Geodesic
A pre-dual for the convolution algebra $\mathcal D^\prime(\Gamma+)$
Trudy Matematicheskogo Instituta imeni V.A. Steklova, Selected problems of mathematical physics and analysis, Tome 203 (1994), pp. 429-440 
V. Schmidt; P. Dierolf. A pre-dual for the convolution algebra $\mathcal D^\prime(\Gamma+)$. Trudy Matematicheskogo Instituta imeni V.A. Steklova, Selected problems of mathematical physics and analysis, Tome 203 (1994), pp. 429-440. http://geodesic.mathdoc.fr/item/TM_1994_203_a32/
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     author = {V. Schmidt and P. Dierolf},
     title = {A~pre-dual for the convolution algebra $\mathcal D^\prime(\Gamma+)$},
     journal = {Trudy Matematicheskogo Instituta imeni V.A. Steklova},
     pages = {429--440},
     year = {1994},
     volume = {203},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/TM_1994_203_a32/}
}
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Let $\mathcal D'(\Gamma+)$ denote the subspace of $\mathcal D(\mathbb R^n)'$ consisting of all distributions whose support is contained in a translate of the cone $\Gamma$. We construct a locally convex space $\mathcal F(\Gamma+)$ consisting of $C^\infty$-functions on $\mathbb R^n$ such that $\mathcal D'(\Gamma+)$ is the dual of $\mathcal F(\Gamma+)$. We then discuss certain natural topologies on $\mathcal F(\Gamma+)$ and on $\mathcal D'(\Gamma+)$.