Riesz – Zigmund means of rational Fourier – Chebyshev seriesand approximations of the function $|x|^s$

Approximations of the function $|x|^s, \ s \in (0,2),$ on the segment $[-1,1]$ by the Sigmund – Riesz means of Fourier series according to the Chebyshev – Markov algebraic fractions are studied. A survey of the basic information related to Sigmund – Riesz summation methods is given. A system of Chebyshev – Markov algebraic fractions is considered and an integral representation of the Sigmund – Riesz means of Fourier series for this orthogonal system is obtained. The approximations of the function $|x|^s, \ s \in (0,2),$ on the segment $[-1,1]$ by the Sigmund – Riesz means are investigated. Estimates of point-wise and uniform approximations, asymptotic equalities for the corresponding majorant of uniform approximations at $n \to \infty$, and the optimal value of the parameter that guarantees the maximal decrease rate for the majorant are found.