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.

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Title

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

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},
MRNUMBER = {MR2656133},
URL = {http://mfat.imath.kiev.ua/article/?id=490},
}