We deduce asymptotic equalities for the least upper bounds of the approximations of functions from the classes \( {W}_{\beta}^r{H}^{\alpha } \) and H α by biharmonic Poisson integrals in the uniform metric.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
A. N. Tikhonov and A. A. Samarskii, Equations of Mathematical Physics [in Russian], Nauka, Moscow (1977).
B. Nagy, “Über gewise Extremalfragen bei transformierten trigonometrischen Entwicklungen, I,” Period. Fall, Ber. Math. Phys. KL. Akad., 90, 103–134 (1938).
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 the deviation of functions from Lip11 from a singular integral,” in: Proc. of the All-Union Mathematical Symp. “Embedding Theorems and Their Applications” [in Russian], Nauka, Alma-Ata (1976), pp. 163–167.
T. I. Amanov and L. P. Falaleev, “Approximation of differentiable functions by Abel–Poisson-type operators,” Proc. of the Fifth Soviet–Czechoslovak Meeting on the Application of 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).
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).
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).
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).
S. Kaniev, “Sharp estimate for the mean deviation of functions biharmonic in a disk from their boundary values,” Dop. Akad. Nauk Ukr. RSR, No. 4, 451–453 (1964).
V. A. Petrov, “Biharmonic Poisson integral,” Lit. Mat. Sb., 7, No. 1, 137–142 (1967).
V. N. Trofimov and A. S. Tsygankov, “Sharp inequalities for the oscillations of harmonic and biharmonic functions in a disk,” Sib. Mat. Zh., 10, No. 2, 398–416 (1969).
L. Gonzales, E. Keller, and G. Wildenhain, “Über das Randverhalten des Poisson-Integrals der polyharmonischen Gleichung,” Math. Nachr., 95, 157–164 (1980).
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).
V. P. Zastavnyi, “Sharp estimate for the approximation of some classes of differentiable functions by convolution operators,” Ukr. Mat. Visn., 7, No. 3, 409–433 (2010).
K. M. Zhyhallo and Yu. I. Kharkevych, “Approximation of functions from the classes \( {C}_{\beta, \infty}^{\psi } \) ,” Ukr. Mat. Zh., 63, No. 7, 939–959 (2011); English translation: Ukr. Math. J., 63, No. 7, 1083–1107 (2011).
L. I. Bausov, “Linear methods of summation of the Fourier series with given rectangular matrices. I,” Izv. Vyssh. Uchebn. Zaved., Ser. Mat., 46, No. 3, 15–31 (1965).
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).
Yu. I. Kharkevich and T. A. Stepanyuk, “Approximation properties of Poisson integrals for the classes \( {C}_{\beta}^{\psi }{H}^{\alpha } \) ,” Math. Notes, 96, No. 5, 1008–1019 (2014).
I. S. Gradshtein and I. M. Ryzhik, Tables of Integrals, Sums, Series, and Products [in Russian], Fizmatgiz, Moscow (1963).
Author information
Authors and Affiliations
Additional information
Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 68, No. 11, pp. 1493–1504, November, 2016.
Rights and permissions
About this article
Cite this article
Kal’chuk, I.V., Kharkevych, Y.I. Approximating Properties of Biharmonic Poisson Integrals in the Classes \( {W}_{\beta}^r{H}^{\alpha } \) . Ukr Math J 68, 1727–1740 (2017). https://doi.org/10.1007/s11253-017-1323-9
Received:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11253-017-1323-9