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.