Inequalities of Jackson type for functions with values in Hilbert space

V.F. Babenko (Oles Honchar Dnipropetrovsk National University, Institute of Applied Mathematics and Mechanics of NAS of Ukraine),
S.V. Savela (Дніпропетровський національний університет імені Олеся Гончара)


We present the generalization of M.I. Chernykh's results about the estimate of the best $$$L_2$$$-approximation of periodic function $$$f$$$ by trigonometric polynomials by its $$$L_2$$$-modulus of continuity, in the case of functions with values in Hilbert space.


Jackson D. Über die Genauigkeit der Annäherung stetiger Funktionen durch ganze rationale Funktionen gegebenen Grades und trigonometrische Summen gegebener Ordnung: Diss., Gottingen, 1911. (in German)

Korneichuk N.P. "On the best approximation on some classes of continuous functions", Doklady AN SSSR, 1961; 140: pp. 748-751. (in Russian)

Korneichuk N.P. "Exact constant in Jackson theorem of the best uniform appproximation of continuous periodic functions", Dokl. AN SSSR, 1962; 145: pp. 514-515. (in Russian)

Korneichuk N.P. "Exact value of the best approximations and widths of some classes of functions", Doklady AN SSSR, 1963; 150: pp. 1218-1220. (in Russian)

Chernykh N.I. "On Jackson inequality in $$$L_2$$$ space", Trudy MIAN SSSR, 1967; 88: pp. 71-74. (in Russian)

Chernykh N.I. "On the best approximation of periodic functions by trigonometrical polynomials in $$$L_2$$$", Matem. zametki, 1967; 2(5): pp. 513-522. (in Russian) doi:10.1007/BF01093942

Grigoryan Yu.I. "Diameters of certain sets in function spaces", Matem. Zametki, 1973; 13(5): pp. 637-646. (in Russian) doi:10.1007/BF01147464

Babenko V.F., Pichugov S.A. "Approximation of continuous vector functions", Ukrainian Math. J., 1994; 46(11): pp. 1435-1448. (in Russian) doi:10.1007/BF01058878

Ivanov V.I., Smirnov O.I. "Jackson and Young constants in $$$L_p$$$ spaces", Tula State Univ., 1995. (in Russian)

Dorogovtsev A.Ya. Periodic and stationary states of infinite-dimensional deterministic and stochastic dynamic systems, Kyiv, 1992. (in Russian)

Korneichuk N.P. Extremum problems in approximation theory, Moscow, 1976; 320 p. (in Russian)




  • There are currently no refbacks.

Copyright (c) 2008 V.F. Babenko, S.V. Savela

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