Extended Gevrey regularity via weighted matrices

The main aim of this paper is to compare two recent approaches for investigating the interspace between the union of Gevrey spaces $\mathcal G_t (U)$ and the space of smooth functions $C^{\infty}(U)$. The first approach in the style of Komatsu is based on the properties of two parameter sequences $M_p=p^{\tau p^{\sigma}}$, $\tau>0$, $\sigma>1$. The other one uses weight matrices defined by certain weight functions. We prove the equivalence of the corresponding spaces in the Beurling case by taking projective limits with respect to matrix parameters, while in the Roumieu case we need to consider a larger space then the one obtained as the inductive limit of extended Gevrey classes.