  For an arbitrary operator A on a Banach space X which is a generator of
C_0-group with certain growth condition at the infinity, the direct theorems on
connection between the smoothness degree of a vector $x\in X$ with respect to
the operator A, the order 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 given. Obtained results allows to acquire Jackson-type
inequalities in many classic spaces of periodic functions and weighted $L_p$
spaces.
