On the projective description of spaces of ultradifferentiable functions of Roumieu type

TL;DR

This paper offers a unified projective description of ultradifferentiable function spaces of Roumieu type, extending previous results to broader classes with weaker assumptions, applicable to various defining weight structures.

Contribution

It provides a new, unified projective description of ultradifferentiable spaces of Roumieu type, accommodating both weight sequences and functions under weaker conditions.

Findings

Unified projective descriptions for different ultradifferentiable classes

Applicable to arbitrary open sets in R^d

Weaker assumptions than previous results

Abstract

We provide a projective description of the space $\mathcal{E}^{\{\mathfrak{M}\}}(\Omega)$ of ultradifferentiable functions of Roumieu type, where $\Omega$ is an arbitrary open set in $\mathbb{R}^d$ and $\mathfrak{M}$ is a weight matrix satisfying the analogue of Komatsu's condition $(M.2)'$. In particular, we obtain in a unified way projective descriptions of ultradifferentiable classes defined via a single weight sequence (Denjoy-Carleman approach) and via a weight function (Braun-Meise-Taylor approach) under considerably weaker assumptions than in earlier versions of these results.