Pointwise Values and Fundamental Theorem in the Algebra of Asymptotic Functions

Todor D. Todorov

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arXiv · math/0601723

Pointwise Values and Fundamental Theorem in the Algebra of Asymptotic Functions

Todor D. Todorov

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Abstract

We show that the algebra of asymptotic functions $^ρ\mathcal{E}(Ω)$ (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 $^ρ\mathbb{C}$. Since the algebra $^ρ\mathcal{E}(Ω)$, contains a copy of the Schwartz distributions $\mathcal{D}^\prime(Ω)$, it follows that every Schwartz distribution is a pointwise function in the field $^ρ\mathbb{C}$. The class of asymptotic functions $^ρ\mathcal{E}(Ω)$ 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 $^ρ\mathbb{C}$.The expected applications are to PDE with variable, possibly discontinuous, coefficients and non-linear PDE with singularities.

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Todor D. Todorov. 2006-01-30. Pointwise Values and Fundamental Theorem in the Algebra of Asymptotic Functions. https://arxiv.org/abs/math/0601723

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