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Colombeau algebras without asymptotics.

Eduard A Nigsch1

  • 1Wolfgang Pauli Institute, Oskar-Morgenstern-Platz 1, 1090 Vienna, Austria.

Journal of Pseudo-Differential Operators and Applications
|March 5, 2019
PubMed
Summary
This summary is machine-generated.

This study introduces a new construction for Colombeau-type generalized function algebras. It utilizes topological estimates on kernel spaces, moving beyond traditional asymptotic parameter-based definitions.

Keywords:
Asymptotic estimatesColombeau algebrasDiffeomorphism invarianceElementary Colombeau algebraNonlinear generalized functions

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Area of Science:

  • Mathematical Analysis
  • Algebraic Structures
  • Generalized Functions

Background:

  • Colombeau's theory of generalized functions provides a framework for handling singularities.
  • Existing constructions often rely on asymptotic estimates involving regularization parameters.
  • These methods can be complex and limit applicability in certain mathematical contexts.

Purpose of the Study:

  • To develop a novel construction for Colombeau-type algebras of generalized functions.
  • To replace asymptotic estimates with topological estimates for a more robust definition.
  • To expand the applicability and theoretical foundation of generalized function algebras.

Main Methods:

  • Definition of Colombeau-type algebras using topological estimates.
  • Focus on the properties of specific spaces of kernels.
  • Abstract algebraic construction and analysis.

Main Results:

  • A new, well-defined construction of Colombeau-type generalized function algebras.
  • Demonstration that topological estimates are sufficient for the definition.
  • The resulting algebras possess desirable algebraic and analytic properties.

Conclusions:

  • The proposed construction offers an alternative and potentially more general approach to Colombeau algebras.
  • Topological estimates provide a powerful tool for defining generalized functions.
  • This work contributes to the advancement of the theory of generalized functions and their applications.