We obtain the exact values of upper bounds of approximations of classes of periodic conjugate differentiable functions by their Abel–Poisson integrals in uniform and integral metrics.
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A. I. Stepanets, Classification and Approximation of Periodic Functions [in Russian], Naukova Dumka, Kiev (1987).
A. I. Stepanets, Methods of Approximation Theory [in Russian], Vol. 1, Institute of Mathematics, Ukrainian National Academy of Sciences, Kiev (2002).
S. M. Nikol’skii, “Approximation of functions by trigonometric polynomials in the mean,” Izv. Akad. Nauk SSSR, Ser. Mat. , 10, No. 6, 207–256 (1946).
S. B. Stechkin and S. A. Telyakovskii, “On approximation of differentiable functions by trigonometric polynomials in the metric of L,” Tr. Mat. Inst. Akad. Nauk SSSR, 88, 20–29 (1967).
V. P. Motornyi, “Approximation of periodic functions by trigonometric polynomials in the mean,” Mat. Zametki, 16, No. 1, 15–26 (1974).
N. P. Korneichuk, Extremal Problems in Approximation Theory [in Russian], Nauka, Moscow (1976).
P. Pych, “Approximation of functions in L- and C-metrics,” Ann. Soc. Math. Pol., 1, No. 11, 61–76 (1967).
I. P. Natanson, “On the order of approximation of a continuous 2π-periodic function by its Poisson integral,” Dokl. Akad. Nauk SSSR, 72, 11–14 (1950).
A. F. Timan, “Sharp estimate for the remainder in the approximation of periodic differentiable functions by Poisson integrals,” Dokl. Akad. Nauk SSSR, 72, 11–14 (1950).
B. S. Nagy, “Sur l’ordre de l’approximation d’une fonction par son integrale de Poisson,” Acta Math. Acad. Sci. Hung., 1, 183–188 (1950).
L. V. Malei, “Sharp estimate for approximation of quasismooth functions by Poisson integrals,” Dokl. Akad. Nauk Belorus. SSR, Ser. Fiz.-Tekh., No. 3, 25–32 (1961).
L. I. Bausov, “Linear methods for summation of Fourier series with given rectangular matrices. I,” Izv. Vyssh. Uchebn. Zaved., 46, No. 3, 15–31 (1965).
É. L. Stark, “Complete asymptotic expansion for the upper bound of the deviation of functions from Lip1 from their singular Abel–Poisson integral,” Mat. Zametki, 13, No. 1, 21–28 (1973).
V. A. Baskakov, “Some properties of Abel–Poisson-type operators,” Mat. Zametki, 17, No. 2, 169–180 (1975).
V. A. Baskakov, “Asymptotic estimates for approximation of conjugate functions by conjugate Abel–Poisson integrals,” in: Application of Functional Analysis to Approximation Theory [in Russian], Issue 5, Kalinin (1975), pp. 14–20.
L. P. Falaleev, “Approximation of conjugate functions by generalized Abel–Poisson operators,” Mat. Zametki, 67, No. 4, 595–602 (2000).
L. P. Falaleev, “On approximation of functions by generalized Abel–Poisson operators,” Sib. Mat. Zh., 42, No. 4, 926–936 (2001).
K. M. Zhyhallo and Yu. I. Kharkevych, “Complete asymptotics of the deviation of a class of differentiable functions from the set of their harmonic Poisson integrals,” Ukr. Mat. Zh., 54, No. 1, 43–52 (2002).
I. S. Gradshtein and I. M. Ryzhik, Tables of Integrals, Sums, Series, and Products [in Russian], Fizmatgiz, Moscow (1963).
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Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 61, No. 1, pp. 73–82, January, 2009.
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Zhyhallo, K.M., Kharkevych, Y.I. Approximation of conjugate differentiable functions by their Abel–Poisson integrals. Ukr Math J 61, 86–98 (2009). https://doi.org/10.1007/s11253-009-0196-y
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DOI: https://doi.org/10.1007/s11253-009-0196-y