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Monatshefte für Mathematik

, 158:195 | Cite as

Pointwise characterizations in generalized function algebras

  • Hans Vernaeve
Article

Abstract

We define the algebra \({\mathcal{G}(\widetilde {\mathbb {R}}^d)}\) of Colombeau generalized functions on \({\widetilde {\mathbb {R}}^d}\) which naturally contains the generalized function algebras \({\mathcal {G}_\fancyscript {S}(\mathbb {R}^d)}\) and \({\mathcal {G}_\tau(\mathbb {R}^d)}\). The subalgebra \({\mathcal {G}_\fancyscript {S}^\infty(\mathbb {R}^d)}\) of \({\mathcal {G}(\widetilde {\mathbb {R}}^d)}\) is characterized by a pointwise property of the generalized functions and their Fourier transforms. We also characterize the equality in the sense of generalized tempered distributions for certain elements of \({\mathcal{G}_\fancyscript {S}(\mathbb {R}^d)}\) (namely those with so-called slow scale support) by means of a pointwise property of their Fourier transforms. Further, we show that (contrary to what has been claimed in the literature) for an open set \({\Omega \subseteq \mathbb {R}^d}\), the algebra of pointwise regular generalized functions \({\dot {\mathcal {G}}^\infty(\upomega)}\) equals \({\mathcal {G}^\infty(\Omega)}\) and give several characterizations of pointwise \({\mathcal {G}^\infty}\)-regular generalized functions in \({\mathcal {G}(\Omega)}\).

Keywords

Colombeau generalized functions Regularity Pointwise properties 

Mathematics Subject Classification (2000)

46F30 

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Copyright information

© Springer-Verlag 2008

Authors and Affiliations

  1. 1.Unit for Engineering MathematicsUniversity of InnsbruckInnsbruckAustria

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