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The purpose of this paper is to give sharp error estimates for regularized versions of A-stable rational approximations of C-regularized semigroups such as the Backward Euler and Crank-Nicolson scheme among others. The main tools used are those developed by P. Brenner and V. Thomee for strongly continuous semigroups together with a regularized version of the Hille-Phillips functional calculus.

Rational approximation schemes for solutions of the first and second order Cauchy problem