Geodesic
Riesz~--~Zigmund means of rational Fourier~--~Chebyshev seriesand approximations of the function $|x|^s$
Trudy Instituta matematiki, Tome 28 (2020) no. 1, pp. 74-90

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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. 
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     author = {Y. A. Rouba and P. G. Patseika},
     title = {Riesz~--~Zigmund means of rational {Fourier~--~Chebyshev} seriesand approximations of the function $|x|^s$},
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     url = {http://geodesic.mathdoc.fr/item/TIMB_2020_28_1_a7/}
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Y. A. Rouba; P. G. Patseika. Riesz~--~Zigmund means of rational Fourier~--~Chebyshev seriesand approximations of the function $|x|^s$. Trudy Instituta matematiki, Tome 28 (2020) no. 1, pp. 74-90. http://geodesic.mathdoc.fr/item/TIMB_2020_28_1_a7/