# Approximation of analytic functions by linear methods of summation of Taylor series

## DOI:

https://doi.org/10.20535/mmtu-2018.1-033## Keywords:

Taylor series, Analytic functions, Linear summation methods, Spaces of functions, Matrix of complex numbers## Abstract

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.

## Downloads

## Issue

## Section

## License

Copyright (c) 2018 Mathematics in Modern Technical University

This work is licensed under a Creative Commons Attribution 4.0 International License.