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

TL;DR

This paper explores the relationship between exponential type entire vectors and spectral subspaces of operators in Banach spaces, establishing inverse theorems linking smoothness, approximation rate, and continuity modules, with applications to weighted L_p spaces.

Contribution

It introduces inverse theorems connecting smoothness, approximation, and spectral properties of operators, generalizing Bernstein inequalities in Banach spaces.

Findings

Relationship between exponential type entire vectors and spectral subspaces established

Inverse theorems linking smoothness and approximation rate proved

Generalized Bernstein-type inequalities obtained for weighted L_p spaces

Abstract

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.