## Bulletin of the Belgian Mathematical Society - Simon Stevin

### 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.

#### Article information

Source
Bull. Belg. Math. Soc. Simon Stevin Volume 14, Number 3 (2007), 425-444.

Dates
First available: 28 September 2007

Permanent link to this document
http://projecteuclid.org/euclid.bbms/1190994204

Zentralblatt MATH identifier
05231820

Mathematical Reviews number (MathSciNet)
MR2387040

#### Citation

Bonet, José; Meise, Reinhold; Melikhov, Sergej N. A comparison of two different ways to define classes of ultradifferentiable functions. Bulletin of the Belgian Mathematical Society - Simon Stevin 14 (2007), no. 3, 425--444. http://projecteuclid.org/euclid.bbms/1190994204.