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# Direct theorems in the theory of approximation of Banach space vectors by exponential type entire vectors

### Abstract

For an arbitrary operator $A$ on a Banach space $X$ which is the generator of a $C_0$-group with certain growth condition at infinity, direct theorems on connection between the degree of smoothness of a vector $x\in X$ with respect to the operator $A$, the rate of convergence to zero of the best approximation of $x$ by exponential type entire vectors for the operator $A$, and the $k$-module of continuity are established. The results allow to obtain Jackson-type inequalities in a number of classic spaces of periodic functions and weighted $L_p$ spaces.

### Article Information

 Title Direct theorems in the theory of approximation of Banach space vectors by exponential type entire vectors Source Methods Funct. Anal. Topology, Vol. 13 (2007), no. 3, 267-278 MathSciNet MR2356759 Copyright The Author(s) 2007 (CC BY-SA)

### Authors Information

Ya. Grushka
Institute of Mathematics, National Academy of Sciences of Ukraine, 3 Teresh\-chenkivs'ka, Kyiv, 01601, Ukraine

S. Torba
Institute of Mathematics, National Academy of Sciences of Ukraine, 3 Teresh\-chenkivs'ka, Kyiv, 01601, Ukraine

### Citation Example

Ya. Grushka and S. Torba, Direct theorems in the theory of approximation of Banach space vectors by exponential type entire vectors, Methods Funct. Anal. Topology 13 (2007), no. 3, 267-278.

### BibTex

@article {MFAT428,
AUTHOR = {Grushka, Ya. and Torba, S.},
TITLE = {Direct theorems in the theory of approximation of Banach space vectors by exponential type entire vectors},
JOURNAL = {Methods Funct. Anal. Topology},
FJOURNAL = {Methods of Functional Analysis and Topology},
VOLUME = {13},
YEAR = {2007},
NUMBER = {3},
PAGES = {267-278},
ISSN = {1029-3531},
URL = {http://mfat.imath.kiev.ua/article/?id=428},
}