Rational approximation schemes for solutions of abstract Cauchy problems and evolution equations

In this dissertation we study time and space discretization methods for approximating solutions of abstract Cauchy problems and evolution equations in a Banach space setting. Two extensions of the Hille-Phillips functional calculus are developed. The first result is the Hille-Phillips functional calculus for generators of bi-continuous semigroups, and the second is a C-regularized version of the Hille-Phillips functional calculus for generators of C-regularized semigroups. These results are used in order to study time discretization schemes for abstract Cauchy problems associated with generators of bi-continuous semigroups as well as C-regularized semigoups. Stability, convergence results, and error estimates for rational approximation schemes for bi-continuous and C-regularized semigroups are presented. We also extend the Trotter-Kato Theorem to the framework of C-regularized semigroups and combine it with the time discretization methods previously mentioned in order to obtain fully discretized schemes, provided by A-stable rational functions. Among the applications, we outline how to use rational approximation schemes to approximate solutions of nonlinear ODE\u27s, and we show the significance of the results for bi-continuous semigroups for obtaining new numerical inversion formulas for the Laplace transform (with sharp error estimates). Furthermore, rational approximation schemes for integrated semigroups are presented with applications to the second order abstract Cauchy problem