Jackson’s rational singular integral on the cut
https://doi.org/10.29235/1561-8323-2019-63-4-398-407
Abstract
The introduction presents the main results of previously known papers on Jackson’s singular integral in polynomial and rational cases. Next, we introduce Jackson’s singular integral on the interval [–1, 1] with the kernel obtained by one system of rational Chebyshev–Markov fractions and establish its basic approximative properties: a theorem on uniform convergence of a sequence of Jackson’s singular integrals for an even function is obtained, and conditions are specified that the parameter must satisfy in order for uniform convergence to take place; the approximative properties of sequences of Jackson’s singular integrals on classes of functions satisfying on the interval [–1, 1] the condition of Lipschitz class with constant M. are investigated. The obtained estimates are asymptotically exact as n → ∞; an estimate of deviation of Jackson’s rational singular integral from the function |x|s, 0 < s < 2 depending on the position of the point on the segment, a uniform estimate of the deviation on the segment [–1, 1] and its asymptotics are found. The optimal value of the parameter is obtained, for which the deviation error of the studied approximation apparatus from the function |x|s, 0 < s < 2 on the interval [–1, 1] has the highest rate of zero; the order of approximation of the function |x| on the interval [–1, 1] byJackson’s considered singular integral is found. It is shown that with a special choice of the parameter, the velocity of the approximation error tending to zero is higher in comparison with the polynomial case. All results of this article are new. The work is both theoretical and applied. It is possible to apply the results to solve specific problems of computational mathematics and to read special courses at mathematical faculties.
About the Authors
Yevgeniy A. Rovba
Yanka Kupala State University of Grodno
Belarus
Belarus
Rovba Yevgeniy Alekseyevich – D. Sc. (Physics and Mathematics), Professor, Head of the Department
22, Ozheshko Str., 230023, Grodno
Pavel G. Potsejko
Yanka Kupala State University of Grodno
Belarus
Belarus
Potsejko Pavel Gennadievich – Postgraduate student
22, Ozheshko Str., 230023, Grodno
References
1. Jackson D. The theory of approximation. American Mathematical Society. Vol. XI. Colloquium Publications, 1930. 184 p. http://dx.doi.org/10.1090/coll/011
2. Schurer F., Steutel F. W. On the Degree of Approximation of Functions in C2π1 with Operators of the Jackson Type. Journal of Approximation Theory, 1979, vol. 27, no. 2, pp. 153–178. https://doi.org/10.1016/0021-9045(79)90117-5
3. Waf A. Saturation of local approximation by linear positive operators of Jackson type. Indian Journal Pure Applied Mathematics, 1980, vol. 11, no. 9, pp. 1194–1201.
4. Safronova G. P. On the summation method for divergent series associated with the singular Jackson integral. Doklady Akademii Nauk SSSR, 1950, vol. 73, no. 2, pp. 277–278 (in Russian).
5. Takenaka S. On the orthogonal functions and a new formula of interpolations. Japanese Journal of Mathematics, 1925, vol. 2, pp. 129–145. https://doi.org/10.4099/jjm1924.2.0_129
6. Malmquist F. Sur la determination d’une classe functions analytiques par leurs dans un ensemble donne de points. Compte Rendus Six. Cong. math. scand. kopenhagen, Denmark, 1925, pp. 253–259.
7. Rusak V. N. Rational functions as approximation apparatus. Minsk, Belarusian State University, 1979. 176 p. (in Russian).
8. Rovba E. A. Rational integral operators on a segment. Doklady Akademii nauk Belarusi = Doklady of the Academy of Sciences of Belarus, 1996, vol. 40, no. 3, p. 42–46 (in Russian).
9. Smatrytski k. A. Approximation of convex functions by rational integral operators on an interval. Vestnik BGU. Seriya 1. Fizika. Matematika. Informatika = Vestnik BSU. Series 1. Physics. Mathematics. Informatics, 2005, no. 3, pp. 64–70 (in Russian).
10. Nikol’skii S. M. On the asymptotic behavior of the remainder in the approximation of functions that satisfy the Lipschitz condition by Fejer sums. Izvestiya Akademii nauk SSSR. Seriya matematicheskaya = Izvestiya: Mathematics, 1940, vol. 4, no. 6, pp. 501–508 (in Russian).
11. Bernstein S. Sur la meilleure approximation de |x| par des polynomes de degres donnes. Acta Mathematica, 1913, vol. 37, pp. 1–57. https://doi.org/10.1007/bf02401828
12. Newman D. J. Rational approximation to |x|. Michigan Mathematical Journal, 1964, vol. 11, no. 1, pp. 11–14. https://doi.org/10.1307/mmj/1028999029
13. Bulanov A. P. Asymptotics for the smallest deviations |x| from rational functions. Mathematics of the USSR-Sbornik, 1968, vol. 5, no. 2, pp. 275–290. https://doi.org/10.1070/sm1968v005n02abeh001006
14. Vyacheslavov N. S. On the approximation of the function |x| rational functions. Matematicheskie zametki = Mathematical Notes, 1974, vol. 16, no. 1, pp. 163–171 (in Russian).
15. Stahl H. Best uniform rational approximation of |x| on [–1, 1]. Russian Academy of Sciences. Sbornik Mathematics, 1993, vol. 76, no. 2, pp. 461–487. http://dx.doi.org/10.1070/SM1993v076n02ABEH003422
16. Bernstein S. Sur la meilleure approximation de |x|p par des polynomes de degres tres eleves. Izvestiya Akademii nauk SSSR. Seriya matematicheskaya = Izvestiya: Mathematics, 1938, vol. 2, no. 2, pp. 169–190 (in Russian).
17. Freud G., Szabados J. Rational approximation to xa. Acta Mathematica Academiae Scientiarum Hungaricae, 1967, vol. 18, no. 3–4, pp. 393–399. https://doi.org/10.1007/bf02280298
18. Gonchar A. A. On the rapidity of rational approximation of continuous functions with characteristic singularities. Mathematics of the USSR-Sbornik, 1967, vol. 2, no. 4, 561–568. http://dx.doi.org/10.1070/SM1967v002n04ABEH002355
19. Vyacheslavov N. S. On approximation of xa by rational functions. Mathematics of the USSR – Izvestiya, 1981, vol. 16, no. 1, pp. 83–101. http://dx.doi.org/10.1070/IM1981v016n01ABEH001297
20. Stahl H. Best uniform rational approximation of xa on [0, 1]. Bulletin of the American Mathematical Society, 1993, vol. 28, no. 1, pp. 116–122. https://doi.org/10.1090/s0273-0979-1993-00351-3
21. Revers M. On the asymptotics of polynomial interpolation to xa at the Chebyshev nodes. Journal of Approximation Theory, 2013, vol. 165, pp. 70–82.
22. Raitsin R. A. Asymptotic properties of uniform approximations of functions with algebraic singularities by partial sums of the Fourier-Chebyshev series. Izvestiya Vysshikh Uchebnykh Zavedenii. Matematika = Soviet Mathematics, 1980, no. 3, pp. 45–49 (in Russian).
23. Potseiko P. G. On one representation of singular integral of Jackson and approximation of the function |x|s on the segment [–1, 1]. Vestnik Grodnenskogo Gosudarstvennogo Universiteta imeni Yanki Kupaly. Seriya 2. Matematika, Fizika, Informatika, Vychislitel’naya Tekhnika i upravlenie = Bulletin of the Grodno State University named after Yanki Kupaly. Series 2. Mathematics, Physics, Informatics, Computing technology and control, 2019, vol. 9, no. 2, pp. 22–38 (in Russian)
24. Rovba y. А., Potsejko P. G. About one system of the Chebyshev–Markov rational fractions. Doklady Natsional’noi akademii nauk Belarusi = Doklady of the National Academy of Sciences of Belarus, 2017, vol. 61, no. 1, pp. 24–29 (in Russian).
25. Rouba y., Patseika P., Smatrytski k. On a system of rational Chebyshev–Markov fractions. Analysis Mathematica, 2018, vol. 44, no. 1, pp. 115–140. https://doi.org/10.1007/s10476-018-0110-7
26. Natanson I. P. Constructive theory of functions. Moscow, State publishing house of technical and theoretical literature, 1949. 684 p. (in Russian).
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