An application of Mittag-Leffler Lemma to the L.F algebra of $\boldsymbol{\mathcal{C}^{(\infty)}}$ $\mathbb{N}$-tempered functions on $\mathbb{R}^+$

Abstract

In this note, we show that a ${\mathcal{C}}^{(\infty)}$ $N$-tempered function $f$ on $\mathbb{R}^+$ can be extended as a ${\mathcal{C}}^{(\infty)}$ function $\widetilde{f}$ in $]-\rho, +\infty[ +\mathbb{R}i$ ($\forall \rho>0)$ such that $$D^{\alpha}\frac{\partial }{\partial \overline{z}}\widetilde{f}(z)|_{\{z=x\}}=0, \quad \forall \alpha =(\alpha_1, \alpha_2)\in \mathbb{N}^2, \quad \forall x\in \mathbb{R}^+.$$ To get this result, we use Mittag-Leffler Lemma.