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Monatshefte für Mathematik

, Volume 170, Issue 2, pp 121–145 | Cite as

Rapidly converging approximations and regularity theory

  • Shantanu DaveEmail author
Article

Abstract

We consider distributions on a closed compact manifold \(M\) as maps on smoothing operators. Thus spaces of maps between \({{\Psi }^{\!-\!\infty }}(M)\) and \(\mathcal{C ^{\infty }}(M)\) are considered as generalized functions. For any collection of regularizing processes we produce various algebras of generalized functions and equivariant embeddings of distributions into such algebras. The regularity for such generalized functions is provided in terms of a certain tameness of maps between graded Frechét spaces. This also recovers the singularity behaviour of distributions (singular support/wavefront sets) in terms of certain subalgebras of the algebra of generalized functions. This notion of regularity is compared with the regularity in Colombeau algebras in the \(\mathcal{G }^{\infty }\) sense.

Keywords

Regularity of distributions Generalized functions  Frechét tame maps 

Mathematics Subject Classification

57J99 46F30 35D10 

Notes

Acknowledgments

I would like to thank Michael Kunzinger for his encouragement and support with this work. I am very grateful to the referee for many valuable comments and corrections.

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Copyright information

© Springer-Verlag Wien 2013

Authors and Affiliations

  1. 1.Faculty of MathematicsUniversity of ViennaViennaAustria

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