What's the idea behind mollifiers?

Question

I'm wondering if someone could explain the idea of mollifiers and how we use them to prove density of some spaces. Up to my knowledge they are used to prove density of test functions. Can we use them to prove the density of much complicated spaces? And Is there any other applications in other contexts?

Thank you for any suggestions or good references.

Answer 1

If $\phi \in C^\infty_c(\Bbb{R}), \int \phi =1,\phi_n(x)=n\phi(nx)$ then for all $\varphi \in C^\infty_c(\Bbb{R}), \varphi \ast \phi_n \to \varphi$ in the $C^\infty_c(\Bbb{R})$ topology, for all distribution $T \ast \phi_n \to T$ in the sense of distributions, and for quite every normed/topological vector space you like ($L^p,H^k,C^k...$) then $f \ast \phi_n \to f$. The usefulness is that $T \ast \phi_n,f \ast \phi_n$ are smooth. The generalization is to take any sequence $\phi_n$ satisfying those properties.

Answer 2

There is this super fancy paper by Meyers&Serrin with this cool title "W=H" in which they have shown that strong derivatives are the same as weak derivatives by using mollifiers, hence shown that two concepts of generalizing derivatives are actually the same !! Strong actually means limits of smooth function in a given norm (roughly) and weak is meant ala Sobolev space.

If that is not an amazing use for a concept, I don't know what is :).

They are also very useful two show regularity, e.g. of the Laplace equation.