Composition in ultradifferentiable classes

Abstract: We characterize stability under composition of ultradifferentiable classes defined by weight sequences $M$, by weight functions $\omega$, and, more generally, by weight matrices $\mathfrak{M}$, and investigate continuity of composition $(g,f) \mapsto f \circ g$. In addition, we represent the Beurling space $\mathcal{E}^{(\omega)}$ and the Roumieu space $\mathcal{E}^{{\omega}}$ as intersection and union of spaces $\mathcal{E}^{(M)}$ and $\mathcal{E}^{{M}}$ for associated weight sequences, respectively.