On the inclusion relations of global ultradifferentiable classes defined by weight matrices

TL;DR

This paper characterizes the inclusion relations among global ultradifferentiable classes defined by weight matrices, exploring growth conditions, classical cases, and constructing oscillating weight sequences to analyze class non-triviality.

Contribution

It provides a comprehensive characterization of inclusion relations in ultradifferentiable classes via weight matrix growth conditions and introduces oscillating weight sequences around critical cases.

Findings

Established growth-based inclusion criteria for classes

Constructed oscillating weight sequences around key weights

Compared classes defined by weight functions and sequences

Abstract

We study and characterize the inclusion relations of global classes in the general weight matrix framework in terms of growth relations for the defining weight matrices. We consider the Roumieu and Beurling cases, and as a particular case we also treat the classical weight function and weight sequence cases. Moreover, we construct a weight sequence which is oscillating around any weight sequence which satisfies some minimal conditions and, in particular, around the critical weight sequence $(p!)^{1/2}$, related with the non-triviality of the classes. Finally, we also obtain comparison results both on classes defined by weight functions that can be defined by weight sequences and conversely.