Vallee Poussin sums of rational Fourier–Chebyshev integral operators and approximations of the Markov function

Rational approximations on the segment $[–1, 1]$ of the Markov function are studied. The Vallee Poussin sums of rational integral Fourier–Chebyshev operators as an approximation apparatus with a fixed number of geometrically distinct poles are chosen. For the constructed rational approximation method, integral representations of approximations and upper estimates of uniform approximations are established.
For a Markov function with a measure whose derivative there is a function that has a power-law singularity on the segment $[–1, 1]$, upper estimates of pointwise and uniform approximations and asymptotic expression of the majorant of uniform approximations are found. The values of the parameters of the approximating function are determined, at which the best uniform rational approximations are provided by this method. It is shown that in this case they have a higher rate of decrease in comparison with the corresponding polynomial analogues.
As a corollary, rational approximations on a segment by Vallee Poussin sums of some elementary functions representable by a Markov function are considered.