Abstract
We obtained the asymptotic equalities for the least upper bounds of the approximation of functions from the classes \( {W}_{\beta}^r{H}^{\alpha } \) by three-harmonic Poisson integrals in the case r + α ≤ 3 in the uniform metric.
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References
A. I. Stepanets, Methods of Approximation Theory, Part I [in Russian], Institute of Mathematics of the NAS of Ukraine, Kiev, 2002.
M. F. Timan, Approximation and Properties of Periodic Functions [in Russian], Naukova Dumka, Kiev, 2009.
V. A. Baskakov, “On some properties of operators of the Abel–Poisson type,” Mat. Zametki, 17(2), 169–180 (1975).
S. Kaniev, “On deviations of the functions biharmonic in a circle from their boundary values,” Dokl. AN SSSR, 153(5), 995–998 (1963).
S. B. Hembars’ka, “On boundary values of three-harmonic Poisson integral on the boundary of a unit disk,” Ukr. Math. J., 70(7), 1012–1021 (2018).
A. F. Timan, “The exact estimate of the residual at the approximation of periodic differentiable functions by Poisson integrals,” Dokl. AN SSSR, 74(1), 17–20 (1950).
É. L. Shtark, “The complete asymptotic expansion for the upper bound of deviations of functions from Lip1 from their singular Abel–Poisson integral,” Mat. Zametki, 13(1), 21–28 (1973).
I. V. Kal’chuk and Yu. I. Kharkevych, “Complete asymptotics of the approximation of function from the Sobolev classes by the Poisson integrals,” Acta Comment. Univ. Tartu. Math., 22(1), 23–36 (2018).
Yu. I. Kharkevich and K. V. Pozharska, “Asymptotics of approximation of conjugate functions by Poisson integrals,” Acta Comment. Univ. Tartu. Math., 22(2), 235–243 (2018).
Yu. I. Kharkevich and T. A. Stepanyuk, “Approximation properties of Poisson integrals for the classes \( {C}_{\beta}^{\psi }{H}^{\alpha } \),” Math. Notes, 96(5–6), 1008–1019 (2014).
I. V. Kal’chuk, Yu. I. Kharkevych, and K. V. Pozharska, “Asymptotics of approximation of functions by conjugate Poisson integrals,” Carpath. Math. Publ., 12(1), 138–147 (2020).
T. V. Zhyhallo and Yu. I. Kharkevych, “Approximating properties of biharmonic Poisson operators in the classes \( {\hat{L}}_{\beta, 1}^{\psi } \),” Ukr. Math. J., 69(5), 757–765 (2017).
I. V. Kal’chuk and Yu. I. Kharkevych, “Approximating properties of biharmonic Poisson integrals in the classes \( {W}_{\beta}^r{H}^{\alpha } \),” Ukr. Math. J., 68(11), 1727–1740 (2017).
U. Z. Hrabova, I. V. Kal’chuk, and T. A. Stepaniuk, “On the approximation of the classes \( {W}_{\beta}^r{H}^{\alpha } \) by biharmonic Poisson integrals,” Ukr. Math. J., 70(5), 719–729 (2018).
F. G. Abdullayev and Yu. I. Kharkevych, “Approximation of the classes \( {C}_{\beta}^{\psi }{H}^{\alpha } \) by biharmonic Poisson integrals,” Ukr. Math. J., 72(1), 21–38 (2020).
K. M. Zhyhallo and Yu. I. Kharkevych, “On the approximation of functions of the Hölder class by triharmonic Poisson integrals,” Ukr. Math. J., 53(6), 1012–1018 (2001).
I. V. Kal’chuk, V. I. Kravets, and U. Z. Hrabova, “Approximation of the classes \( {W}_{\beta}^r{H}^{\alpha } \) by three-harmonic Poisson integrals,” J. Math. Sci., 246(1), 9–50 (2020).
U. Z. Hrabova and I. V. Kal’chuk, “Approximation of the classes \( {W}_{\beta, \infty}^r \) by three-harmonic Poisson integrals,” Carpath. Math. Publ., 11(2), 321–324 (2019).
L. I. Bausov, “Linear methods of summation of the Fourier series with given rectangular matrices, II,” Izv. Vyssh. Ucheb. Zav. Mat., 6, 3–17 (1966).
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Translated from Ukrains’kiĭ Matematychnyĭ Visnyk, Vol. 17, No. 4, pp. 538–548, October–December, 2020.
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Hrabova, U.Z., Kal’chuk, I.V. & Filozof, L.I. Approximative properties of the three-harmonic Poisson integrals on the classes \( {W}_{\beta}^r{H}^{\alpha } \). J Math Sci 254, 397–405 (2021). https://doi.org/10.1007/s10958-021-05311-8
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DOI: https://doi.org/10.1007/s10958-021-05311-8