2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/169553For 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.13 pages, typos correctedFunctional AnalysisSpectral Theory41A25; 41A17; 41A65text