List of issues > Series «Mathematics». 2017. Vol. 21
Summation of the Universal Series on the Chebyshev Polynomials
The Universal functional series have been studied by many authors since 1906, the year when the Hungarian mathematician, Michael Fekete first considered the universal power series in the real domain. Trigonometric universal series were built by D. E. Menshov (1945), they were also researched, for example, by J. Edge (1970), and N. Pogosyan (1983). In the complex domain the existence of the universal power series was proved by A. I. Seleznyov, C.K. Chui, M.N. Parnes (1971), V. Nestoridis (1996), and by other authors. Various authors have also studied other universal functional series.
The property of universality of a functional series is the approximation of the function of a certain class by partial sums of this series. This property is a generalization of the well-known S.N. Mergelyan theorem (1952) on the approximation of analytic functions by polynomials on compact sets.
The present work demonstrates the existence of the universal series on the Chebyshev polynomials. W. Luh (1976) summarized the universal property of a power series in case of its matrix transformations. In some sense the analogues of this generalization were obtained by the first author of this work (1990, coauthored: 2012, 2013) for certain functional series. In this paper the aforementioned generalization is extended to series on the Chebyshev polynomials by summation of the universal series. Namely, we have constructed special sums related to the series on the Chebyshev polynomials possessing the property of universality, that is, any function of a certain class on compact sets, taken in a specific way, uniformly approach by these sums. The construction of the sums is carried out by the method of matrix transformation that was previously used by the first author in the construction of specific sums for other series but unlike them, in researched sams in the matrix transformation the columns are taken without gaps.
1. Gosteva N.V., Dodunova L.K. A generalization of the universal property of power series with gaps (in Russian). Izvestiya VUZ. Matematika, 2012, no 3, pp. 3-8.
2. Dodunova L.K. A generalization of the universality property of Faber polynomial series (in Russian). Izvestiya VUZ. Matematika, 1990, no 12, pp. 31-34.
3. Dodunova L.K. On the overconvergence of universal series (in Russian). Izvestiya VUZ. Matematika, 1988, no 2, pp. 19-22.
4. Dodunova L.K., Tyutyulina O.V. Approximation of functions by universal sums of Hermite polynomial the subsystems series (in Russian). Izvestiya VUZ. Matematika, 2013, no 9, pp. 16-20.
5. Dodunova L.K., Okhatrina D.D. The generalization of the universal series in Chebyshev polynomials (in Russian). Chebyshev sbornic, 2017, vol. 18, issue1, pp. 65-72.
6. Men′shov D.E. On universal trigonometric series (in Russian). Dokl. Akad. Nauk SSSR, 1945, vol. 49, no 2, pp. 79-82.
7. Paszkovski C. Computational applications Chebyshev polynomials and series. Trans. from Polish by S. N. Kiro, V. I., Lebedev edited (in Russian). Moscow, Nauka, 1983.
8. Pogosyan N.B. On universal Fourier series (in Russian). UMN, 1983, vol. 38, no 1, pp. 185-186. https://doi.org/10.1070/RM1983v038n01ABEH003410
9. Seleznev A.I. On universal power series (in Russian). Mat. Sb., 1951, vol. 28, no 2, pp. 453-460.
10. Seleznev A.I., Dodunova L.K. Some classes of universal series (in Russian). Izvestiya VUZ. Matematika, 1977, no 12, pp. 92-98.
11. Seleznev A.I., Dodunova L.K. On two theorems of A. F. Leontev on the completeness of subsystems of Faber and Jacobi polynomials (in Russian). Izvestiya VUZ. Matematika, 1982, no 4, pp. 51-55.
12. Seleznev A.I., Motova I.V., Volokhin V.A. The completeness of systems of functions and universal series (in Russian). Izvestiya VUZ. Matematika, 1977, no 11, pp. 84-90.
13. Chashchina N.S. On the theory of a universal dirichlet series (in Russian). Izvestiya VUZ. Matematika, 1963, no 4, pp. 165-167.
14. Chebyshev P.L. The complete collection of the works of Chebyshev (in Russian). Moscow, Akad. Nauk SSSR, 1947, vol. 2.
15. Chui C. K. Parnes M. N. Approximation by overconvergence of a power series. J. Math. Anal. Appl., 1971, vol. 36, no 3, pp. 693-696. https://doi.org/10.1016/0022-247X(71)90049-7
16. Edge J. J. Universal trigonometric series. J. Math. Anal. Appl., 1970, no 29, pp. 507-511. https://doi.org/10.1016/0022-247X(70)90064-8
17. Katsoprinakis E. Nestoridis, V. Papadoperakis I. Universal Faber series. Analysis (Munich), 2001, no 21, pp. 339-363.
18. Luh W. Uber den Sats von Mergelyan. J. Approxim. Theory, 1976, vol. 16, no 2, pp. 194-198. https://doi.org/10.1016/0021-9045(76)90048-4
19. Nestoridis V. Universal Taylor series. Ann. Inst. Fourier (Grenoble), 1996, no 46, pp. 1293-1306.