Convolvability and regularization of distributions

TL;DR

This paper uses Schwartz's theory of vector-valued distributions to unify and generalize the understanding of convolvability, regularization, and topological properties of distributions, providing new insights and simplified proofs.

Contribution

It introduces a unified framework for analyzing convolvability and regularization of distributions using vector-valued distribution theory, extending previous results.

Findings

Unified approach to convolvability and regularization

Simplified proofs of distribution properties

Characterization of multiplier and convolutor spaces

Abstract

We apply L.~Schwartz' theory of vector valued distributions in order to simplify, unify and generalize statements about convolvability of distributions, their regularization properties and topological properties of sets of distributions. The proofs rely on propositions on the multiplication of vector-valued distributions and on the characterization of the spaces $\mathcal{O}_{M}(E,F)$ and $\mathcal{O}_{C}'(E,F)$ of multipliers and convolutors for distribution spaces $E$ and $F$.