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

S.B. Vakarchuk; M.B. Vakarchuk

doi:10.15421/241602

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

S.B. Vakarchuk (Alfred Nobel Dnipropetrovsk University), https://orcid.org/0000-0002-2562-8844
M.B. Vakarchuk (Oles Honchar Dnipropetrovsk National University)

Abstract

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 $$$L_2$$$.

Keywords

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

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References

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.

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

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

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

Samko S.G., Kilbas A.A., Marichev O.I. Integrals and derivatives of fractional order and their applications, Nauka i tekhnika, 1987.

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

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.

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.

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.

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.

Shevchuk I.A. Approximation by polynomials and traces of functions being continuous on the segment, Naukova dumka, 1992.