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

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

2008-09-24

arxiv.org/abs/0809.4030v1

Methods Funct. Anal. Topology 16 (2010), no. 1, 69-82

Mathematics

Functional Analysis

14 pages

Scientific paper

Arbitrary operator A on a Banach space X which is the generator of C_0-group with certain growth condition at infinity is considered. The relationship between its exponential type entire vectors and its spectral subspaces is found. Inverse theorems on connection between the degree of smoothness of vector $x\in X$ with respect to 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 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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