A comparison of two different ways to define classes of ultradifferentiable functions

Abstract

We characterize the weight sequences $(M_p)_p$ such that the class of ultra-differentiable functions ${\mathcal E}_{(M_p)}$ defined by imposing conditions on the derivatives of the function in terms of this sequence coincides with a class of ultradifferentiable functions ${\mathcal E}_{(\omega)}$ defined by imposing conditions on the Fourier Laplace transform. As a corollary, we characterize the weight functions $\omega$ for which there exists a weight sequence $(M_p)_p$ such that the classes ${\mathcal E}_{(\omega)}$ and ${\mathcal E}_{(M_p)}$ coincide. These characterizations also hold in the Roumieu case. Our main results are illustrated by several examples.