A mollifier is defined as a function $\varphi: \mathbb{R}^n \rightarrow \mathbb{R}$ such that
- $$\int_{\mathbb{R}^n} \varphi(x) dx = 1$$
- $\varphi$ has compact support
- $$\lim_{\epsilon \rightarrow 0} \varphi_\epsilon = \lim_{\epsilon \rightarrow 0} \frac{1}{\epsilon^{n}} \varphi\big(\frac{x}{\epsilon}\big) = \delta_0$$
Thus we may mollify any function $f: \mathbb{R}^n \rightarrow \mathbb{R}$ by computing the convolution $f * \varphi_\epsilon$. I would like to do the same for a function $g: \mathbb{R}^n \rightarrow \mathbb{R}^m$ where $m > 1$. To do this would require the mollifiers $\varphi$ to be extended to higher dimensions, that is $\varphi: \mathbb{R}^n \rightarrow \mathbb{R}^m$ for $m > 1$.
Does such an object exist? If not, what about the three criteria above fails in higher dimensions? I could not find it in any textbooks.