The best polynomial approximation, derivatives of fractional order, and widths of classes of functions in $$$L_2$$$

S.B. Vakarchuk (Alfred Nobel Dnipropetrovsk University),
M.B. Vakarchuk (Oles Honchar Dnipropetrovsk National University)


On the classes of $$$2\pi$$$-periodic functions $$${\mathcal{W}}^{\alpha} (K_{\beta}, \Phi)$$$, where $$$\alpha, \beta \in (0;\infty)$$$, defined by $$$K$$$-functionals $$$K_{\beta}$$$, fractional derivatives of order $$$\alpha$$$, and majorants $$$\Phi$$$, the exact values of different $$$n$$$-widths have been computed in the space $$$L_2$$$.


the best polynomial approximation; trigonometric polynom; K-functional; fractional derivative of order $$$\alpha$$$; n-width


Samko S.G., Kilbas A.A., Marichev O.I. Integrals and derivatives of fractional order and their applications, Nauka i tekhnika, Minsk, 1987; 688 p. (in Russian)

Butzer P.L., Dyckhoff H., Gorlich E., Stens R.L. "Best trigonometric approximation, fractional order derivatives and Lipschitz classes", Canad. J. Math., 1977; 29(4): pp. 781-793. doi:10.4153/CJM-1977-081-6

Butzer P.L., Westphal U. "An introduction to fractional calculus", Applications of Fractional Calculus in Physics (ed. by R. Hilfer), Singapure, 2000; pp. 1-85. doi:10.1142/3779

Ivanov K.G. "On the rate of convergence of two moduli of functions", Pliska Stud. Math. Bulg., 1983; 5: pp. 97-104.

Ponomarenko V.G. "Moduli of smoothness of fractional order and the best approximations in $$$L_p$$$ ($$$1 < p < \infty$$$)", Proc. of International Conference on Constructive Function Theory (Varna, 1-5 June 1981), Sofia, 1983; pp. 129-133. (in Russian)

Tikhonov S. "On moduli of smoothness of fractional order", Real Analysis Exchange, 2004/2005; 30(2): pp. 507-518.

Vakarchuk S.B. "K-functionals and exact values of n-widths of certain classes from $$$L_2$$$", Matem. zametki, 1999; 66(4): pp. 494-499. (in Russian) doi:10.1007/BF02679087

Vakarchuk S.B. "On K-functionals and exact values of n-widths of certain classes in $$$C(2\pi)$$$ and $$$L_1(2\pi)$$$", Matem. zametki, 2002; 71(4): pp. 522-531. (in Russian) doi:10.1023/A:1014823613463

Vakarchuk S.B. "Mean approximation on the real domain by algebraic polynomials with Chebyshev-Hermite weight and widths of classes of functions", Matem. zametki, 2014; 95(5): pp. 666-684. (in Russian) doi:10.1134/S0001434614050046

Shabozov M.Sh., Tukhliev K. "K-functionals and exact values of n-widths of certain classes of functions in $$$L_2\bigl((1 - x^2)^{-1/2};[-1;1]\bigr)$$$ space", Izvestiya TulGU, 2014; 1(1): pp. 83-97. (in Russian)

Shevchuk I.A. Approximation by polynomials and traces of functions being continuous on the segment, Naukova dumka, Kyiv, 1992; 225 p. (in Russian)




  • There are currently no refbacks.

Copyright (c) 2016 S.B. Vakarchuk, M.B. Vakarchuk

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

Registered in


ISSN (Online): 2664-5009
ISSN (Print): 2664-4991