We consider a function $f : [0 ; 1] \mapsto \mathbb{R}$, which is $\mathcal{C}^{\infty}$ on the open interval, and positive. For example $f(x) = 1 + x$. ($f$ is a nice function, not a nasty beast.)
I would like to perform a regularization of this function using mollifiers. I want to build $f_{\varepsilon} : \mathbb{R} \to \mathbb{R}$, $\mathcal{C}^{\infty}$ on $\mathbb{R}$, such that
$$f_{\varepsilon} = \begin{cases} 0 \,\, \text{ if } x \in ] -\infty ; - \epsilon ] \cup [1 + \varepsilon ; + \infty] \\ f(x) \,\, \text{ if } x \in [0;1] \end{cases}$$
I know that the solution to this problem requires to use mollifiers and smooth unity partitions, but I want the continuation to be $\mathcal{C}^{\infty}$.
My questions are as follows :
How could I proceed ? As I want then to do other estimations with this smoothed function, the more explicit the continuation is, the better it will be. (If having a $\mathcal{C}^{\infty}$ continuation is too complex, a $\mathcal{C}^{2}$ continuation might be enough.)
Can I have a control on the maximum height reached by $f_{\varepsilon}$ ? As $\varepsilon \to 0$, I would like $\sup (f_{\varepsilon})$ to remain uniformly bounded.
How would you handle the multidimensionnal case where $f$ is defined on $[0 ; 1]^{n}$ ?
Finally, would you know any reference books that deals with this subject ?
This doesn't answer all your question, but at anyway:
You (clearly) need the condition that all the derivatives of $f$ have (finite) limits at the endpoints $x=0$ an $x=1$. If this condition is satisfied ($f$ is nice:), use Borel's theorem saying that there is a smooth function around $0$ with these derivatives, and similarly around $1$. Now use these functions to extend $f$ to a larger interval: keep $f$ on $[0,1]$, and use those functions outside. So now you have $f$ on say $(-1,2)$. Multiply it with a smooth $\phi$ s.t. $\phi=1$ on $[0,1]$ and $\phi=0$ outside $[-\epsilon,1+\epsilon]$ - and you have an extension of $f$ you want.