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Inversion of Colombeau generalized functions

Part of: Distributions, generalized functions, distribution spaces Equations and inequalities involving nonlinear operators

Published online by Cambridge University Press:  21 March 2013

Evelina Erlacher
Evelina Erlacher*
Affiliation:
Institute for Statistics and Mathematics, Vienna University of Economics and Business, Augasse 2–6, 1090 Vienna, Austria (evelina.erlacher@wu.ac.at)
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Abstract

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We introduce different notions of invertibility for generalized functions in the sense of Colombeau. Several necessary conditions for (left, right) invertibility are derived, giving rise to the concepts of compactly asymptotic injectivity and surjectivity. We analyse the extent to which these properties are also sufficient to guarantee the existence of a (left, right) inverse of a generalized function. Finally, we establish several Inverse Function Theorems in this setting and study the relation to their classical counterparts.

Keywords

Colombeau algebrainverse generalized functionInverse Function Theoremlocal existence result

MSC classification

Secondary: 46F30: Generalized functions for nonlinear analysis (Rosinger, Colombeau, nonstandard, etc.) 47J07: Abstract inverse mapping and implicit function theorems
Type
Research Article
Information
Proceedings of the Edinburgh Mathematical Society , Volume 56 , Issue 2 , June 2013 , pp. 469 - 500
Copyright
Copyright © Edinburgh Mathematical Society 2013

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