42 citations · 107 across the 22 of their papers we have counts for
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Generalized functions with infinitesimals
Hans Vernaeve
We give a survey of the use of infinitesimals within mathematical analysis to rigorously deal with the delta-function from physics, and more generally, with distributions in the se…
A Grothendieck topos of generalized functions I: basic theory
Paolo Giordano, Michael Kunzinger, Hans Vernaeve
The main aim of the present work is to arrive at a mathematical theory close to the historically original conception of generalized functions, i.e. set theoretical functions define…
Filter-dependent versions of the Uniform Boundedness Principle
Ben De Bondt, Hans Vernaeve
For every filter on , we introduce and study corresponding uniform -boundedness principles for locally convex topological vector spaces. These p…
A non-linear theory of infrahyperfunctions
Andreas Debrouwere, Hans Vernaeve, Jasson Vindas
We develop a nonlinear theory for infrahyperfunctions (also referred to as quasianalytic (ultra)distributions by L. Hörmander). In the hyperfunction case our work can be summarized…
Regularity of nonlinear generalized functions: a counterexample in the nonstandard setting
H. Vernaeve
Regularity theory in generalized function algebras of Colombeau type is largely based on the notion of -regularity, which reduces to -regularity when…
An application of internal objects to microlocal analysis in generalized function algebras
Hans Vernaeve
We illustrate the use of internal objects in the nonlinear theory of generalized functions by means of an application to microlocal analysis in Colombeau algebras.
Optimal embeddings of ultradistributions into differential algebras
Andreas Debrouwere, Hans Vernaeve, Jasson Vindas
We construct embeddings of spaces of non-quasianalytic ultradistributions into differential algebras enjoying optimal properties in view of a Schwartz type impossibility result, al…
Microlocal analysis in generalized function algebras based on generalized points and generalized directions
Hans Vernaeve
We develop a refined theory of microlocal analysis in the algebra of Colombeau generalized functions. In our approach, the wave front is a set of generalized poin…
Strongly internal sets and generalized smooth functions
Paolo Giordano, Michael Kunzinger, Hans Vernaeve
Based on a refinement of the notion of internal sets in Colombeau's theory, so-called strongly internal sets, we introduce the space of generalized smooth functions, a maximal exte…
Characterization of distributions having a value at a point in the sense of Robinson
Hans Vernaeve, Jasson Vindas
We characterize Schwartz distributions having a value at a single point in the sense introduced by means of nonstandard analysis by A. Robinson. They appear to be distributions con…
Nonstandard principles for generalized functions
Hans Vernaeve
Someone knowledgeable in nonstandard analysis may get the feeling that in the nonlinear theory of generalized functions, too often one works directly on the nets and spends effort…
Topological properties of regular generalized function algebras
Hans Vernaeve
We investigate density of various subalgebras of regular generalized functions in the special Colombeau algebra of generalized functions.
Banach -algebras
Hans Vernaeve
We study Banach -algebras, i.e., complete ultra-pseudo-normed algebras over the ring of Colombeau generalized complex numbers. We dev…
Generalized analytic functions on generalized domains
Hans Vernaeve
We define the algebra of Colombeau generalized functions on a subset A of the space of d-dimensional generalized points. If the domain A is open, such generalized functions can be…
Pointwise characterizations in generalized function algebras
Hans Vernaeve
We define the algebra of Colombeau generalized functions on the space of generalized points of {\mathbb R}^d which naturally contains the tempered generalized functions. The subalg…
Full Algebra of Generalized Functions and Non-Standard Asymptotic Analysis
Todor D. Todorov, Hans Vernaeve
We 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…
Weak homogeneity in generalized function algebras
Hans Vernaeve
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 chara…
Hilbert $\widetilde{\C}$-modules: structural properties and applications to variational problems
Claudia Garetto, Hans Vernaeve
We develop a theory of Hilbert $\widetilde{\C}$-modules by investigating their structural and functional analytic properties. Particular attention is given to finitely generated su…
Internal sets and internal functions in Colombeau theory
M. Oberguggenberger, H. Vernaeve
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 in…
Isomorphisms of algebras of Colombeau generalized functions
Hans Vernaeve
We show that for smooth manifolds X and Y, any isomorphism between the special algebra of Colombeau generalized functions on X, resp. Y is given by composition with a unique Colomb…
The Local Structure of Nonstandard Representatives of Distributions
Hans Vernaeve
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 infinite…
Group invariant Colombeau generalized functions
Hans Vernaeve
Colombeau generalized functions invariant under smooth (additive) one-parameter groups are characterized. This characterization is applied to generalized functions invariant under…