Polynomial ultradistributions on R+d

Abstract

Let $G_{+}^{\prime }=G^{\prime }\left(R_{+}^d\right)$ stand for Roumieu ultradistributions with supports in the positive cone $R_{+}^d$. Throughout $P\left(G_{+}^{\prime }\right)$ denotes the algebra of continuous scalar polynomials on the space $G_{+}^{\prime }$. We investigate the dual pair $〈P^{\prime }\left(G_{+}^{\prime }\right)\mid P\left(G_{+}^{\prime }\right)〉$ generated by the algebra $P\left(G_{+}^{\prime }\right)$ and by its strong dual $P^{\prime }\left(G_{+}^{\prime }\right)$. Properties of the polynomially extended operational calculus and the semigroups of shifts along the cone $R_{+}^d$ are considered.