Abstract
We offer an axiomatic definition of a differential algebra of generalized functions over an algebraically closed non-Archimedean field. This algebra is of Colombeau type in the sense that it contains a copy of the space of Schwartz distributions. We study the uniqueness of the objects we define and the consistency of our axioms. Next, we identify an inconsistency in the conventional Laplace transform theory. As an application, we offer a free of contradictions alternative in the framework of our algebra of generalized functions. The article is aimed at mathematicians, physicists and engineers who are interested in the nonlinear theory of generalized functions, but who are not necessarily familiar with the original Colombeau theory. We assume, however, some basic familiarity with the Schwartz theory of distributions.
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Acknowledgements
The author thanks the anonymous referees for the numerous constructive remarks which improved the quality of the manuscript. The author also thanks Günther Hörmann for the discussion of some topics of the manuscript. The work is partly supported by START-project Y237 of the Austrian Science Fund.