In this paper we continue the study of the spaces {mathcal O}_{M,omega }({mathbb R}^N) and {mathcal O}_{C,omega }({mathbb R}^N) undertaken in Albanese and Mele (J Pseudo-Differ Oper Appl, 2021). We determine new representations of such spaces and we give some structure theorems for their dual spaces. Furthermore, we show that {mathcal O}'_{C,omega }({mathbb R}^N) is the space of convolutors of the space {mathcal S}_omega ({mathbb R}^N) of the omega -ultradifferentiable rapidly decreasing functions of Beurling type (in the sense of Braun, Meise and Taylor) and of its dual space {mathcal S}'_omega ({mathbb R}^N). We also establish that the Fourier transform is an isomorphism from {mathcal O}'_{C,omega }({mathbb R}^N) onto {mathcal O}_{M,omega }({mathbb R}^N). In particular, we prove that this isomorphism is topological when the former space is endowed with the strong operator lc-topology induced by {mathcal L}_b({mathcal S}_omega ({mathbb R}^N)) and the last space is endowed with its natural lc-topology.