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# Vanishing Theorems in Colombeau Algebras of Generalized Functions

Published:2008-12-01
Printed: Dec 2008
• V. Valmorin
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## Abstract

Using a canonical linear embedding of the algebra ${\mathcal G}^{\infty}(\Omega)$ of Colombeau generalized functions in the space of $\overline{\C}$-valued $\C$-linear maps on the space ${\mathcal D}(\Omega)$ of smooth functions with compact support, we give vanishing conditions for functions and linear integral operators of class ${\mathcal G}^\infty$. These results are then applied to the zeros of holomorphic generalized functions in dimension greater than one.
 Keywords: Colombeau generalized functions, linear integral operators, generalized holomorphic functions
 MSC Classifications: 32A60 - Zero sets of holomorphic functions 45P05 - Integral operators [See also 47B38, 47G10] 46F30 - Generalized functions for nonlinear analysis (Rosinger, Colombeau, nonstandard, etc.)

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