Approximation by algebraic polynomials in metric spaces $$$L_{\psi}$$$

T.A. Agoshkova (Dnipropetrovsk National University of Railway Transport named after Academician V. Lazaryan),


In the space $$$L_{\psi}[-1;1]$$$ of non-periodic functions with metric $$$\rho(f,0)_{\psi} = \int\limits_{-1}^1 \psi(|f(x)|)dx$$$, where $$$\psi$$$ is a function of the type of modulus of continuity, we study Jackson inequality for modulus of continuity of $$$k$$$-th order in the case of approximation by algebraic polynomials. It is proved that the direct Jackson theorem is true if and only if the lower dilation index of the function $$$\psi$$$ is not equal to zero.


the direct Jackson theorem; modulus of continuity; modulus of continuity of k-th order; the lower dilation index; an algebraic polynomial


Krein S.G., Petunin Yu.I., Semyonov Ye.M. Interpolation of linear operators, Nauka, Moscow, 1978; 400 p. (in Russian)

Storozhenko E.A., Krotov V.G., Osval'd P. "Direct and inverse theorems of Jackson type in $$$L_p$$$ spaces, $$$0 < p < 1$$$", Mat. sb., 1975; 98(3): pp. 395-415. (in Russian)

Ivanov V.I. "Direct and inverse theorems of approximation theory in the metric of $$$L_p$$$ when $$$0 < p < 1$$$", Mat. zametki, 1975; 18(5): pp. 641-658. (in Russian)

Storozhenko E.A., Osval'd P. "Jackson theorem in $$$L_p({\mathbb{R}}^k)$$$ spaces, $$$0 < p < 1$$$", Dokl. AN SSSR, 1976; 229(3): pp. 554-557. (in Russian)

Storozhenko E.A., Osval'd P. "Jackson theorem in $$$L_p({\mathbb{R}}^k)$$$ spaces, $$$0 < p < 1$$$", Sib. mat. zhurn., 1978; 19(4): pp. 888-901. (in Russian)

Pichugov S.A. "On Jacksom theorem for periodic functions in spaces with integral metric", Ukrainian Math. J., 2000; 52(1): pp. 122-133. (in Russian)

Pichugov S.A. "On Jacksom theorem for periodic functions in metric spaces with integral metric. II", Ukrainian Math. J., 2011; 63(11): pp. 1524-1533. (in Russian)

Storozhenko E.A. "Approximation by algebraic polynomials in $$$L_p$$$, $$$0 < p < 1$$$", Vestnik MGU. Seriya: Matematika, Mehanika, 1978; 4: pp. 87-92. (in Russian)

Hodak L.B. "On approximation of functions by alebraic polynomials in $$$L_p$$$ metric when $$$0 < p < 1$$$", Mat. zametki, 1981; 30(3): pp. 321-332. (in Russian)

Shvedov A.S. "Jackson theorem in $$$L_p$$$, $$$0 < p < 1$$$, for algebraic polynomials, and the orders of comonotone approximations", Mat. zametki, 1979; 25(1): pp. 107-117. (in Russian)

Runovski K. "On Jackson’s type inequalities in Orlicz classes", Revista Mat. Comp., 2001; 14(2): pp. 394-404.

Brudnyi Yu.A. "Approximation of functions by algebraic polynomials", Izv. AN SSSR, 1968; 32(4): pp. 780-787. (in Russian)

Timan A.F. Approximation theory for real-variable functions, Fizmatgiz, Moscow, 1960; 624 p. (in Russian)


  • There are currently no refbacks.

Copyright (c) 2013 T.A. Agoshkova

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

Registered in

Not much? See «Misleading Metrics» [web-archive] (by J. Beall), «The story of fake impact factor companies ...» (by M. Jalalian); on the other hand, see «Citation Statistics» (by R. Adler, J. Ewing, P. Taylor)

ISSN (Online): 2518-7996
ISSN (Print): 2312-9557