An axiomatic approach to the non-linear theory of generalized functions and consistency of Laplace transforms

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 non-linear 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.

Keywords:

• totally ordered field
• non-Archimedean valuation field
• real closed field
• saturated field
• Schwartz distributions
• Colombeau algebra
• Laplace transform

AMS Subject Classifications :

• 46F30
• 44A10
• 46S10
• 46F10
• 12J15

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CORRIGENDUM

Acknowledgements

The research reported in this paper was partly supported by START-project Y237 of the Austrian Science Fund. The author thanks the anonymous referees for the numerous constructive remarks which improved the quality of the manuscript.