  We study the structural and linear topological properties of the space
$\dot{\mathcal{B}}^{\prime \ast}_{\omega}$ of ultradistributions vanishing at
infinity (with respect to a weight function $\omega$). Particularly, we show
the first structure theorem for $\dot{\mathcal{B}}^{\prime \ast}_{\omega}$
under weaker hypotheses than were known so far. As an application, we determine
the structure of the S-asymptotic behavior of ultradistributions.

Functional Analysis