On the approximation of conjugate functions and their derivatives on the segment by partial sums of Fourier - Chebyshev series

P. G. Potseiko ; E. A. Rovba ; K. A. Smotritskii

Journal of the Belarusian State University. Mathematics and Informatics, Tome 2 (2024), pp. 6-18.

Résumé

In this paper, we study the approximation of conjugate functions with the density $\mathit{f}\in \mathit{H}^{(\alpha)}[-1,1], \alpha\in (0,1],$ on the segment $[-1,1]$ by the conjugate Fourier - Chebyshev series. We establish the order estimations of the approximation depending on the location of a point on the segment. It is noted that approximation at the endpoints of the segment has a higher rate of decrease in comparison with the whole segment. We introduce classes of functions, which, in a certain sense, can be associated with the derivative of a conjugate function on the segment $[-1,1]$, and the approximation of functions from these classes by partial sums of the Fourier - Chebyshev series is studied. An integral representation of the approximation is found. In the case when the density $\mathit{f}\in \mathit{W}^{1}\mathit{H}^{(\alpha)}[-1,1], \alpha\in (0,1],$ the order estimations of the approximation, depending on the location of the point on the segment, are established. The case, when the density $\mathit{f} (t)=|t|^{s}, s>1$, is considered. In this case, an integral representation of the approximation, estimations for pointwise and uniform approximations, as well as an asymptotic estimation for the uniform approximation are obtained. It is noted that the order of the uniform approximations of the function under study by partial sums of the Fourier - Chebyshev series and the corresponding conjugate function by conjugate sums coincide.

Détail

Keywords: singular integral on a segment; conjugate function; Lipschitz condition; Fourier - Chebyshev series; uniform estimations; asymptotic estimations

Bibliographie

[1] F. D. Gakhov, Kraevye zadachi, Gosudarstvennoe izdatelstvo fiziko-matematicheskoi literatury, Moskva, 1958, +545 pp.

[2] N. I. Muskhelishvili, Singulyarnye integralnye uravneniya. 3-e izdanie, Nauka, Moskva, 1968, +600 pp.

[3] P. L. Butzer, R. L. Stens, “The operational properties of the Chebyshev transform. II. Fractional derivatives”, Teoriya priblizheniya funktsii. Trudy Mezhdunarodnoi konferentsii po teorii priblizheniya funktsii (Kaluga, Rossiya), Nauka, Moskva, 1977, 49–61

[4] J. Priwaloff, “Sur les fonctions conjuguees”, Bulletin de la Societe Mathematique de France, 44 (1916), 100–103 | DOI

[5] I. I. Privalov, “K teorii sopryazhennykh trigonometricheskikh ryadov”, Matematicheskii sbornik, 31(2) (1923), 224–228

[6] A. Kolmogoroff, “Sur les fonctions harmoniques conjuguees et les series de Fourier”, Fundamenta Mathematicae, 7 (1925), 24–29 | DOI

[7] M. Riesz, “Les fonctions conjuguees et les series de Fourier”, Comptes rendus de lAcademie des Sciences, 178 (1924), 1464–1467

[8] M. Riesz, “Sur les fonctions conjuguees”, Mathematische Zeitschrift, 27 (1928), 218–244 | DOI

[9] S. M. Nikolskii, “Priblizhenie periodicheskikh funktsii trigonometricheskimi mnogochlenami”, Trudy Matematicheskogo instituta imeni VA Steklova, 15 (1945), 3–76

[10] V. P. Motornyi, “Priblizhenie nekotorykh klassov singulyarnykh integralov algebraicheskimi mnogochlenami”, Ukrainskii matematicheskii zhurnal, 53(3) (2001), 331–345

[11] V. P. Motornyi, “Priblizhenie odnogo klassa singulyarnykh integralov algebraicheskimi mnogochlenami s uchetom polozheniya tochki na otrezke”, Trudy Matematicheskogo instituta imeni VA Steklova, 232 (2001), 268–285

[12] V. R. Misyuk, A. A. Pekarskii, “Sopryazhennye funktsii na otrezke i sootnosheniya dlya ikh nailuchshikh ravnomernykh polinomialnykh priblizhenii”, Izvestiya Natsionalnoi akademii nauk Belarusi. Seriya fiziko-matematicheskikh nauk, 2 (2015), 37–40

[13] N. K. Bari, “O nailuchshem priblizhenii trigonometricheskimi polinomami dvukh sopryazhennykh funktsii”, Izvestiya Akademii nauk SSSR. Seriya matematicheskaya, 19(5) (1955), 285–302

[14] S. B. Stechkin, “O nailuchshem priblizhenii sopryazhennykh funktsii trigonometricheskimi polinomami”, Izvestiya Akademii nauk SSSR. Seriya matematicheskaya, 20(2) (1956), 197–206

[15] S. M. Nikolskii, “O nailuchshem priblizhenii mnogochlenami funktsii, udovletvoryayuschikh usloviyu Lipshitsa”, Izvestiya Akademii nauk SSSR. Seriya matematicheskaya, 10(4) (1946), 295–322

[16] A. F. Timan, “Priblizhenie funktsii, udovletvoryayuschikh usloviyu Lipshitsa, obyknovennymi mnogochlenami”, Doklady Akademii nauk SSSR, 77(6) (1951), 969–972

[17] I. M. Ganzburg, “Obobschenie nekotorykh rezultatov SM Nikolskogo i AF Timana”, Doklady Akademii nauk SSSR, 116(5) (1957), 727–730

[18] Yu. I. Rusetskii, “O priblizhenii nepreryvnykh na otrezke funktsii summami Abelya – Puassona”, Sibirskii matematicheskii zhurnal, 9(1) (1968), 136–144

[19] I. M. Ganzburg, A. F. Timan, “Lineinye protsessy priblizheniya funktsii, udovletvoryayuschikh usloviyu Lipshitsa, algebraicheskimi mnogochlenami”, Izvestiya Akademii nauk SSSR. Seriya matematicheskaya, 22(6) (1958), 771–810

[20] E. A. Rovba, P. G. Potseiko, “Priblizheniya sopryazhennykh funktsii chastichnymi summami sopryazhennykh ryadov Fure po odnoi sisteme algebraicheskikh drobei Chebysheva – Markova”, Izvestiya vysshikh uchebnykh zavedenii. Matematika, 9 (2020), 68–84

[21] A. I. Stepanets, “Priblizhenie periodicheskikh funktsii summami Fure”, Trudy Matematicheskogo instituta imeni VA Steklova, 180 (1987), 202–204

[22] I. V. Kalchuk, T. A. Stepanyuk, U. Z. Grabova, “Priblizhenie differentsiruemykh funktsii bigarmonicheskimi integralami Puassona”, Vesnik Brestskaga universiteta. Seryya 4, Matematyka. Fizika, 1 (2010), 83–92

[23] S. B. Stechkin, “O nailuchshem priblizhenii nekotorykh klassov periodicheskikh funktsii trigonometricheskimi mnogochlenami”, Izvestiya Akademii nauk SSSR. Seriya matematicheskaya, 20(5) (1956), 643–648

[24] V. K. Dzyadyk, “O nailuchshem priblizhenii na klassakh periodicheskikh funktsii, opredelyaemykh integralami ot lineinoi kombinatsii absolyutno monotonnykh yader”, Matematicheskie zametki, 16(5) (1974), 691–701

[25] K. M. Zhigallo, Yu. І. Kharkevich, “Nablizhennya bigarmoniinimi integralami Puassona klasiv, diferentsiiovanikh funktsii v integralnii metritsi. Problemi teorii nablizhennya ta sumizhni pitannya”, Prats Іnstitutu matematiki NAN Ukraini, 1(1) (2004), 144–170

[26] Yu. I. Kharkevich, T. A. Stepanyuk, “Approksimativnye svoistva integralov Puassona na klassakh”, Matematicheskie zametki, 96(6) (2014), 939–952 | DOI

[27] O. V. Besov, “Otsenka priblizheniya periodicheskikh funktsii summami Fure”, Matematicheskie zametki, 79(5) (2006), 784–787 | DOI

[28] O. V. Besov, Lektsii po matematicheskomu analizu. 4-e izdanie, Fizmatlit, Moskva, 2020, +476 pp.

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

[30] M. V. Fedoryuk, Asimptotika. Integraly i ryady, Nauka, Moskva, 1987, +544 pp.

[31] P. G. Potseiko, “O sopryazhennykh summakh Abelya – Puassona na otrezke i ikh approksimatsionnykh svoistvakh”, Vesnik Grodzenskaga dzyarzhaunaga universiteta imya Yanki Kupaly. Seryya 2, Matematyka. Fizika. Іnfarmatyka, vylichalnaya tekhnika i kiravanne, 11(2) (2021), 15–29