A discontinuous Colombeau differential calculus
type: Item
instance of: scholarly article
title: A discontinuous Colombeau differential calculus
zbMATH Open document ID: 1081.46029
author: S. O. Juriaans
author: J. Aragona
author: Roseli Fernandez
published in: Monatshefte für Mathematik
publication date: 2005-03-17T00:00:00Z
review text: This paper introduces a new differential calculus for maps \(\varphi: \widetilde{\mathbb R}\rightarrow \widetilde{\mathbb R}\), where \(\widetilde{\mathbb R}\) is the ring of generalized numbers endowed with the sharp valuation. This setup serves for an analysis of generalized functions (here, elements of Colombeau's special construction) from the point of view of (compactly supported) generalized points. The latter form the ``minimal set'' of points on which evaluation determines generalized functions uniquely [cf. \textit{M. Kunzinger} and \textit{M. Oberguggenberger}, Math. Nachr. 203, 147--157 (1999; Zbl 0935.46041)]. The difficulty in defining a notion of differentiability for such maps \(\varphi\) lies on the one hand in the existence of zero divisors inside the generalized numbers; on the other hand, the sharp valuation on \(\widetilde{\mathbb R}\) fails to be multiplicative (however, there is a subring with dense valuation in \(\mathbb R^+_0)\) for which multiplicativity of the valuation holds). The authors of this paper overcome these obstacles successfully and show that their notion of differentiability is compatible with differentiability in the algebra of generalized functions in the sense that differentiation commutes with evaluation on generalized points. It also turns out that locally constant generalized functions are constant (see Proposition 4.7) which shows that scenarios which the (totally disconnected) topology allowed can be excluded by the ``smoothness'' of generalized functions. The authors also introduce and discuss analyticity of generalized functions and the notion of a generalized manifold.
reviewed by: Eberhard Mayerhofer
Mathematics Subject Classification ID: 46F30
Mathematics Subject Classification ID: 46T20
zbMATH DE Number: 2146823
zbMATH Keywords: Colombeau algebra
zbMATH Keywords: generalized function
zbMATH Keywords: sharp topologies
zbMATH Keywords: generalized points
zbMATH Keywords: point value
MaRDI profile type: Publication
full work available at URL: s00605-004-0257-0
OpenAlex ID: W1999278558
DOI: 10.1007/S00605-004-0257-0
description: scientific article; zbMATH DE number 2146823