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On the approximations of derivatives of integrated semigroups

This paper is devoted to regularization in the process of derivative’s approximation of integrated semigroups in time variables. We consider the direct method and also the method based on A. N. Tikchonov’s approach. It is shown that discrete derivative converges in strong sense and the order of convergence in general Banach space is obtained. The presentation is given in the abstract framework of discrete approximation scheme, which includes finite element methods, finite difference schemes and projection methods.