# Kernel Theorems in Spaces of Tempered Generalized Functions

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1 AOC - Analyse Optimisation Controle

Abstract : In analogy to the classical isomorphism between $\mathcal{L}\left \mathcal{S}\left \mathbb{R}^{n} ight ,\mathcal{S}^{\prime}\left \mathbb{R}^{m} ight ight$ and $\mathcal{S}^{\prime}\left \mathbb{R}^{n+m} ight$, we show that a large class of moderate linear mappings acting between the space $\mathcal{G} {\mathcal{S}}\left \mathbb{R}^{n} ight$ of Colombeau rapidly decreasing generalized functions and the space $\mathcal{G} {\tau}\left \mathbb{R}^{n} ight$ of temperate ones admits generalized integral representations, with kernels belonging to $\mathcal{G} {\tau}\left \mathbb{R}^{n+m} ight$. Furthermore, this result contains the classical one in the sense of the generalized distribution equality.

Keywords : kernel Theorem Colombeau temperate generalized functions integral operator temperate distributions

Autor: Antoine Delcroix -

Fuente: https://hal.archives-ouvertes.fr/

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