Pointwise Values and Fundamental Theorem in the Algebra of Asymptotic Functions

Abstract

We show that the algebra of asymptotic functions ${}^{\rho }\mathcal{E}(\Omega )$ (introduced in another paper by the author of this article jointly with M. Oberguggenberger) is isomorphic to a class of pointwise functions in the field of A. Robinson asymptotic numbers ${}^{\rho }\mathbb{C}$. Since the algebra ${}^{\rho }\mathcal{E}(\Omega )$, contains a copy of the Schwartz distributions ${\mathcal{D}}^{\prime }(\Omega )$, it follows that every Schwartz distribution is a pointwise function in the field ${}^{\rho }\mathbb{C}$. The class of asymptotic functions ${}^{\rho }\mathcal{E}(\Omega )$ is an algebra of generalized functions of Colombeau's type. The field of the constants of this algebra (the functions with zero gradient) coincides with Robinson field ${}^{\rho }\mathbb{C}$.The expected applications are to PDE with variable, possibly discontinuous, coefficients and non-linear PDE with singularities.

arXiv comment15 pages