approximate identity- common compact set

Question

let us consider a sequence of smooth functions with compact support $\phi_n$, which approximate Dirac measure at zero, such that $\int_{\mathbb{R}}\phi_n(x)\,dx=1$. Okay, I can consider something like that, but can somebody give me some example of such sequence? I know it will be something like $\phi(x)=\exp(-x^2)$ but could somebody write me it down really precisely? If not, I have one question: do this sequence has one common compact support or it changes with the change of $n$?

Answer 1

In general an approximation to the identity just needs to be some family $\{\phi_n\}$ of $L^1$ functions with $\int \phi_n \, dx = 1$ for each $n$. In many applications we want these $\phi_n$ to be 1) smooth and 2) compactly supported.

To achieve (1) your example of $\phi(x) = \frac{1}{\sqrt{\pi}}e^{-x^2}$ works, if we then set $\phi_n(x) = n\phi(nx)$. To achieve goal (2) we want to pick $\phi$ to be some bump function, usually $$ \phi(x) = \begin{cases} c\exp((1 - |x|^2)^{-1}) & x \in (-1, 1) \\ 0 & x \not\in (-1, 1). \end{cases} $$ Here $c$ is whatever positive constant needed to make the integral equal to 1. Then set $\phi_n(x) = n\phi(nx)$ as above. In this case the support of $\phi$, and $\phi_n$ for all $n \geq 1$, is contained within $B(0, 1)$.