On approximations of Riemann--Liouville integral on a segement by rational Fourier--Chebyshev integral operators

P. G. Patseika ; E. A. Rovba

Trudy Instituta matematiki, Tome 32 (2024) no. 1, pp. 38-56.

Résumé

Approximations of Riemann–Liouville integral on a segment by rational integral operators Fourier–Chebyshev are investigated. An integral representation of the approximations is found. Rational approximations Riemann–Liouville integral with density $\varphi_\gamma(x) = (1-x)^\gamma,$ $\gamma \in (0,+\infty)\backslash\mathbb{N},$ are studied, estimates of pointwise and uniform approximations are established. In the case of one pole in an open complex plane, an asymptotic expression is obtained for the approximating function majorants of uniform approximations and the optimal value of the parameter at which the majorant has the asymptotically highest rate of decrease. As a consequence, estimates of approximations of Riemann–Liouville integral with density belonging to some classes of continuous functions on the segment by partial sums of the polynomial Fourier–Chebyshev series are obtained.

Détail

Keywords: Riemann–Liouville integral, rational Fourier–Chebyshev integral operator, functions with power singularity, uniform approximations, asymptotic estimates, Laplace method.

Bibliographie

[1] S. G. Samko, A. A. Kilbas, O. I. Marichev, Integrals and derivatives of fractional order and some of their applications, Nauka i tekhnika Publ, Minsk, 1987 (in Russian)

[2] S. M. Nikol'skii, “On the best approximation of a function whose s-th derivative has discontinuities of the first kind”, Dokl. Academy of Sciences of the USSR, 55:2 (1947), 99–102 (in Russian) | Zbl

[3] A. A. Tyleneva, “Approximation of the Riemann-Liouville integrals by algebraic polynomials on the segment”, Izv. Saratov University. Ser. Math. Fur. Inform, 14:3 (2014), 305–311 (in Russian)

[4] I. M. Ganzburg, “On the approximation of functions with a given modulus of continuity by P. L. Chebyshev sums”, Dokl. Academy of Sciences of the USSR, 91:6 (1953), 1253–1256 (in Russian) | Zbl

[5] V. M. Badkov, “Approximation of functions in the uniform metric by Fourier sums over orthogonal polynomials”, Trudy MIAN, 145, 1980, 20–62 | Zbl

[6] S. G. Selivanova, “Asymptotic estimates for approximations of differentiable non-periodic functions by Chebyshev sums”, Dokl. Academy of Sciences of the USSR, 105:4 (1955), 648–651 (in Russian) | Zbl

[7] A. F. Timan, L. I. Tuchinskii, “Approximation of differentiable functions defined on a finite interval by algebraic polynomials”, Dokl. Academy of Sciences of the USSR, 111:4 (1956), 771–773 (in Russian)

[8] R. Raitsin, “A The best approximation of a certain class of differentiable functions by algebraic polynomials”, Soviet Math. (Iz. VUZ), 20:1 (1976), 54–62 | MR

[9] R. A. Raitsin, “Fourier-Chebyshev series of a class of functions”, Soviet Math. (Iz. VUZ), 32:10 (1988), 123–126 | MR

[10] E. A. Rovba, “Approximation of functions differentiable in the Riemann-Liouville sense by rational operators”, Doklady of the National Academy of Sciences of Belarus, 40:6 (1996), 18–22 (in Russian) | MR | Zbl

[11] K. A. Smotritskii, “Approximation by rational operators of Valle Poussin on a segment”, Trudy Instituta matematiki, 9 (2001), 110–114 (in Russian)

[12] K. A. Smotritskii, “On the approximation of functions differentiable in the sense of Riemann-Liouville”, Vestsi NAN Belarusi. Ser. fiz. mat. navuk, 2002, no. 4, 42–47 (in Russian) | MR | Zbl

[13] I. V. Rybachenko, “Rational interpolation of functions with the Riemann-Liouville derivative from $L_p$”, Vestnik BGU. Ser. 1, 2006, no. 2, 69–74 (in Russian) | MR | Zbl

[14] A. P. Starovoitov, “Comparison of the rates of rational and polynomial approximations of differentiable functions”, Math. Notes, 44:4 (1988), 770–774 | DOI | MR | Zbl

[15] A. P. Starovoitov, “Rational approximations of Riemann-Liouville and Weyl fractional integrals”, Math. Notes, 78:3 (2005), 391–402 | DOI | DOI | MR | Zbl

[16] S. Pashkovskii, Computational applications of polynomials and Chebyshev series, Nauka. Gl. red. fiz. mat. lit-ry, M., 1983 (in Russian)

[17] N. I. Vasiliev, Yu. A. Klokov, A. Ya. Shkerstena, Application of Chebyshev polynomials in numerical analysis, Zinatne Publ, Riga, 1984 (in Russian) | MR

[18] P. K. Suetin, Classical orthogonal polynomials, 3rd ed., revised. and additional, FIZMATLIT, M., 2005

[19] S. V. Marfitsyn, V. P. Marfitsyn, “The use of Chebyshev's polynomials of the first type for a description of the steady-state conditions of metal under constant and variable loading”, Vestnik Kurganskogo gosuniversiteta. Ser. tekhn. nauki, 2016, no. 1(3), 96–98 (in Russian)

[20] T. Miyakoda, “Direct discretization of the fractional-order differential by using Chebyshev series expansion”, PAMM Proc. Appl. Math. Mech., 7 (2007), 2020011–2020012 | DOI

[21] E. A. Rovba, “On one direct method in rational approximation”, Dokl. NAS BSSR, 23:11 (1979), 968–971 (in Russian) | MR | Zbl

[22] 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

[23] P. G. Potseiko, Y. A. Rovba, “Approximations on classes of Poisson integrals by Fourier-Chebyshev rational integral operators”, Siberian Math. J., 62:2 (2021), 292–312 | DOI | MR | Zbl

[24] V. N. Rusak, Rational functions as an apparatus of approximation, Belorusskii gosudarstvennyi universitet im. V. I. Lenina, Minsk, 1979, 179 pp. (in Russian)

[25] S. Nikolsky, “On the best approximation of functions satisfying Lipschitz's conditions by polynomials”, Izvestia Akad. Nauk SSSR, 10:4 (1946), 295–322 (in Russian) | Zbl

[26] M. A. Evgrafov, Asymptotic estimates and entire functions, Nauka, M., 1979, 320 pp. (in Russian) | MR

[27] Fedoryuk M. V., Asymptotics. Integrals and series, Nauka, M., 1987, 544 pp. (in Russian) | MR

[28] W. Pinkewitch, “Sur l'ordre du reste de la serie de Fourier des fonctions derivables au sens de Weyl”, Izvestia Akad. Nauk SSSR, 4:6 (1940), 521–528 (in Russian)