A non-archimedean Algebra and the Schwartz impossibility theorem

TL;DR

This paper constructs a non-archimedean algebra of ultrafunctions that embeds distributions and extends the derivative, circumventing Schwartz's impossibility theorem by relaxing certain algebraic requirements.

Contribution

It introduces a new algebra of restricted ultrafunctions over a non-archimedean field that embeds distributions and satisfies a weak Leibnitz rule, avoiding Schwartz's classical impossibility.

Findings

Embedded distributions in a non-archimedean algebra.

Constructed an algebra with a derivative extending distributional derivatives.

Achieved a weak Leibnitz rule in the algebra.

Abstract

In the 1950s L. Schwartz proved his famous impossibility result: for every k in N there does not exist a differential algebra (A,+,*,D) in which the distributions can be embedded, where D is a linear operator that extends the distributional derivative and satisfies the Leibnitz rule (namely D(u*v)=Du*v+u*Dv) and * is an extension of the pointwise product on the continuous functions. In this paper we prove that, by changing the requests, it is possible to avoid the impossibility result of Schwartz. Namely we prove that it is possible to construct an algebra of functions (A,+,*,D) such that (1) the distributions can be embedded in A in such a way that the restriction of the product to the C^{1} functions agrees with the pointwise product, namely for every f,g in C^{1} we have {\Phi}(fg)={\Phi}(f)*{\Phi}(g), and (2) there exists a linear operator D:A\rightarrow A that extends the distributional derivative and satisfies a weak form of the Leibnitz rule. The algebra that we construct is an algebra of restricted ultrafunction, which are generalized functions defined on a subset {\Sigma} of a non-archimedean field K (with {\Sigma}\subseteq R\subseteq K) and with values in K. To study the restricted ultrafunctions we will use some techniques of nonstandard analysis.