Find the distributional limit of the sequence of distributions
$$F_n=n\delta_{-\frac{1}{n}}-n\delta_{\frac{1}{n}}$$
Hint: $F_n$ is the distributional derivative of some function
So far I've tried kind of working backwards towards the definition of the distributional derivative to make use of the hint:
$$F_n=n\delta_{-\frac{1}{n}}-n\delta_{\frac{1}{n}}$$
$$F_n=<n\delta_{-\frac{1}{n}},\phi>-<n\delta_{\frac{1}{n}},\phi>$$
$$n\int_{-1/n}^{1/n}\phi'(x) \ dx$$
$$-\int_{\mathbb{R}}f(x)\phi'(x) \ dx$$
$$-<f,\phi'> \ = \ <f',\phi>$$
where
$$f(x) =
\begin{cases}
-n & \text{$-\frac{1}{n}\le x\le\frac{1}{n}$} \\
0 & \text{otherwise}
\end{cases}$$
But I'm not sure how to proceed from here and how following this hint got me closer to the answer.
Any help is greatly appreciated!
Best Answer
Take some test function $\phi$, then
$$\langle F_n, \phi\rangle = \int n\big(\delta(x+\tfrac 1n) - \delta(x-\tfrac 1n)\big)\phi(x) d x = n\big(\phi(-\tfrac 1n)-\phi(+\tfrac 1n)\big) \overset{n\to\infty}{\longrightarrow} -2\phi'(0) $$
So the limit distribution $F$ is the linear functional $F(\phi) = -2\phi'(0)$. In particular $F=2\delta'$.