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For an arbitrary operator A on a Banach space X which is a generator ofC_0-group with certain growth condition at the infinity, the direct theorems onconnection between the smoothness degree of a vector $x\in X$ with respect tothe operator A, the order of convergence to zero of the best approximation of xby exponential type entire vectors for the operator A, and the k-module ofcontinuity are given. Obtained results allows to acquire Jackson-typeinequalities in many classic spaces of periodic functions and weighted $L_p$spaces.

Direct Theorems in the Theory of Approximation of the Banach Space Vectors by Entire Vectors of Exponential Type