We obtain asymptotic equalities for the least upper bounds of deviations of the biharmonic Poisson integrals of functions of the classes \( {W}_{\beta}^r{H}^{\alpha } \) in the case where r > 2, 0 ≤ α < 1.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
B. Nagy, “Sur l’ordre de l’approximation d’une fonction par son integrale de Poisson,” Acta Math. Acad. Sci. Hung., 1, 183–188 (1950).
S. B. Hembars’ka, “Tangential limit values of a biharmonic Poisson integral in a disk,” Ukr. Mat. Zh., 49, No. 9, 1171–1176 (1997); English translation: Ukr. Math. J., 49, No. 9, 1317–1323 (1997).
A. I. Stepanets, Methods of Approximation Theory [in Russian], Vol. 1, Institute of Mathematics, Ukrainian National Academy of Sciences, Kiev (2002).
S. Kaniev, “On the deviation of functions biharmonic in a disk from their boundary values,” Dokl. Akad. Nauk SSSR, 153, No. 5, 995–998 (1963).
P. Pych, “On a biharmonic function in unit disc,” Ann. Pol. Math., 20, No. 3, 203–213 (1968).
L. P. Falaleev, “Complete asymptotic decomposition for the upper bound of deviations of functions in Lip11 from a singular integral,” in: Proc. of the All-Union Mathematical Symp. “Embedding Theorems and Their Applications” [in Russian], Nauka Kaz. SSR, Alma-Ata (1976), pp. 163–167.
T. I. Amanov and L. P. Falaleev, “Approximation of differentiable functions by Abel–Poisson-type operators,” in: Proc. of the Fifth Soviet–Czechoslovak Meeting on the Application of the Methods of the Theory of Functions and Functional Analysis to Problems of Mathematical Physics (Alma-Ata, 1976) [in Russian], Novosibirsk (1979), pp. 13–16.
K. M. Zhyhallo and Yu. I. Kharkevych, “On the approximation of functions of the Hölder class by biharmonic Poisson integrals,” Ukr. Mat. Zh., 52, No. 7, 971–974 (2000); English translation: Ukr. Math. J., 52, No. 7, 1113–1117 (2000).
S. B. Hembars’ka and K. M. Zhyhallo, “Approximative properties of biharmonic Poisson integrals on Hölder classes,” Ukr. Mat. Zh., 69, No. 7, 925–932 (2017); English translation: Ukr. Math. J., 69, No. 7, 1075–1084 (2017).
K. M. Zhyhallo and Yu. I. Kharkevych, “Approximation of differentiable periodic functions by their biharmonic Poisson integrals,” Ukr. Mat. Zh., 54, No. 9, 1213–1219 (2002); English translation: Ukr. Math. J., 52, No. 7, 1462–1470 (2002).
K. M. Zhyhallo and Yu. I. Kharkevych, “Approximation of conjugate differentiable functions by biharmonic Poisson integrals,” Ukr. Mat. Zh., 61, No. 3, 333–345 (2009); English translation: Ukr. Math. J., 61, No. 3, 399–413 (2009).
Yu. I. Kharkevych and I. V. Kal’chuk, “Asymptotics of the values of approximations in the mean for classes of differentiable functions by using biharmonic Poisson integrals,” Ukr. Mat. Zh., 59, No. 8, 1105–1115 (2007); English translation: Ukr. Math. J., 59, No. 8, 1224–1237 (2007).
A. I. Stepanets, Classification and Approximation of Periodic Functions [in Russian], Naukova Dumka, Kiev (1987).
Yu. I. Kharkevych and T. V. Zhyhallo, “Approximation of functions from the class \( {\widehat{C}}_{\beta, \infty}^{\psi } \) by Poisson biharmonic operators in the uniform metric,” Ukr. Mat. Zh., 60, No. 5, 669–693 (2008); English translation: Ukr. Math. J., 60, No. 5, 769–798 (2008).
K. M. Zhyhallo and Yu. I. Kharkevych, “Approximation of functions from the classes \( {C}_{\beta, \infty}^{\psi } \) by biharmonic Poisson integrals,” Ukr. Mat. Zh., 63, No. 7, 939–959 (2011); English translation: Ukr. Math. J., 63, No. 7, 1083–1107 (2011).
K. M. Zhyhallo and Yu. I. Kharkevych, “Approximation of (ψ, β)-differentiable functions of low smoothness by biharmonic Poisson integrals,” Ukr. Mat. Zh., 63, No. 12, 1602–1622 (2011); English translation: Ukr. Math. J., 63, No. 12, 1820–1844 (2012).
I. V. Kal’chuk and Yu. I. Kharkevych, “Approximating properties of biharmonic Poisson integrals in the classes \( {W}_{\beta}^r{H}^{\alpha } \) ,” Ukr. Mat. Zh., 68, No. 11, 1493–1504 (2016); English translation: 68, No. 11, 1727–1740 (2017).
L. I. Bausov, “Linear methods of summation of Fourier series with given rectangular matrices. II,” Izv. Vyssh. Uchebn. Zaved., Ser. Mat., 46, No. 3, 15–31 (1965).
Yu. I. Kharkevich and T. A. Stepanyuk, “Approximating properties of Poisson integrals in the classes \( {W}_{\beta}^{\psi }{H}^{\alpha } \) ,” Mat. Zametki, 96, No. 6, 939–952 (2014).
Author information
Authors and Affiliations
Additional information
Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 70, No. 5, pp. 625–634, May, 2018.
Rights and permissions
About this article
Cite this article
Hrabova, U.Z., Kal’chuk, I.V. & Stepanyuk, T.A. On the Approximation of the Classes \( {W}_{\beta}^r{H}^{\alpha } \) by Biharmonic Poisson Integrals. Ukr Math J 70, 719–729 (2018). https://doi.org/10.1007/s11253-018-1528-6
Received:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11253-018-1528-6