Proof of Dirac delta function identity involving exponential

Question

Is the identity

$$\delta(x-a)=\lim_{\epsilon \to 0}\frac{1}{\sqrt{2\pi \epsilon}}\exp{[-\frac{1}{2\epsilon}(x-a)^2]}$$

correct? If yes, what is the proof?

Answer 1

Pretend for the moment that $a = 0$. Show that these functions give a family of approximate identities, that is, a family $(f_n)$ of functions that

1. All integrate to 1,
2. Have uniformly bounded $L^1$ norm, and
3. Satisfy $$\lim \limits_{n \rightarrow \infty} \int_{\vert x \vert > \epsilon} \vert f_n \vert \rightarrow 0$$ for all $\epsilon > 0$. It is not quite sufficient that the $f_n$ converge pointwise to zero away from the origin.

(In fact, any nonnegative $L^1$ function rescaled in the way you describe will give such a family.)

A family of approximate identities will act as a delta function when integrated any sufficiently nice function, such as the smooth, compactly supported functions. That is, they will satisfy $$\lim \limits_{n \rightarrow \infty} \int f_n(x) g(x) \, dx = g(0)$$ for such $g$.