Pointwise Values and Fundamental Theorem in the Algebra of Asymptotic Functions

Details Pointwise Values and Fundamental Theorem in the Algebra of Asymptotic Functions Pointwise Values and Fundamental Theorem in the Algebra of Asymptotic Functions

2006-01-30

arxiv.org/abs/math/0601723v1

Chapman & Hall/CRC Research Notes in Mathematics, 401, 1999, p. 369-383.

Mathematics

Functional Analysis

15 pages

Scientific paper

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.

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