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Abstract

We consider the multiplicative algebra $\mathsf P(\mathcal G_+')$ of continuous scalar polynomials on the space $\mathcal G_+'$ of Roumieu ultradistributions on $[0,\infty)$ as well as its strong dual $\mathsf P'(\mathcal G_+')$. The algebra $\mathsf P(\mathcal G_+')$ is densely embedded into $\mathsf P'(\mathcal G_+')$ and the operation of multiplication possesses a unique extension to $\mathsf P'(\mathcal G_+')$, that is, $\mathsf P'(\mathcal G_+')$ is also an algebra. The operation of differentiation on these algebras is investigated. The polynomially extended Laplace transformation and its connections with the differentiation are also studied.