Division by Flat Ultradifferentiable Functions and Sectorial Extensions
Abstract
Description
We consider classes $ \mathcal{A}_M(S) $ of functions holomorphic in an open plane sector $ S $ and belonging to a strongly non-quasianalytic class on the closure of $ S $. In $ \mathcal{A}_M(S) $, we construct functions which are flat at the vertex of $ S $ with a sharp rate of vanishing. This allows us to obtain a Borel-Ritt type theorem for $ \mathcal{A}_M(S) $ extending previous results by Schmets and Valdivia. We also derive a division property for ideals of flat ultradifferentiable functions, in the spirit of a classical $ C^\infty $ result of Tougeron.
Slight update of the published version. The definition of closedness in subsections 4.1 and 4.2 is less restrictive. One minor typo corrected
Slight update of the published version. The definition of closedness in subsections 4.1 and 4.2 is less restrictive. One minor typo corrected