Composition in ultradifferentiable classes

We characterize stability under composition of ultradifferentiable classes defined by weight sequences $M$, by weight functions $ω$, 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 $\mathcal{E}^{(ω)}$ and the Roumieu space $\mathcal{E}^{\{ω\}}$ as intersection and union of spaces $\mathcal{E}^{(M)}$ and $\mathcal{E}^{\{M\}}$ for associated weight sequences, respectively.

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