSHEV POLYNOMIALS
https://doi.org/10.22405/2226-8383-2017-18-1-65-72
Abstract
Chebyshev polynomials are widely used in theoretical and practical studies. Recently, they have become more signicant, particularly in quantum chemistry. In research [1] their important properties are described to "provide faster convergence of expansions of functions in series of Chebyshev polynomials, compared with their expansion into a power series or in a series of other special polynomials or functions"([1], p. 6). In this paper, a result associated with an approximation theory is presented. To some extent, the analogues of this result were obtained from other studies, such as in [2] [4], respectively for the power series, as well as the series in Hermite and Faber polynomials. With regard to the denition of the signicance of the series in Chebyshev polynomials listed above, the result of this research is of particular signicance in contrast to these analogues. More precisely, we can assume that the practical solution to the particular problems, can be solved much faster with the use of Chebyshev polynomials rather than the usage of such amounts related to power series [2] and the series in Hermite polynomials [3]. In addition, it is considered the rst synthesis of the universal series for polynomials with a density of one. The concept of a universal series of functions is associated with the notion of approximation of functions by partial sums of the corresponding rows. In [2] [19] the universal property of certain functional series are reviewed. In [2] [4], [18] a generalization of this property is considered. This paper generalizes the universality series properties in Chebyshev polynomials.
About the Authors
L. K. Dodunova
N. I. Lobachevsky State University of Nizhny Novgorod
Russian Federation
Russian Federation
Candidate of Physico-Mathematical Sciences, Docent, Associate Professorof the Institute of Information Technology, Mathematics and Mechanics
D. D. Okhatrina
N. I. Lobachevsky State University of Nizhny Novgorod
Russian Federation
Russian Federation
4th year student of the Institute of Information Technology, Mathematics and Mechanics
References
1. Paskovskij, C. 1983, ``Vychislitel'nye primeneniya mnogochlenov i ryadov Chebysheva. ``, [Computational applications of polynomials and Chebyshev series], trans. from Pol. Kiro, S. N. ; editeur scientifique Lebedev, V. I., Nauka, Moscow, 384 pp. (Russian)
2. Luh, W. 1976, ``Uber den Sats von Mergelyan``, J. Approxim. Theory, vol. 16, no. 2, pp. 194–198.
3. Dodunova, L. K. & Tyutyulina, O. V. 2013, ``Approximation of functions of universal sums of series in the sub-systems of polynomials Hermite``, Izvestiya VUZ. Matematika, no. 9, pp. 16–20. (Russian)
4. Dodunova, L. K. 1990, ``On a generalization of the universal property series of Faber polynomials``, Izvestiya VUZ. Matematika, no. 12, pp. 31–34. (Russian)
5. Men'shov, D. E. 1945, ``On universal trigonometric series``, Dokl. Akad. Nauk, vol. 49, no. 2, pp. 79–82. (Russian)
6. Chashina, N. C. 1963, ``By the universal theory of Dirichlet series``, Izvestiya VUZ. Matematika, no. 4, pp. 165–167. (Russian)
7. Seleznev, A. I. 1951, ``On universal power series``, Mathematics collection, vol. 28, no. 2, pp. 453–460. (Russian)
8. Edge, J. J. 1970, ``Universal trigonometric series``, J. Math. Anal. Appl., no. 29, pp. 507–511.
9. Chui, C. K. & Parnes, M. N. 1971, ``Approximation by overconvergence of a power series``, J. Math. Anal. Appl., vol. 36, no. 3, pp. 693–696.
10. Seleznev, A. I., Motova, I. V. & Volokhin, V. A. 1977, ``On the completeness of systems of functions and universal series``, Izvestiya VUZ. Matematika, no. 11, pp. 84–90. (Russian)
11. Seleznev, A. I. & Dodunova, L. K. 1977, ``Certain classes of universal series``, Izvestiya VUZ. Matematika, no. 12, pp. 92–98. (Russian)
12. Poghosyan, N. B. 1983, ``Universal Fourier series``, Russian Math. Surveys, vol. 38, no. 1, pp. 185–186. (Russian)
13. Buczolich, Z. 1987, ``On universal functions and series``, Acta Math. Hungar., no. 49, pp. 403–414.
14. Dodunova, L. K. 1988, ``About convergence over universal series``, Izvestiya VUZ. Matematika, no. 2, pp. 19–22. (Russian)
15. Nestoridis,V. 1996, ``Universal Taylor series``, Ann. Inst. Fourier, Grenoble, no. 46, pp. 1293--1306.
16. Melas, A. & Nestoridis, V. 2001, ``On various types of universal Taylor series``, Complex Variables Theory Appl, no. 44, pp. 245–258.
17. Katsoprinakis, E., Nestoridis, V. & Papadoperakis, I. 2001, ``Universal Faber series``, Analysis, Munich, no. 21, pp. 339–363.
18. Gostevа, N. V. & Dodunova, L. K. 2012, ``A generalization of the universal property of power series with gaps``, Izvestiya VUZ. Matematika, no. 3, pp. 3–8. (Russian)
19. Dodunova, L. K. & Savikhin, S. A. 2012, ``The completeness of subsystems of Faber polynomials``, Izvestiya VUZ. Matematika, no. 9, pp. 3–7. (Russian)
20. Aleksich, G. 1963, ``Problemy sxodimosti ortogonal'nyx ryadov. ``, [Convergence problems of orthogonal series], Trans. from English. Efimova, A. V., ed. Ulyanova, P. L., Publishing House of Foreign literature, Moscow, 359 pp. (Russian)
21. Chebyshev, P. L. 1947, ``Polnoe sobranie sochinenij. ``, [The complete collection of the works of Chebyshev], Akad. Nauk SSSR, vol. 2, 520 pp. (Russian)
Review
For citations:
Dodunova L.K., Okhatrina D.D. THE GENERALIZATION OF THE UNIVERSAL SERIES IN CHEBYSHEV POLYNOMIALS. Chebyshevskii Sbornik. 2017;18(1):65-72. (In Russ.) https://doi.org/10.22405/2226-8383-2017-18-1-65-72
Views:
738
Email this article 