On Rational Approximation of Markov Functions by Partial Sums of Fourier Series on a Chebyshev–Markov System

Approximations on the closed interval $[-1,1]$ of functions that are combinations of classical Markov functions by partial sums of Fourier series on a system of Chebyshev–Markov rational fractions are considered. Pointwise and uniform estimates for approximations are established. For the case in which the derivative of the measure is weakly equivalent to a power function, an asymptotic expression for the majorant of uniform approximations and an optimal parameter value ensuring the greatest rate of approximation by the method used in the paper are found. In the case of the even multiplicity of the poles of the approximating function, the asymptotic estimate is sharp. Examples of approximations of concrete functions are given.