Consider the following problem. I have an arbitrary function $f : \Omega \subset \mathbb{R}^n \rightarrow \mathbb{R} $ with $\Omega$ a compact set, and f is of class $C^k$ over $\Omega$. I want to extend the value of $f$ all over $\mathbb{R}^n$ using a function decreasing exponentially fast toward minus infinity in all direction, and preserving the smoothness of $f$ :
$$\lim_{||x||_2 \rightarrow + \infty} g(x) = - \infty$$
$$ g(x) = f(x), \forall x \in \Omega, \quad g \in C^k$$
Do you know what extension of f could satisfy those conditions ?