The generalization of the universal series in Chebyshev polynomials

Chebyshev polynomials are widely used in theoretical and practical studies. Recently, they have become more significant, 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 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 [2] and the series in Hermite polynomials [3]. 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 [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.
This work is devoted to the seventieth Doctor of Physical and Mathematical Sciences, Professor Vasily Ivanovich Bernik. In her curriculum vitae, a brief analysis of his scientific work and educational and organizational activities. The work included a list of 80 major scientific works of V. I. Bernik.
Bibliography: 21 titles.