Locally convex topological algebras of generalized functions: Compactness and nuclearity in a nonlinear context

Aragona, J.; Juriaans, S.; Colombeau, J.

doi:10.1090/S0002-9947-2015-06213-8

Locally convex topological algebras of generalized functions: Compactness and nuclearity in a nonlinear context
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by J. Aragona, S. O. Juriaans and J. F. Colombeau PDF

Trans. Amer. Math. Soc. 367 (2015), 5399-5414 Request permission

Abstract:

In this paper we introduce Hausdorff locally convex algebra topologies on subalgebras of the whole algebra of nonlinear generalized functions. These topologies are strong duals of Fréchet-Schwartz space topologies and even strong duals of nuclear Fréchet space topologies. In particular, any bounded set is relatively compact and one benefits from all deep properties of nuclearity. These algebras of generalized functions contain most of the classical irregular functions and distributions. They are obtained by replacing the mathematical tool of $\mathcal {C}^\infty$ functions in the original version of nonlinear generalized functions by the far more evolved tool of holomorphic functions. This paper continues the nonlinear theory of generalized functions in which such locally convex topological properties were strongly lacking up to now.

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