Hans Vernaeve

Internal sets and internal functions in Colombeau theory

This paper is devoted to the study of generalized functions as pointwise functions (so-called internal functions) on certain sets of generalized points (so-called internal sets). We treat the case of the Colombeau algebras of generalized functions, for which these notions have turned out to constitute a fundamental technical tool. We provide general foundations for the notion of internal functions and internal sets and prove a saturation principle. Various applications to Colombeau algebras are given.

Ideals in the Ring of Colombeau Generalized Numbers

The structure of the ideals in the ring of Colombeau generalized numbers is investigated. Connections with the theories of exchange rings, Gelfand rings and lattice-ordered rings are given. Characterizations for prime, projective, pure, and topologically closed ideals are given, answering in particular the questions about prime ideals in [1]. Also z-ideals [23] are characterized. It is shown that the quotient rings modulo maximal ideals are canonically isomorphic with nonstandard fields of asymptotic numbers and that the Hahn–Banach extension property does not hold for a large class of topological modules over the ring of Colombeau generalized numbers.

Full algebra of generalized functions and non-standard asymptotic analysis

AbstractWe construct an algebra of generalized functions endowed with a canonical embedding of the space of Schwartz distributions.We offer a solution to the problem of multiplication of Schwartz distributions similar to but different from Colombeau’s solution.We show that the set of scalars of our algebra is an algebraically closed field unlike its counterpart in Colombeau theory, which is a ring with zero divisors. We prove a Hahn–Banach extension principle which does not hold in Colombeau theory. We establish a connection between our theory with non-standard analysis and thus answer, although indirectly, a question raised by Colombeau. This article provides a bridge between Colombeau theory of generalized functions and non-standard analysis.

Pointwise characterizations in generalized function algebras

We define the algebra

\widetilde{\mathbb{C}}

{\mathcal{G}(\widetilde {\mathbb {R}}^d)}

Isomorphisms of algebras of generalized functions

of Colombeau generalized functions on

Group invariant Colombeau generalized functions

{\widetilde {\mathbb {R}}^d}