Approximation of analytic functions by universal Vallee-Poussin sums on the Chebyshev polynomials

As it is known, Chebyshev polynomials provide the best uniform approach of a function. They are a special case of Faber polynomials. A. I. Shvay (1973) proved that the Vallee-Poussin sums are the best approach apparatus in comparison with the partial sums of the series in the Faber polynomials. Therefore, from the point of view of the best approach, it is natural to consider the approach of functions by means of the Vallee-Poussin sums over Chebyshev polynomials. The study of these sums from any point of view is of definite interest. As O.G. Rovenskaya and O. A. Novikov (2016) note, “during the last decades, the Vallee-Poussin sums and their special cases (Fourier sums and Fejer sums) have been extensively studied by many outstanding specialists in the theory of functions.”
The authors (2017) of this article proved a theorem on the summability of a universal series in Chebyshev polynomials. In this paper we find a subsequence of transformed Vallee-Poussin sums satisfying the conditions of this theorem, that is, these sums are a special case of special sums constructed in the theorem of the authors mentioned above. Thus, the above subsequence of the Vallee-Poussin sums has the universality property. With the help of so-called matrix transformations, a generalization of this property is also obtained for these sums, which consists of the following: on the basis of the selected subsequence, sums are constructed, which uniformly approaches any function from a certain class on the specially defined compact sets. Thus, the sums constructed have the property of universality, which many authors have studied over the years for the functional series. In particular, it was studied for the Fourier series, Dirichlet, Faber, Hermite and other series. Then generalizations of this property were studied. For example, W. Luh (1976) generalized the universality property of a power series.
The existence of universal series and their generalizations was proved in various ways, depending on the specifics of the functions under consideration and the applicability of the methods. The first author (1990) developed a method of matrix transformations, which was subsequently used to solve similar problems (1997, 2012, 2013, 2017). The same method is used to prove the main result of this paper. W. Luh used another method.