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# Inverse theorems in the theory of approximation of vectors in a Banach space with exponential type entire vectors

### Abstract

An arbitrary operator $A$ on a Banach space $X$ which is a generator of a $C_0$-group with a certain growth condition at infinity is considered. A relationship between its exponential type entire vectors and its spectral subspaces is found. Inverse theorems on the 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 operator $A$, and the $k$-module of continuity with respect to $A$ are established. Also, a generalization of the Bernstein-type inequality is obtained. The results allow to obtain Bernstein-type inequalities in weighted $L_p$ spaces.

### Article Information

 Title Inverse theorems in the theory of approximation of vectors in a Banach space with exponential type entire vectors Source Methods Funct. Anal. Topology, Vol. 16 (2010), no. 1, 69-82 MathSciNet MR2656133 Copyright The Author(s) 2010 (CC BY-SA)

### Authors Information

S. Torba
Institute of Mathematics, National Academy of Sciences of Ukraine, 3 Tereshchenkivs'ka, Kyiv, 01601, Ukraine

### Citation Example

S. Torba, Inverse theorems in the theory of approximation of vectors in a Banach space with exponential type entire vectors, Methods Funct. Anal. Topology 16 (2010), no. 1, 69-82.

### BibTex

@article {MFAT490,
AUTHOR = {Torba, S.},
TITLE = {Inverse theorems in the theory of approximation of vectors in a Banach space with exponential type entire vectors},
JOURNAL = {Methods Funct. Anal. Topology},
FJOURNAL = {Methods of Functional Analysis and Topology},
VOLUME = {16},
YEAR = {2010},
NUMBER = {1},
PAGES = {69-82},
ISSN = {1029-3531},
URL = {http://mfat.imath.kiev.ua/article/?id=490},
}