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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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/

[1] Riesz M., “Une méthode de sommation équivalente á la méthode des moyennes arithmétique”, C.R. Acad. Sci. Paris, 152 (1911), 1651–1654

[2] Riesz M., “Sur la sommation des séries de Dirichlet”, C.R. Acad. Sci. Paris, 149 (1909), 18–21

[3] Hardy G. H., Riesz M., The general theory of Dirichlet's series, Cambridge University Press, 1915 | MR

[4] Vinogradov I. M., Matematicheskaya entsiklopediya, Sovetskaya entsiklopediya, M., 1977–1985

[5] Oberchoff N., “Applications de la sommation par les moyennes arithmétiques dans la théorie des séries de Fourier, des séries sphŕiques et ultrasphériques”, Bulletin mathématique de la Société Roumaine des Sciences, 40:1/2, Actes du deuxiéme congrés interbalkanique des mathématiciens (1938), 27–38

[6] Kozlov V. V., Madsen T. V., “Chisla vrascheniya Puankare i srednie Rissa i Voronogo”, Matem. zametki, 74:2 (2003), 314–315 | DOI | Zbl

[7] Kwee B., “The approximation of continuous functions by Riesz typical means of their Fourier series”, Journal of the Australian Mathematical Society, 7:4 (1967), 539–544 | DOI | MR | Zbl

[8] Dodonov N. Y., Zhuk V. V., “On approximation of periodic functions by Riesz sums”, Journal of Mathematical Sciences, 166:2 (2010), 134–144 | DOI | MR | Zbl

[9] Zygmund A., “The approximation of functions by typical means of their Fourier series”, Duke Math. J., 12:4 (1945), 695–704 | DOI | MR | Zbl

[10] Geit V. E., “Kharakterizatsiya posledovatelnosti priblizhenii srednimi Zigmunda”, Izv. vuzov. Matem., 1996, no. 6, 78–79 | MR | Zbl

[11] Ilyasov N. A., “Priblizhenie periodicheskikh funktsii srednimi Zigmunda”, Matem. zametki, 39:3 (1986), 367–382 | MR | Zbl

[12] Ilyasov N. A., “O priblizhenii periodicheskikh funktsii srednimi Feiera – Zigmunda v raznykh metrikakh”, Matem. zametki, 48:4 (1990), 48–57 | Zbl

[13] Stepanets A. I., “Approksimatsionnye svoistva metoda Zigmunda”, Ukra\"inskii matematichnii zhurnal, 51:4 (1999), 493–518 | MR | Zbl

[14] Chikina T. S., “Approximation by Zygmund – Riesz means in the p-variation metric”, Analysis Mathematica, 39:1 (2013), 29–44 | DOI | MR

[15] Volosivets S. S., Likhacheva T. V., “Nekotorye voprosy priblizheniya polinomami po multiplikativnym sistemam v vesovykh prostranstvakh $L_p$”, Izv. Sarat. un-ta. Nov. ser. Ser. Matematika. Mekhanika. Informatika, 15:3 (2015), 251–258 | Zbl

[16] Bernstein S., “Sur meilleure approximation de $|x|$ par des polynomés de degrés donnés”, Acta Math., 37 (1913), 1–57 | DOI | MR

[17] Newman D. J., “Rational approximation to $|x|$”, Mich. Math. J., 11:1 (1964), 11–14 | DOI | MR | Zbl

[18] Bulanov A. P., “Asimptotika dlya naimenshikh uklonenii $|x|$ ot ratsionalnykh funktsii”, Matem. sb., 76(118):2 (1968), 288–303 | MR | Zbl

[19] Vyacheslavov N. S., “O priblizhenii funktsii $|x|$ ratsionalnymi funktsiyami”, Matem. zametki, 16:1 (1974), 163–171 | MR | Zbl

[20] Shtal G., “Nailuchshie ravnomernye ratsionalnye approksimatsii $|x|$ na $[-1,1]$”, Matem. sb., 183:8 (1992), 85–118

[21] Bernstein S., “Sur la meilleure approximation de $|x|^p$ par des polynomés de degrés trés eléves”, Izv. Akad. Nauk SSSR Ser. Mat., 2:2 (1938), 169–190 | MR | Zbl

[22] Freud G., Szabados J., “Rational approximation to $x^\alpha$”, Acta Mathematica Academiae Scientiarum Hungaricae, 18:3–4 (1967), 393–399 | DOI | MR | Zbl

[23] Gonchar A. A., “O skorosti ratsionalnoi approksimatsii nepreryvnykh funktsii s kharakternymi osobennostyami”, Matem. sb., 73:4 (1967), 630–638 | Zbl

[24] Vyacheslavov N. S., “Ob approksimatsii $x^\alpha$ ratsionalnymi funktsiyami”, Izv. AN SSSR. Ser. matem., 44:1 (1980), 92–109 | MR | Zbl

[25] Shtal G., “Best uniform rational approximation of $x^\alpha$ on $[0,1]$”, Am. math. Society, 28:1 (1993), 116–122 | DOI | MR

[26] Revers M., “On the asymptotics of polynomial interpolation to $x^\alpha$ at the Chebyshev nodes”, Journal of Approximation Theory, 165 (2013), 70–82 | DOI | MR | Zbl

[27] Raitsin R. A., “Asimptoticheskie svoistva ravnomernykh priblizhenii funktsii s algebraicheskimi osobennostyami chastichnymi summami ryada Fure – Chebysheva”, Izv. vuzov. Matem., 1980, no. 3, 45–49 | MR

[28] Rouba Y., Patseika P., Smatrytski K., “On one system of rational Chebyshev – Markov fractions”, Analysis Math., 44:1 (2018), 115–140 | DOI | MR | Zbl

[29] Rovba E. A., Potseiko P. G., “Approksimatsiya funktsii $|x|^s$ na otrezke $[-1,1]$ chastichnymi summami ratsionalnogo ryada Fure – Chebysheva”, Vesnik Grodzenskaga dzyarzhaŭnaga ŭniversiteta imya Yanki Kupaly. Seryya 2. Matematyka. Fizika. Infarmatyka, vylichalnaya tekhnika i kiravanne, 9:3 (2019), 16–28 | MR

[30] Potseiko P. G., Rovba E. A., “Summy Feiera ratsionalnogo ryada Fure – Chebysheva i approksimatsii funktsii $|x|^s$”, Zhurnal Belorusskogo gosudarstvennogo universiteta. Matematika. Informatika, 2019, no. 3, 18–34 | MR | Zbl

[31] Sidorov, Yu. V., Fedoryuk M. V., Shabunin M. I., Lektsii po teorii funktsii kompleksnogo peremennogo, Nauka, Gl. red. Fiz-mat. lit-ry, M., 1989 | MR

[32] Evgrafov M. A., Asimptoticheskie otsenki i tselye funktsii, Nauka, M., 1979 | MR

[33] Fedoryuk M. V., Asimptotika. Integraly i ryady, Gl. red. Fiz-mat lit-ry, M., 1987 | MR