On a rational Jackson singular integral and approximations of Markov functions
EDN: RBMQLB
Abstract
The rational Jackson singular integral is introduced, which is a linear combination of Fourier-Chebyshev rational integral operators with a corresponding triangular matrix of coefficients and a fixed number of geometrically different poles. Its integral representation is established. Rational approximations of Markov functions on the segment $[-1,1]$ are investigated by the introduced method. An integral representation of approximations and an upper bound of uniform approximations are established. Approximations of Markov functions with an absolutely continuous measure whose derivative is asymptotically equal to a function with a power singularity are studied. In this case, top-down estimates of pointwise and uniform approximations and an asymptotic expression of the majorant of uniform approximations are found. Optimal values of the parameters of rational Jackson singular integrals are established, at which the best uniform approximations of Markov functions are provided by this method. For this purpose, the corresponding extreme problem is solved. It is shown that with a special choice of parameters, uniform rational approximations have a higher rate of decrease in comparison with the corresponding polynomial analogues. As a corollary, approximations of some elementary functions represented by Markov functions on the segment $[-1,1]$ are considered.
About the Authors
P. G. Patseika
Yanka Kupala State University of Grodno
Belarus
Belarus
Grodno
Ya. A. Rouba
Yanka Kupala State University of Grodno
Belarus
Belarus
Grodno
References
1. Markov A. A. Selected Works on the Theory of Continued Fractions and the Theory of Functions Deviating Least from Zero. Moscow, Gostehizdat, 1948, pp. 106–119 (in Russian).
2. Gonchar A. A. On the speed of rational approximation of some analytic functions Math. USSR-Sb., 1978. vol. 34, iss. 2, pp. 131–145 (in Russian).
3. Ganelius Т. Ortogonal polynomials and rational approximation of holomorphic function. Studies in Pure Mathematics (To the Memory of Paul Turan), ed. P. Erdos. Basel, Birkhauser Verlag, 1978, pp. 237–243.
4. Andersson J.-E. Best Rational Approximation to Markov Functions. J. Approx. Theory, 1994, vol. 76, iss. 1, pp. 219–232.
5. Pekarskii A. A. Best uniform rational approximations of Markov functions. St. Petersburg Math. J., 1996, vol. 7, iss. 2, pp. 277–285.
6. Vyacheslavov N. S., Mochalina E. P. Rational approximations of functions of Markov–Stieltjes type in Hardy spaces. Mosc. Univ. Math. Bull., 2008, vol. 63, iss. 4, pp. 125–134.
7. Starovoitov A. P., Labych Yu. A. Rational approximation of Markov functions generated by Borel measures of power type. Problemy fiziki, matematiki i tekhniki, 2009, vol. 1, iss. 1, pp. 69–73 (in Russian).
8. Mardvilko T. S. Uniform rational approximation of the odd and even Cauchy transforms. Sbornik: Mathematics, 2025, vol. 216, iss. 2, pp. 239–256. https://doi.org/10.4213/sm10116e
9. Pekarskii A. A., Rovba E. A. Uniform approximations of Stieltjes functions by orthogonal projection on the set of rational functions. Mathematical Notes, 1999, vol. 65, pp. 302–307.
10. Takenaka S. On the orthogonal functions and a new formula of interpolations. Jpn. J. Math., 1925, vol. 2, iss. 1, pp. 129–145.
11. Malmquist F. Sur la de´termination d’une classe functions analytiques par leurs dans un ensemble donne´ de points. Compte Rendus Sixie´me Congre´s math. scand. Kopenhagen, Denmark, 1925, pp. 253–259.
12. Dzhrbashyan M. M., Kitbalyan A. A. On a generalization of Chebyshev polynomials. Doklady AN Armyanskoi SSR, 1964, vol. 38, iss. 5, pp. 263–270 (in Russian).
13. Rovba E. A., Mikulich E. G. Constants in rational approximation of Markov–Stieltjes functions with fixed number of poles. Vesnik of Yanka Kupala State University of Grodno. Series 2. Mathematics. Physics. Informatics, Сomputer Technology and its Сontrol, 2013, vol. 1, iss. 148, pp. 12–20.
14. Lungu K. N. On the best approximations by rational functions with a fixed number of poles. Mathematics of the USSR-Sbornik, 1971, vol. 86(128), iss. 2(10), pp. 314–324 (in Russian).
15. Lungu K. N. On the best approximations by rational functions with a fixed number of poles. Siberian Mathematical Journal, 1984. vol. XXV, iss. 2. pp. 150–159 (in Russian).
16. Jackson D. The theory of approximation. Colloq. Publ. Amer. Math. Soc. Vol. XI, 1930. 184 p.
17. Schurer F., Steutel F. W. On the degree of approximation of functions in with operators of the Jackson type. J. Approx. Theory, 1979. vol. 27, iss. 2, pp. 153–178.
18. Wafi A. Saturation of local approximation by linear positive operators of Jackson type. Indian J. pure appl. Math., 1980, vol. 11, iss. 9, pp. 1194–1201.
19. Alekseev V. G. Jackson-type kernels and their application to the construction of low-pass filters. Problemy peredachi informatsii, 1994, vol. 30, iss. 1, pp. 97–102 (in Russian).
20. Alekseev V. G. Jackson and Jackson–Vallee Poussin-type kernels and their probabilistic applications. Teoriya veroyatnosti i ee primenenie, 1996, vol. 41, iss. 1, pp. 170–177 (in Russian).
21. Safronova G. P. On the method of summation of divergent series related to the singular Jackson integral. Doklady AN SSSR, 1950, vol. 73, iss. 2, pp. 277–278 (in Russian).
22. Pokalo A. K. On one class of linear methods for summing Fourier series of differentiable functions. Vestsi Akademii navuk Belaruskai SSR. Ser. fiz.-mat. navuk, 1969, vol. 2, pp. 48–49 (in Russian).
23. Potseiko P. G. On one representation of the singular Jackson integral and the approximation of the function x s on the segment [ 1, 1]. Vesnik of Yanka Kupala State University of Grodno. Series 2. Mathematics. Physics. Informatics, Сomputer Technology and Сontrol, 2019, vol. 9, iss. 2, pp. 22–38 (in Russian).
24. Rusak V. N. On approximation by rational functions on the real axis. Izv. AN BSSR. Ser. fiz.-tekh. nauk, 1974, vol. 1, pp. 22–28 (in Russian).
25. Rusak V. N. On the order of approximation by positive rational operators. Izv. AN BSSR. Ser. fiz.-tekh. nauk, 1975, iss. 3. pp. 39–46 (in Russian).
26. Pekarskii A. A. Chebyshev rational approximations in a circle, on a circle and on a segment. Mathematics of the USSR-Sbornik, 1988, vol. 61, iss. 1, pp. 87–102.
27. Rovba E. A. Rational integral operators on a segment. Vestn. BGU. Ser. 1, Fizika. Matematika. Informatika, 1996, no. 1, pp. 34–39 (in Russian).
28. Smotritskii K. A. On the approximation of convex functions by rational integral operators on a segment. Vesnik BGU. Ser. 1., 2005, no. 3, pp. 64–70 (in Russian).
29. Rovba E. A., Potseiko P. G. On a rational singular Jackson integral on a segment. Dokl. NAN Belarusi, 2019, vol. 63, no. 4, pp. 398–407 (in Russian).
30. Rovba E. A. On one direct method in rational approximation. Docl. AN BSSR, 1979, vol. 23, iss. 11, pp. 968–971 (in Russian).
31. Patseika P. G., Rouba Y. A., Smatrytski K. A. On one rational integral operator of Fourier– Chebyshev type and approximation of Markov functions. J. BSU, Math. Inform, 2020, vol. 2, pp. 6–27.
32. Potseiko P. G., Rovba E. A. On rational Abel–Poisson sums on a segment and approximations of Markov functions. J. BSU, Math. Inform, 2021, vol. 3, pp. 6–24.
33. Potseiko P. G., Rovba E. A. On rational approximations of the Markov function on a segment by Fejer sums with a fixed number of poles. Trudy Instituta matematiki, 2022, vol. 30, iss. 1–2, pp. 57–77 (in Russian).
34. Potseiko P. G., Rovba E. A. Vallee´ Poussin sums of Fourier–Chebyshev rational integral operators and approximations of the Markov function. St. Petersburg Mathematical Journal, 2024, vol. 35, iss. 5, pp. 879–896. https://doi.org/10.1090/spmj/1834.
35. Sidorov Yu. V., Fedoryuk M. V., Shabunin M. I. Lectures on the Theory of Functions of a Complex Variable. Moscow, Nauka, Gl. red. Fiz.-mat. lit-ry, 1989. 480 p. (in Russian).
36. Evgrafov M. A. Asymptotic Estimates and Integer Functions. Moscow, Nauka, 1979. 320 p. (in Russian).
37. Fedoryuk M. V. Asymptotics. Integrals and Series. Moscow, Gl. red. Fiz.-mat. lit-ry, 1987. 544 p. (in Russian).
38. Bernshtein S. N. Extreme Properties of Polynomials and the Best Approximation of Continuous Functions of One Real Variable. Part 1. Moscow, Leningrad, The main editorial office of general technical literature, 1937. 200 p. (in Russian).
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For citations:
Patseika P.G., Rouba Ya.A. On a rational Jackson singular integral and approximations of Markov functions. Proceedings of the Institute of Mathematics of the NAS of Belarus. 2025;33(2):73-89. (In Russ.) EDN: RBMQLB
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