Approximation of analytic functions by linear methods of summation of Taylor series
DOI:
https://doi.org/10.20535/mmtu-2018.1-033Keywords:
Taylor series, Analytic functions, Linear summation methods, Spaces of functions, Matrix of complex numbersAbstract
In this paper we find estimates for the deviation of polynomials generated by general linear methods of summation of Taylor series on the spaces of analytic functions $ H^{\psi}_\infty $ (Kolmogorov–Nikolsky problem). These classes, generated by the sequence $ \{\psi (k)\}^\infty_{k = 0}, \psi (k) = \psi_1 (k) + i \psi_2 (k) $, are analogues of classes of differentiable functions, which were introduced by A. I. Stepanets. In the sequence $ \{\psi_i (k)\}, i = 1,2 $, the Boas–Telyakovskii conditions are superimposed. The deviation of polynomials is considered in a uniform metric.References
Fomin, G. A. (1978). A class of trigonometric series. Mathematical Notes, 23 , 117–123. https://doi.org/10.1007/BF01153150
Goluzin, G. M. (1969). Geometric theory of functions of a complex variable (Vol. 26). Providence, RI: American Mathematical Society.
Haevskyi, M. V., & Zaderei, P. V. (2016). Approximation of analytic functions by partial sums of their Taylor series. Ukrainian Mathematical Journal, 67 (12), 1810–1830. https://doi.org/10.1007/s11253-016-1192-7
Privalov, I. I. (1950). Boundary properties of analytic functions [in Russian] (2nd ed.). Moscow: Gosudarstv. Izdat. Tehn.-Teor. Lit.
Savchuk, V. V. (2008). Approximation of $2pi$-periodic and holomorphic functions by some linear methods [in Russian]. Zbirnyk Prats Instytutu Matematyky NAN Ukrainy, 12 (5), 309–323.
Savchuk, V. V., Savchuk, M. V., & Chaichenko, S. O. (2010). Approximation of analytic functions by de la Vallèe Poussin sums. Matematychni Studii, 34 (2), 207–219. http://matstud.org.ua/texts/2010/34_2/207-219.pdf
Scheick, J. T. (1966). Polynomial approximation of functions analytic in a disk. Proceedings of the American Mathematical Society, 17 (6), 1238–1243. https://doi.org/10.1090/S0002-9939-1966-0206303-8
Shvetsova, A. M. (2000). Approximation by partial Taylor sums and the best approximation for certain classes of functions by analytic functions in the unit polydisk [in Russian]. Visnyk of Karazin Kharkiv National University. Ser. Mathematics, Applied Mathematics and Mechanics, 475 (49), 208–217.
Stepanets, A. I. (2002). Approximation theory methods [in Russian]. Kyiv: Institute of Mathematics NAS of Ukraine.
Telyakovskii, S. A. (1964). Integrability conditions for trigonometrical series and their application to the study of linear summation methods of Fourier series [in Russian]. Izv. Akad. Nauk SSSR Ser. Mat., 28 , 1209–1236. http://www.mathnet.ru/links/e107f7dc3ecea04e62f0c2ef0b543847/im3051.pdf
Telyakovskii, S. A. (1967). Asymptotic estimate of the integral of the modulus of a function given by a series of sines. Sibirskii matematiceskii zhurnal, 8 (6), 1416–1422. https://gdz.sub.uni-goettingen.de/id/PPN394039319_0008
Telyakovskii, S. A. (1971). An estimate of the norm of a function by its Fourier coefficientsthat is suitable in problems of approximation theory [in Russian]. Proceedings of the Steklov Institute of Mathematics, 109, 65–97. http://www.mathnet.ru/links/80497a228da6c8f735f714d40ba309fe/tm2982.pdf
Zaderei, P. V. (1990). On the deviation of $(psi,bar{beta })$-differentiable periodic functions from the linear mean of their Fourier series [in Russian]. Theory of real functions: XXXIV semester in Banach center, 482, Preprint of Institute of mathematics Polish Academy of Sciences (pp. 96–100).
Zygmund, A. (2002). Trigonometric series (3rd ed.). Cambridge, England: Cambridge university press.
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