On rational approximations of conjugate function on an interval by conjugate Vall\'ee Poussin sums

P. G. Patseika

Problemy fiziki, matematiki i tehniki, no. 3 (2024), pp. 59-70.

Résumé

The approximations of the conjugate function on the segment $[-1, 1]$ by Vallée Poussin sums of conjugate rational integral Fourier – Chebyshev operators with restrictions on the number of geometrically different poles are investigated. An integral representation of the corresponding approximations is established. An integral representation of approximations, the estimation of pointwise approximations and uniform approximations with a certain majorant are obtained for a conjugate function with density $(1-x)^\gamma$, $\gamma\in(0,1)$. Its asymptotic expression for $n\to\infty$, depending on the parameters of the approximating function, is established. The optimal values of the parameters at which the highest rate of decreasing majorant is provided are found. As a consequence, the estimates of approximations of conjugate function on the segment $[-1, 1]$ by Vallée Poussin sums of conjugate polynomial Fourier – Chebyshev series are found.

Détail

Keywords: conjugate function, Fourier – Chebyshev series, function with power singularity, pointwise and uniform approximations, best approximations, asymptotic estimates.
Mots-clés : Vallée Poussin sums

Bibliographie

[1] F.D. Gakhov, Kraevye zadachi, Gos. izd-vo fiz.-mat. lit-ry, M., 1958, 543 pp.

[2] N.I. Muskhelishvili, Singulyarnye integralnye uravneniya, 3-e izd., Nauka, M., 1968, 513 pp.

[3] N.K. Bari, Trigonometricheskie ryady, Fizmatlit, M., 1961, 936 pp.

[4] A. Zigmund, Trigonometricheskie ryady, V 2-kh tomakh, v. 1, Mir, M., 1965, 616 pp.

[5] V.N. Rusak, I.V. Rybachenko, “Ravnomernaya ratsionalnaya approksimatsiya sopryazhennykh funktsii”, Vestnik BGU. Ser. 1. Matematika i informatika, 3 (2013), 83-86

[6] E.A. Rovba, P.G. Potseiko, “Priblizheniya sopryazhennykh funktsii chastichnymi summami sopryazhennykh ryadov Fure po odnoi sisteme algebraicheskikh drobei Chebysheva-Markova”, Izvestiya vuzov. Matematika, 2020, no. 9, 68-84

[7] E.A. Rovba, “Ob odnom pryamom metode v ratsionalnoi approksimatsii”, Doklady AN BSSR, 23:11 (1979), 968-971 | Zbl

[8] K.A. Smotritskii, “O priblizhenii differentsiruemykh v smysle Rimana-Liuvillya funktsii”, Izvestiya NAN Belarusi. Ser. fiz.-mat. nauk, 4 (2002), 42-47

[9] P.G. Patseika, Y.A. Rouba, K.A. Smatrytski, “On one rational integral operator of Fourier-Chebyshev type and approximation of Markov functions”, Journal of the Belarusian State University. Mathematics and Informatics, 2 (2020), 6-27 | DOI

[10] P.G. Potseiko, E.A. Rovba, “Priblizheniya na klassakh integralov Puassona ratsionalnymi integralnymi operatorami Fure-Chebysheva”, Sibirskii matematicheskii zhurnal, 62:2 (2021), 362-386 | Zbl

[11] P.G. Potseiko, E.A. Rovba, “Sopryazhennyi ratsionalnyi operator Fure-Chebysheva i ego approksimatsionnye svoistva”, Izvestiya vuzov. Matematika, 2022, no. 3, 44-60

[12] Ch.-J. de La Vallée Poussin, “Sur la meilleure approximation des function d'une variable reelle par des expressions d'ordre donne”, Comptes Rendus Acad. sci. Paris, 166 (1918), 799-802

[13] Ch.-J. de La Vallée Poussin, Lecons sur l'approximation des fonctions d'une variable réelle, GAUTHIER VILLARS ET CIE, Paris, 1919, 150 pp. | MR

[14] S.M. Nikolskii, “O nekotorykh metodakh priblizheniya trigonometricheskimi summami”, Izvestiya Akademii nauk SSSR. Ser. matem., 4:6 (1940), 509-520 | Zbl

[15] S.B. Stechkin, “O summakh Valle–Pussena”, Doklady AN SSSR, 80:4 (1951), 545-548 | Zbl

[16] S.A. Telyakovskii, “Priblizhenie differentsiruemykh funktsii summami Valle–Pussena”, Doklady AN SSSR, 121:3 (1958), 426-429 | Zbl

[17] A.V. Efimov, “O priblizhenii periodicheskikh funktsii summami Valle–Pussena”, Izvestiya Akademii nauk SSSR. Ser. matem., 23:5 (1959), 737-770 | Zbl

[18] V.I. Rukasov, “Priblizhenie summami Valle–Pussena klassov analiticheskikh funktsii”, Ukrainskii matematicheskii zhurnal, 55:6 (2003), 806-816 | Zbl

[19] A.S. Serdyuk, “Nablizhennya integraliv Puassona sumami Valle–Pussena v rivnomirnii ta integralnikh metrikakh”, Reports of the National Academy of Sciences of Ukraine, 6 (2009), 34-39 | Zbl

[20] L.M. Abramov, “Ob asimptoticheskom povedenii funktsii Lebega nekotorykh metodov summirovaniya ryadov Chebysheva”, Doklady AN SSSR, XCVIII:2 (1954), 173-176 | Zbl

[21] I.M. Ganzburg, “Obobschenie nekotorykh rezultatov S.M. Nikolskogo i A.F. Timana”, Doklady AN SSSR, 116:5 (1957), 727-730 | Zbl

[22] I.M. Ganzburg, A.F. Timan, “Lineinye protsessy priblizheniya funktsii, udovletvoryayuschikh usloviyu Lipshitsa, algebraicheskimi mnogochlenami”, Izvestiya Akademii nauk SSSR. Ser. matem., 22:6 (1958), 771-810 | Zbl

[23] T.O. Omataev, “O priblizhenii nepreryvnykh na otrezke funktsii usechennymi summami Valle–Pussena”, Izvestiya vysshikh uchebnykh zavedenii. Matematika, 1977, no. 6, 99-106

[24] A.D. Scherbina, “Ob odnom metode summirovaniya ryadov, sopryazhennykh ryadam Fure”, Matematicheskii sbornik, 27 (69):2 (1950), 157-170 | Zbl

[25] A.F. Timan, “Approksimativnye svoistva lineinykh metodov summirovaniya ryadov Fure”, Izvestiya AN SSSR. Ser. matem., 17:2 (1953), 99-134 | Zbl

[26] S.A. Telyakovskii, “O priblizhenii differentsiruemykh funktsii lineinymi srednimi ikh ryadov Fure”, Izvestiya AN SSSR. Ser. matem., 24:2 (1960), 213-242 | Zbl

[27] V.A. Dudas, “Priblizhenie sopryazhennykh periodicheskikh funktsii summami Valle–Pussena”, Ukrainskii matematicheskii zhurnal, 30:4 (1978), 522-528 | Zbl

[28] S.P. Baiborodov, “Priblizhenie funktsii summami Valle–Pussena”, Matematicheskie zametki, 27:1 (1980), 33-48 | Zbl

[29] V.N. Rusak, “Ob odnom metode priblizheniya ratsionalnymi funktsiyami na veschestvennoi osi”, Matematicheskie zametki, 22:3 (1977), 375-380

[30] E.A. Rovba, “Priblizhenie funktsii, differentsiruemykh v smysle Rimana–Liuvillya, ratsionalnymi operatorami”, Doklady Natsionalnoi akademii nauk Belarusi, 40:6 (1996), 18-22

[31] K.A. Smotritskii, “O priblizhenii funktsii ogranichennoi variatsii ratsionalnymi operatorami na otrezke”, Vesnik Grodzenskaga dzyarzhaŭnaga ŭniversiteta imya Yanki Kupaly. Seryya 2. Matematyka. Fizika. infarmatyka, vylichalnaya tekhnika i kiravanne, 2:2 (2005), 60-68

[32] P.G. Potseiko, E.A. Rovba, “Summy Valle–Pussena ratsionalnykh integralnykh operatorov Fure-Chebysheva i approksimatsii integralov Puassona na otrezke”, Sibirskii matematicheskii zhurnal, 64:1 (2023), 162-183 | Zbl

[33] P.G. Potseiko, E.A. Rovba, “Summy Valle–Pussena ratsionalnykh integralnykh operatorov Fure-Chebysheva i approksimatsii funktsii Markova”, Algebra i analiz, 35:5 (2023), 183-208

[34] P.G. Potseiko, “O ratsionalnykh sopryazhennykh summakh Feiera na otrezke i approksimatsiyakh sopryazhennoi funktsii”, Problemy fiziki, matematiki i tekhniki, 2023, no. 2 (55), 56-67

[35] K.N. Lungu, “O nailuchshikh priblizheniyakh ratsionalnymi funktsiyami s fiksirovannym chislom polyusov”, Matematicheskii sbornik, 86 (128):2 (10) (1971), 314-324 | Zbl

[36] K.N. Lungu, “O nailuchshikh priblizheniyakh ratsionalnymi funktsiyami s fiksirovannym chislom polyusov”, Sibirskii matematicheskii zhurnal, 15:2 (1984), 151–160

[37] V.N. Rusak, Ratsionalnye funktsii kak apparat priblizheniya, BGU, Minsk, 1979, 153 pp.

[38] Yu.V. Sidorov, M.V. Fedoryuk, M.I. Shabunin, Lektsii po teorii funktsii kompleksnogo peremennogo, Nauka, Gl. red. fiz.-mat. lit-ry, M., 1989, 480 pp.

[39] M.A. Evgrafov, Asimptoticheskie otsenki i tselye funktsii, Nauka, M., 1979, 320 pp.

[40] M.V. Fedoryuk, Asimptotika. Integraly i ryady, Gl. red. fiz.-mat. lit-ry, M., 1987, 544 pp.

[41] E.A. Rovba, E.G. Mikulich, “Konstanty v priblizhenii $|x|$ ratsionalnymi interpolyatsionnymi protsessami”, Doklady Natsionalnoi akademii nauk Belarusi, 53:6 (2009), 11-15

[42] E.G. Mikulich, “Tochnye otsenki ravnomernykh priblizhenii funktsii $|\sin x|$ chastnymi summami ryadov Fure po ratsionalnym funktsiyam”, Vestnik BGU. Ser. 1. Fizika. Matematika. Informatika, 1 (2011), 84-90