On rational Abel – Poisson means on a segment and approximations of Markov functions

Approximations on the segment $[-1,1]$, of Markov functions by Abel – Poisson sums of a rational integral operator of Fourier type associated with the Chebyshev – Markov system of algebraic fractions in the case of a fixed number of geometrically different poles are investigated. An integral representation of approximations and an estimate of uniform approximations are found. Approximations of Markov functions in the case when the measure $\mu$ satisfies the conditions $supp\mu =[1,a], a>1, d\mu(t)=\phi(t)dt$ and $\phi (t)\asymp (t-1)^{\alpha}$ on $[1,a]$ are studied and estimates of pointwise and uniform approximations and the asymptotic expression of the majorant of uniform approximations are obtained. The optimal values of the parameters at which the majorant has the highest rate of decrease are found. As a corollary, asymptotic estimates of approximations on the segment $[-1,1]$, are given by the method of rational approximation of some elementary Markov functions under study.