Fejer means of rational Fourier – Chebyshev series and approximation of function $|x|^{s}$

Approximation properties of Fejer means of Fourier series by Chebyshev – Markov system of algebraic fractions and approximation by Fejer means of function $|x|^{s}, 0<s<2$, on the interval $[-1,1]$, are studied. One orthogonal system of Chebyshev – Markov algebraic fractions is considers, and Fejer means of the corresponding rational Fourier – Chebyshev series is introduce. The order of approximations of the sequence of Fejer means of continuous functions on a segment in terms of the continuity module and sufficient conditions on the parameter providing uniform convergence are established. A estimates of the pointwise and uniform approximation of the function $|x|^{s}, 0<s<2$, on the interval $[-1,1]$ , the asymptotic expressions under $n\rightarrow \infty$ of majorant of uniform approximations, and the optimal value of the parameter, which provides the highest rate of approximation of the studied functions are sums of rational use of Fourier – Chebyshev are found.