Analytic extension of ultradifferentiable Whitney jets

Let $\omega$ be a weight and $F$ be a closed proper subset of $\mathbb{R}^n$. Then for every function $f$ on $\mathbb{R}^n$ belonging to the non quasi-analytic ($\omega$)-class of Beurling (resp. Roumieu) type, there is an element $g$ of the same class which is analytic on $\mathbb{R}^n \setminus F$ and such that $D^\alpha f(x) = D^\alpha g(x)$ for every $\alpha\in\mathbb{N}^n_0$ and $x\in F$.