Geodesic
On some modifications of Jackson's generalized theorem for the best approximations of periodic functions
Vestnik Sankt-Peterburgskogo universiteta. Prikladnaâ matematika, informatika, processy upravleniâ, no. 1 (2014), pp. 40-50 Cet article a éte moissonné depuis la source Math-Net.Ru

Voir la notice de l'article

Let us consider the space of continuous periodic functions endowed with the uniform norm. The structural properties of the functions are commonly characterized by moduli of continuity of various orders. In 1911, D. Jackson established a number of fundamental theorems that give estimates for the best approximation by the modulus of continuity of the first order for the function and its derivatives. These results were later extended to the case when the estimates of the best approximations are produced by the moduli of continuity of arbitrary order. Inequalities of this type play an important role in the theory of approximation and is studied (variously ways) in a large number of works of many authors. Similar relations are called direct theorems or generalized Jackson inequalities. Inequalities that contain the estimates of norm for intermediate derivative by the rules norms of the function and its derivatives of a higher order than the ones being estimated also play an important role in approximation theory. They are called the inequalities of Landau–Kolmogorov. In this paper the non-standard modification of Jackson-type inequalities with respect to the direction suggested by inequalities of Landau–Kolmogorov are obtained for a wide class of spaces obtained. The main instruments used in the work are approximation methods built on the basis of functions of V. A. Steklov. Bibliogr. 8.
Keywords: best approximation, moduli of a continuity, Jackson's generalized theorem. 
@article{VSPUI_2014_1_a4,
     author = {V. V. Zhuk and O. A. Tumka},
     title = {On some modifications of {Jackson's} generalized theorem for the best approximations of periodic functions},
     journal = {Vestnik Sankt-Peterburgskogo universiteta. Prikladna\^a matematika, informatika, processy upravleni\^a},
     pages = {40--50},
     year = {2014},
     number = {1},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/VSPUI_2014_1_a4/}
}
TY  - JOUR
AU  - V. V. Zhuk
AU  - O. A. Tumka
TI  - On some modifications of Jackson's generalized theorem for the best approximations of periodic functions
JO  - Vestnik Sankt-Peterburgskogo universiteta. Prikladnaâ matematika, informatika, processy upravleniâ
PY  - 2014
SP  - 40
EP  - 50
IS  - 1
UR  - http://geodesic.mathdoc.fr/item/VSPUI_2014_1_a4/
LA  - ru
ID  - VSPUI_2014_1_a4
ER  - 
%0 Journal Article
%A V. V. Zhuk
%A O. A. Tumka
%T On some modifications of Jackson's generalized theorem for the best approximations of periodic functions
%J Vestnik Sankt-Peterburgskogo universiteta. Prikladnaâ matematika, informatika, processy upravleniâ
%D 2014
%P 40-50
%N 1
%U http://geodesic.mathdoc.fr/item/VSPUI_2014_1_a4/
%G ru
%F VSPUI_2014_1_a4
V. V. Zhuk; O. A. Tumka. On some modifications of Jackson's generalized theorem for the best approximations of periodic functions. Vestnik Sankt-Peterburgskogo universiteta. Prikladnaâ matematika, informatika, processy upravleniâ, no. 1 (2014), pp. 40-50. http://geodesic.mathdoc.fr/item/VSPUI_2014_1_a4/

[1] Timan A. F., Theory of approximation of functions of a real variable, Fizmatgiz, M., 1960, 624 pp.

[2] De Vore R. A., Lorentz G. G., Constructive Approximation, Springer-Verlag, Berlin–Heidelberg–New York, 1993, 450 pp. | MR

[3] Zhuk V. V., Approximations of periodic functions, Izd-vo Leningr. un-ta, L., 1982, 366 pp. | MR | Zbl

[4] Foucart S., Kryakin Y., Shadrin A., “On the exact constant in the Jackson–Stechkin inequality for the uniform metric”, Constr. Approx., 29 (2009), 157–179 | DOI | MR | Zbl

[5] Zhuk V. V., “Some sharp inequalities between uniform the best approximations of periodic functions”, Dokl. Akad. Nauk SSSR, 201:2 (1971), 263–265 | Zbl

[6] Zhuk V. V., “Some inequalities between the best approximations of periodic functions”, Izv. Vysŝ. Uĉebn. Zaved., Mathematics, 1973, no. 9(136), 18–26 | Zbl

[7] Vinogradov O. L., Zhuk V. V., “Sharp inequalities between the modules of continuity of various orders and the best approximations of periodic functions by trigonometric polinomials and splines”, Dokl. Akad. Nauk, 389:5 (2003), 588–591 | MR | Zbl

[8] Vinogradov O. L., Zhuk V. V., “Additions to sharp inequalities of Jackson and Landau–Kolmogorov type for derivatives of small orders”, Vestnik S.-Peterb. un-ta, ser. 1: Matematika, mekhanika, astronomiya, 2003, no. 4 (25), 11–19 | Zbl