Internal sets and internal functions in Colombeau theory

DOI10.1016/j.jmaa.2007.10.030zbMath1173.46024arXiv0707.0994OpenAlexW1976983551MaRDI QIDQ2474967

Hans Vernaeve, Michael Oberguggenberger

Publication date: 6 March 2008

Published in: Journal of Mathematical Analysis and Applications (Search for Journal in Brave)

Full work available at URL: https://arxiv.org/abs/0707.0994

zbMATH Keywords

Colombeau generalized functionsinternal sets

Mathematics Subject Classification ID

Generalized functions for nonlinear analysis (Rosinger, Colombeau, nonstandard, etc.) (46F30) Nonstandard functional analysis (46S20) Methods of nonstandard analysis for ordinary differential equations (34E18)

Related Items (21)

Microlocal analysis in generalized function algebras based on generalized points and generalized directions ⋮ New topologies on Colombeau generalized numbers and the Fermat-Reyes theorem ⋮ The classical theory of calculus of variations for generalized functions ⋮ Differential calculus and integration of generalized functions over membranes ⋮ Inverse Function Theorems for Generalized Smooth Functions ⋮ An Application of Internal Objects to Microlocal Analysis in Generalized Function Algebras ⋮ A Convenient Notion of Compact Set for Generalized Functions ⋮ Algebraic theory of Colombeau's generalized numbers ⋮ Characterization of distributions having a value at a point in the sense of Robinson ⋮ Strongly internal sets and generalized smooth functions ⋮ Regular generalized solutions to semilinear wave equations ⋮ Hilbert $\widetilde{\mathbb{C}}$-modules: Structural properties and applications to variational problems ⋮ An axiomatic approach to the non-linear theory of generalized functions and consistency of Laplace transforms ⋮ Symplectic Modules Over Colombeau-Generalized Numbers ⋮ Nonstandard principles for generalized functions ⋮ Pointwise characterizations in generalized function algebras ⋮ Supremum, infimum and hyperlimits in the non-Archimedean ring of Colombeau generalized numbers ⋮ Calculus of variations and optimal control for generalized functions ⋮ Hyperseries in the non-Archimedean ring of Colombeau generalized numbers ⋮ A nonlinear theory of generalized functions ⋮ Locally convex topological algebras of generalized functions: Compactness and nuclearity in a nonlinear context

Cites Work

• Unnamed Item
• Unnamed Item
• Unnamed Item
• Unnamed Item
• Unnamed Item
• Unnamed Item
• Eine Erweiterung der Infinitesimalrechnung
• A nonlinear theory of generalized functions.
• An embedding of Schwartz distributions in the algebra of asymptotic functions
• Topological structures in Colombeau algebras: Topological \(\widetilde{\mathbb C}\)-modules and duality theory
• Topological structures in Colombeau algebras: Investigation of the duals of \(\mathcal G_c(\Omega)\), \(\mathcal G(\Omega)\) and \(\mathcal G_{\mathcal S}(\mathbb R^n)\)
• Borel's theorem for generalized functions
• A contribution to the theory of generalized functions
• Characterization of colombeau generalized functions by their pointvalues
• Local properties of Colombeau generalized functions
• Group Analysis of Differential Equations and Generalized Functions
• Non-standard analysis

This page was built for publication: Internal sets and internal functions in Colombeau theory