We define the algebra GÌƒ(A) of Colombeau generalized functions on a subset A of the space of generalized points RÌƒd. If A is an open subset of RÌƒd, such generalized functions can be identified withâ€¦ (More)

Inspired by nonstandard analysis, we define and study internal subsets and internal functions in algebras of Colombeau generalized functions. We prove a saturation principle for internal sets andâ€¦ (More)

In this paper, 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 areâ€¦ (More)

Colombeau generalized functions invariant under smooth (additive) one-parameter groups are characterized. This characterization is applied to generalized functions invariant under orthogonal groupsâ€¦ (More)

We construct an algebra of generalized functions endowed with a canoniÂ cal embedding of the space of Schwartz distributions. We offer a solution to the problem of multiplication of Schwartzâ€¦ (More)

We show that principles from nonstandard analysis hold to some extent for nonlinear generalized functions. The generalized functions under consideration are constructed as families of functionsâ€¦ (More)

We study Banach CÌƒ-algebras, i.e., complete ultra-pseudo-normed algebras over the ring CÌƒ of Colombeau generalized complex numbers. We develop a spectral theory in such algebras. We show by explicitâ€¦ (More)

In this paper, weakly homogeneous generalized functions in the special Colombeau algebras are determined up to equality in the sense of generalized distributions. This yields characterizations thatâ€¦ (More)

We develop a theory of Hilbert Ìƒ C-modules which forms the core of a new functional analytic approach to algebras of generalized functions. Particular attention is given to finitely generatedâ€¦ (More)

It is shown that the nonstandard representatives of Schwartz-distributions, as introduced by K. D. Stroyan and W. A. J. Luxemburg in their book Introduction to the theory of infinitesimals [5], areâ€¦ (More)