Convolution of Distribution-Valued Functions. Applications.

In this article we examine products and convolutions of vector-valued functions. For nuclear normal spaces of distributions Proposition 25 in \cite[p. 120]{MR0117544} yields a vector-valued product or convolution if there is a continuous product or convolution mapping in the range of the vector-valued functions. For specific spaces, we generalize this result to hypocontinuous bilinear maps at the expense of generality with respect to the function space. We consider holomorphic, meromorphic and differentiable vector-valued functions and state propositions that contain assertions on products and convolutions of distribution-valued functions in literature as particular cases. Moreover we consider the general convolution of analytic distribution-valued functions and give an approach different to \cite{MR2088667}