Rational approximations of continuous functions and functions with a power-law singularity on a closed interval are studied by means of integral Fejér-type operators. Upper estimates of approximations of continuous functions on a closed interval are derived; the estimates are expressed in terms of the modulus of continuity and depend on the position of a point in the interval. Rational approximations of the function $(1-x)^\gamma$, $\gamma\in (0,1)$, on the interval $[-1,1]$ are studied. Upper estimates of uniform approximations in terms of the corresponding majorant and an asymptotic expression as $n\to\infty$ of this majorant are found. In the case of a fixed number of poles of the approximating function, optimal values of the parameters are obtained, for which the majorant of the uniform approximations decreases at the highest rate. A consequence of the results obtained is asymptotic estimates of approximations of some specific functions by Fejér sums of polynomial Fourier-Chebyshev series.
Keywords: rational approximations, Fejér integral operator, pointwise and uniform estimates of approximations, modulus of continuity, function with a power-law singularity, asymptotic estimates
Received May 15, 2023
Revised December 18, 2023
Accepted December 25, 2023
Funding Agency: This work was supported by the National Program for Scientific Research of the Republic of Belarus “Convergence 2020” (project no. 20162269).
Pavel G. Potseiko, Ph.D., Faculty of Mathematics and Informatics Yanka Kupala State University of Grodno (Belarus) Ozheshko St., 22, 230023, Grodno, Belarus; e-mail: pahamatby@gmail.com
Yevgeniy A. Rovba, Dr. Phys.-Math. Sci., Prof., Faculty of Mathematics and Informatics Yanka Kupala State University of Grodno (Belarus) Ozheshko St., 22, 230023, Grodno, Belarus; e-mail: rovba.ea@gmail.com
REFERENCES
1. Fejér L. Untersuchungen über Fouriersche Reihen. Math. Ann., 1904, vol. 58, pp. 51–69. doi: 10.1007/BF01447779
2. Lebesgue H. Sur les intégrales singuliéres. Annales de la faculté des sciences de Toulouse 3e série, 1909, vol. 1, pp. 25–117.
3. Bernstein S. Sur l’ordre de la meilleure approximation des fonctions continues par des polynomés de degré donné. Bruxelles : Hayez, imprimeur des academies royales, 1912. 104 p.
4. Privalov I.I. On approximation of functions satisfying Lipschitz condition by Fejér sums. Math. Sb., 1918, vol. 30, no. 4, pp. 521–526 (in Russian).
5. Nikol’skii S.M. On asymptotic behaviour of remainder in approximation of functions satisfying Lipschitz condition by Fejér sums. Izvestiya Akad. Nauk SSSR, Ser. Matem., 1940, vol. 4, no. 6, pp. 501–508 (in Russian).
6. Zygmund A. On the degree of approximation of functions by Fejér means. Bull. Amer. Math. Soc., 1945, vol. 51, no. 4, pp. 274–278. doi: 10.1090/S0002-9904-1945-08332-3
7. Novikov O.A., Rovenskaya O.G. Approximation of classes of Poisson integrals by Fejér sums. Computer Research and Modeling, 2015, vol. 7, no. 4, pp. 813–819. doi: 10.20537/2076-7633-2015-7-4-813-819
8. Efimov A. V. On approximation of some classes of continious functions by Fourier sums and Fejér sums. Izvestiya Akad. Nauk SSSR, Ser. Matem., 1958, vol. 22, no. 1, pp. 81–116 (in Russian).
9. Lebed’ G.K., Avdeenko A.A. Approximation of periodic functions by Fejér sums. Math. USSR-Izv., 1971, vol. 5, no. 1, pp. 86–96. doi: 10.1070/IM1971v005n01ABEH001014
10. Savchuk V.V. Approximation of functions of Dirichlet class by Fejér means. Math. Notes, 2007, vol. 81, no. 5, pp. 665–670. doi: 10.1134/S0001434607050124
11. Dzhrbash’an M.M. To the theory of Fourier series in rational functions. Izvestiya Akad. Nauk Armyan. SSR. Ser. Fiz.-Mat., 1956, vol. 9, no. 7, pp. 1–27 (in Russian).
12. Rusak V.N. On approximation by rational fractions. Dokl. Akad Nauk BSSR, 1964, vol. 8, no. 7, pp. 432–435 (in Russian).
13. Rusak V.N. On approximation by rational functions on the real line. Izvestiya Akad. Nauk BSSR, Ser. Fiz.-Mat., 1974, no. 1, pp. 22–28 (in Russian).
14. Rusak V. N. Ratsional’nye funktsii kak apparat priblizheniya [Rational functions as approximation apparatus], Minsk, Belarus State Univ. Publ., 1979, 179 p.
15. Kitbalyan A.A. On a generalization of the Fejér kernel. Dokl. Akad. Nauk Arm. SSR, 1979, vol. 69, pp. 8–14 (in Russian).
16. Rovba E.A. Rational integral operators on a segment. Vestnik Beloruss. Gos. Univ., 1996, vol. 1, no. 1, pp. 34–39 (in Russian).
17. Rovba E.A. On the approximation of functions of bounded variation by Fejér and Jackson rational operators. Dokl. Nats. Akad. Nauk Belarusi, 1998, vol. 42, no. 4, pp. 13–17 (in Russian).
18. Smotritskii K.A. On approximation of convex functions by rational integral operators on a segment. Vestnik Beloruss. Gos. Univ., 2005, vol. 1, no. 3, pp. 64–70 (in Russian).
19. Bernstein S. N. Ekstremal’nye svoistva polinomov i nailuchshee priblizhenie nepreryvnykh funktsii odnoi veshchestvennoi peremennoi [Extremal properties of polynomials and the best approximation of continuous functions of one real variable], Part 1, Moscow–Leningrad, Ob’edinennoe Nauch. Tekh. Izd. Narod. Komiss. Tyazheloi Promyshl. SSSR, 1937, 203 p.
20. Ibragimov I.I. On asymptotic value of best approximations of functions having real singular point. Izvestiya Akad. Nauk SSSR. Ser. Matem., 1946, vol. 10, no. 5, pp. 429–460 (in Russian).
21. Nikol’skii S.M. On the best approximation in the mean to the function $|a-x|^s$ by polynomials. Izvestiya Akad. Nauk SSSR. Ser. Matem., 1947, vol. 11, no. 2, pp. 139–180 (in Russian).
22. Reddy A.R. A note on rational approximation to $(1 - x)^{1/2}$. J. Approx. Theory, 1979, vol. 25, no. 1, pp. 31–33. doi: 10.1016/0021-9045(79)90030-3
23. Bundschuh P.A. Remark on Reddy’s paper on the rational approximation of $(1 - x)^{1/2}$. J. Approx. Theory, 1981, vol. 32, no. 3, pp. 167–169. doi: 10.1016/0021-9045(81)90113-1
24. McD. Mercer A. A note on rational approximation to $(1 -x)^\alpha$. J. Approx. Theory, 1983, vol. 38, no. 1, pp. 101–103. doi: 10.1016/0021-9045(83)90146-6
25. Reddy A.R. A note on rational approximation to $(1 - x)^\sigma$. J. Approx. Theory, 1987, vol. 49, no. 4, pp. 404–407. doi: 10.1016/0021-9045(87)90078-5
26. Alzer H. On rational approximation of $(1 - x)^\sigma$. Archiv Math., 1996, vol. 67, no. 2, pp. 134–137. doi: 10.1007/BF01268927
27. Andersson J.-E. Best rational approximation to Markov functions. J. Approx. Theory, 1994, vol. 76, no. 2, pp. 219–232. doi: 10.1006/jath.1994.1015
28. Pekarskii A.A. Best uniform rational approximations to Markov functions. St. Petersburg Math. J., 1996, vol. 7, no. 2, pp. 277–285.
29. Patseika P.G., Rouba Y.A., Smatrytski K.A. On one rational integral operator of Fourier–Chebyshev type and approximation of Markov functions. J. Belarusian State Univ. Math. Inform., 2020, vol. 2, pp. 6–27. doi: 10.33581/2520-6508-2020-2-6-27
30. Rovba E.A. One direct method in the rational approximation. Dokl. Nats. Akad. Nauk Belarusi, 1979, vol. 23, no. 11, pp. 968–971 (in Russian).
31. Patseika P.G., Rouba Y.A. On rational Abel–Poisson means on a segment and approximations of Markov functions. Zhurnal Beloruss. Gos. Univ. Matem. Inform., 2021, vol. 3, pp. 6–24 (in Russian). doi: 10.33581/2520-6508-2021-3-6-24
32. Potseiko P.G., Rovba E.A. On rational approximations of Markov function on a segment by Fejér sums with fixed number of poles. Trudy Inst. Mat., 2022, vol. 30, no. 1–2, pp. 63–83 (in Russian).
33. Misyuk V.R. On the best approximation of power function in Bergman space. In: Proc. 13th Int. Conf. Kazan Summer Scientific School-Conf. “Function theory, its applications and related questions”. Kazan: Kazan Math. Soc. publ., Acad. Sci. Resp. Tatarstan publ., 2017, vol. 54, pp. 257–259 (in Russian).
34. Timan A.F. Theory of approximation of functions of a real variable, Oxford, Pergamon Press, 1963, 631 p. doi: 10.1016/C2013-0-05307-8 Original Russian text was published in Timan A.F., Teoriya priblizheniya funktsii deistvitel’nogo peremennogo, Moscow, Gos. Izd-vo Fiz. Mat. Lit., 1960, 624 p.
35. Akhiezer N.I. Lektsii po teorii approksimatsii [Lectures on the Theory of Approximation], Moscow, Nauka Publ., 1965, 409 p.
36. Sidorov Yu. V., Fedoryuk M. V., Shabunin M. I. Lektsii po teorii funktsii kompleksnogo peremennogo [Lections on complex variable functions theory], Moscow, Nauka Publ., Glav. Red. Fiz. Mat. Lit., 1989, 480 p.
37. Patseika P.G., Rouba Y.A. Fejér means of rational Fourier–Chebyshev series and approximation of function $|x|^s$. Zhurnal Beloruss. Gos. Univ. Matem. Inform., 2019, vol. 3, pp. 18–34 (in Russian). doi: 10.33581/2520-6508-2019-3-18-34
Cite this article as: P.G. Potseiko, E.A. Rovba. A Fejér rational integral operator on a closed interval and approximation of functions with a power-law singularity. Trudy Instituta Matematiki i Mekhaniki UrO RAN, 2024, vol. 30, no. 1, pp. 170–189.
