What's the use for mollifiers?
I understand that mollification is used in conjunction with convolution. However, I don't understand how is this useful. Perhaps one can apply more sophisticated rules to a convolution integral?
Does it enable removing of discontinuities?
I guess the coolest thing you can do with Mollifiers is to get an explicit sequence of nice functions which converge back to some ugly function. Let me give you an example for you to get the feel of this stuff:
Let's say you have a functional (not necessarily linear) $A:L^p\to\mathbb R$ for which you can only tell its behaviour if $f$ is differentiable. For example, lets say that you manage to prove that if $f$ is differentiable then: $$ |A(f)|\leq C\|f\|_p $$
Probably you will use the differentiability when proving this fact, in order to make things easier. But the result you really wanted to show that the same thing holds true for all $f\in L^p$. Now, assume you can also prove that $A$ is continuous in $L^p$. Then you can do the following:
Pick your favorite mollifying sequence $\{\phi_n, n\in\mathbb N\}\subset \mathcal C^\infty_c$ and consider $f_n = \phi_n * f$. Then, by the magic of convolution, $f_n\in \mathcal C^\infty$ (you should try to prove this), and even still, $f_n\to f$ in $L^p$ (you should really try to prove this). This means you can use the bound on the $f_n$'s and get: $$ |A(f)| = |A(\lim_{n\to\infty}f_n)| = \lim_{n\to \infty}|A(f_n)|\leq\lim_{n\to\infty}C\|f_n\|_p = C\|f\|_p $$ where the first equality is the fact that $f_n\to f$, the second is the fact that $A$ is continuous in $L^p$, the third is by using the fact that $f_n$ are differentiable and the fourth again that $f_n\to f$.
Now, the cool part about this is that, if you only know how to calculate $A$ if $f$ is differentiable, then this gives you a way of calculating when $f$ is just $L^p$: You just have: $$ A(f) = \lim_{n\to\infty} A(f*\phi_n) $$
These are some uses i could think of mollifiers. I'm sure there are tons of more interesting applications.
For the fact that $C_{0}^{\infty}$ is dense in $L^{p}$, $1\leq p<\infty$, one uses mollifiers to do the estimation. In some sense, convolution with mollifiers makes functions better.
The Riemann-Lebesgue lemma is a good example of something that can be easily proved using a mollifer. In this setting the goal is to show that $$ \lim_{n\rightarrow\infty}\int_{0}^{2\pi}f(t)\sin(nt+\phi)dt=0,\;\;\; f\in L^2(0,2\pi). $$ To this end, let $\epsilon > 0$ be given, and choose $f_s(t)$ to be a smooth function that vanishes identically near the endpoints of $[0,2\pi]$ such that $\|f-f_s\|_{L^2} < \epsilon$. Then $\int_{0}^{2\pi}f_{s}(t)\sin(nt+\phi)dt$ is easily shown to tend to $0$ as $n\rightarrow\infty$ by using integration by parts, leaving behind a term that is bounded by $\epsilon$. The Riemann-Lebesgue Lemma then follows rather easily.
