On the foundations of nonlinear generalized functions II

1999/12/28 by Michael Grosser
Mathematics · Physics and Astronomy · #math.FA #math-ph #math.MP #msc:46F30 #msc:26E15 #msc:46E50 #msc:35D05

paper · pdf

published as Mem. Amer. Math. Soc. 153, No. 729, 2001. · 43 pages, LaTeX

arxiv created 1999/12/28 · arxiv updated 2009/11/30

Abstract

This paper gives a comprehensive analysis of algebras of Colombeau-type generalized functions in the range between the diffeomorphism-invariant quotient algebra G^d = E_M/N introduced in part I and Colombeau's original algebra G^e. Three main results are established: First, a simple criterion describing membership in N (applicable to all types of Colombeau algebras) is given. Second, two counterexamples demonstrate that G^d is not injectively included in G^e. Finally, it is shown that in the range ``between'' G^d and G^e only one more construction leads to a diffeomorphism invariant algebra. In analyzing the latter, several classification results essential for obtaining an intrinsic description of G^d on manifolds are derived.

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• A new approach to diffeomorphism invariant algebras of generalized functions
• Diffeomorphism invariant Colombeau algebras. Part II: Classification
• Point value characterizations and related results in the full Colombeau algebras G^e and G^d
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• Topological structures in Colombeau algebras: investigation of the duals of \Gc(\Om), \G(\Om) and \GS(\Rⁿ)