On the Inclusion Relations of Global Ultradifferentiable Classes Defined by Weight Matrices.

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.

Nuclear global spaces of ultradifferentiable functions in the matrix weighted setting.

We prove that the Hermite functions are an absolute Schauder basis for many global weighted spaces of ultradifferentiable functions in the matrix weighted setting and we determine also the corresponding coefficient spaces, thus extending the previous work by Langenbruch. As a consequence, we give very general conditions for these spaces to be nuclear. In particular, we obtain the corresponding results for spaces defined by weight functions.