A discontinuous Colombeau differential calculus
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.
scientific article; zbMATH DE number 2146823