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The generalization of the universal series in Chebyshev polynomials. (Russian. English summary) Zbl 1379.30024

Summary: Chebyshev polynomials are widely used in theoretical and practical studies. Recently, they have become more significant, particularly in quantum chemistry. In [S. Paszkowski, Numerical applications of Chebyshev polynomials and series. Transl. from the Polish (Russian). Moskva: Izdat. “Nauka” (1983; Zbl 0527.65008)] 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” [loc. cit., 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 [W. Luh, J. Approx. Theory 16, 194–198 (1976; Zbl 0336.30016); the first author and O. V. Tyutyulina, Russ. Math. 57, No. 9, 12–15 (2013; Zbl 1295.30088); translation from Izv. Vyssh. Uchebn. Zaved., Mat. 2013, No. 9, 16–20 (2013); the first author, Sov. Math. 34, No. 12, 37–40 (1990; Zbl 0737.30002); translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1990, No. 12(343), 31–34 (1990)], respectively for the power series, as well as the series in Hermite and Faber polynomials.
With regard to the definition of the significance of the series in Chebyshev polynomials listed above, the result of this research is of particular significance 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 [Zbl 1295.30088] and the series in Hermite polynomials [Zbl 0737.30002]. In addition, it is considered the first 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 the literature the universal property of certain functional series are reviewed. In [Zbl 0336.30016; Zbl 1295.30088; Zbl 0737.30002], [N. V. Gosteva and the first author, Russ. Math. 56, No. 3, 1–5 (2012; Zbl 1264.30002); translation from Izv. Vyssh. Uchebn. Zaved., Mat. 2012, No. 3, 3–8 (2012)] a generalization of this property is considered. This paper generalizes the universality series properties in Chebyshev polynomials.

MSC:

30E10 Approximation in the complex plane
30C10 Polynomials and rational functions of one complex variable
30K99 Universal holomorphic functions of one complex variable
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