Composition in ultradifferentiable classes

TL;DR

This paper characterizes the stability and continuity of composition in ultradifferentiable classes defined by various weight structures, and relates different function spaces through intersections and unions.

Contribution

It provides a comprehensive analysis of composition stability in ultradifferentiable classes and links Beurling and Roumieu spaces to weighted sequence spaces.

Findings

Characterization of stability under composition for classes defined by weights.

Representation of Beurling and Roumieu spaces as intersections and unions.

Continuity properties of the composition operation.

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.