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Arbitrary operator A on a Banach space X which is the generator of C 0 -group with certain growth condition at infinity is considered. The relationship between its exponential type entire vectors and its spectral subspaces is found. Inverse theorems on connection between the degree of smoothness of vector x ∈ X with respect to operator A, the rate of convergence to zero of the best approximation of x by exponential type entire vectors for operator A, and the k-module of continuity are established. Also, a generalization of the Bernstein-type inequality is obtained. The results allow to obtain Bernstein-type inequalities in weighted Lp spaces.

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We construct the logarithmic extension for real numbers in which the numbers, less then −∞ exist. Using this logarithmic extension we give the single formula for hyperbolic representation of generalized tachyon Lorentz transforms.