Direct theorems in the theory of approximation of Banach space vectors by exponential type entire vectors - Torba, S. M.; Grushka, Ya. I. (Methods of Functional Analysis and Topology, 2007)
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 xX 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.