Factorization in Denjoy-Carleman classes associated to representations of (R^d,+)

For two types of moderate growth representations of (R^d, +) on sequen- tially complete locally convex Hausdorff spaces (including F-representations [14]), we introduce Denjoy-Carleman classes of ultradifferentiable vectors and show a strong factorization theorem of Dixmier-Malliavin type for them. In particular, our fac- torization theorem solves [14, Conjecture 6.4] for analytic vectors of representations of G = (R^d,+). As an application, we show that various convolution algebras and modules of ultradifferentiable functions satisfy the strong factorization property.