DIRECT THEOREMS IN THE THEORY OF APPROXIMATION OF VECTORS IN A BANACH SPACE WITH EXPONENTIAL TYPE ENTIRE VECTORS

For an arbitrary operator A on a Banach space X which is the generator of a C-0-group with certain growth condition at infinity, direct theorems on connection between the degree of smoothness of a vector x is an element of 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 the operator A, and the k-module of continuity are established. The results allow to obtain Jackson-type inequalities in a number of classic spaces of periodic functions and weighted L-p spaces.