Separation of Sequences and Multipliers in the Space of Tempered Distributions

Abstract

We consider the notions of $v$-separation and $(N,v)$-separation for increasing sequences that tend to infinity. We study several of the connections between the properties of a multiplier in the space $S'(\mathbb{R})$ and in other related spaces and the separation properties of the sequence of its zeros.
We also prove that a distributional division problem $$Fh=f,$$always has tempered solutions $h$ for any tempered data $f$ if and only if the non integrable function $1/F$ admits regularizations that are tempered, and that this holds if and only if the pseudofunction $\mathcal{P} f (1/F)$ is tempered.