On the class of real-valued functions which extensible to entire functions

Question

Let $\mathcal{D}$ be the class of real-valued functions $f(x)$ defined on $\mathbb{R}$, which are extensible to entire functions $\bar{f}(z)$ on $\mathbb{C}$ such that $\bar{f}(x)=f(x)$, for all $x \in \mathbb{R}$. Is $\mathcal{D}$ dense in $C^{\infty}(\mathbb{R})$? Any references?