| Summary: | <p>This dissertation refines and further develops numerical methods for the inversion of the classical Laplace transform and explores the effectiveness of these methods when applied (<i>a</i>) to an asymptotic generalization of the Laplace transform for generalized functions and (<i>b</i>) to the numerical approximation of solutions of ill-posed evolution equations (e.g. backwards in time problems). </p>
<p>Chapter 1 of the dissertation reviews some of the key features of asymptotic Laplace transform theory and its application to evolution equations. Although some of the statements and results contain slight modifications and improvements, the material presented in Chapter 1 is known to the experts in the field. The main contributions of this work is in Chapter 2 where an attempt is made to help clarify and determine the size of the constant in the celebrated Hersh-Kato and Brenner-Thomèe approximation theorem of semigroup theory. In particular, by improving an earlier estimate, we are able to show that matrix semigroups <i>e<sup>tA</sup></i> can be approximated "without scaling and squaring" in terms of the resolvent <i>R</i>(λ,<i>A</i>)<i>=</i>(λ<i>I -A</i>)<sup>-1</sup> of the generating matrix <i>A </i>(see Theorem 2.3.2). Also, our estimate of the Brenner-Thomèe constant given in Section 2.4 improves earlier estimates given by Neubrander, Özer, and Sandmaier in [28]. The techniques used in Section 2.4 open the door to Theorem 2.5.1, a first attempt to lift the matrix result (Theorem 2.3.2) to the general semigroup setting. Finally, in (2.31) we present a new approach on how to approximate the continuous representatives <i>f=k*u</i> of a generalized function u in terms of its Laplace transform <i>û</i>. </p>
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