On the impossibility of the convolution of distributions

Abstract

Certain incompatibilities are proved related to the prolongation of an associative derivation convolution algebra, defined for a subset of distributions, to a larger subset of distributions containing a derivation and the one distribution. This result is a twin of Schwartz‘ impossibility theorem, stating certain incompatibilities related to the prolongation of the multiplication product from the set of continuous functions to a larger subset of distributions containing a derivation and the delta distribution. The presented result shows that the non-associativity of a recently constructed derivation convolution algebra of associated homogeneous distributions with support in R cannot be avoided.